<?xml version="1.0" encoding="UTF-8"?>
<!DOCTYPE proofdatabase
SYSTEM "http://www.logic.at/ceres/xml/4.0/proofdatabase.dtd">
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 </axiomset>
 <proof symbol="p" calculus="LK">
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                  <formulalist>
                   <constantatomformula symbol="≤" definition="≤">
                    <variable symbol="x_0"/>
                    <variable symbol="z_0"/>
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                  </formulalist>
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                   <formulalist>
                    <constantatomformula symbol="≤" definition="≤">
                     <variable symbol="y_0"/>
                     <variable symbol="z_0"/>
                    </constantatomformula>
                    <constantatomformula symbol="≤" definition="≤">
                     <variable symbol="x_0"/>
                     <variable symbol="y_0"/>
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                   </formulalist>
                   <formulalist>
                    <constantatomformula symbol="≤" definition="≤">
                     <variable symbol="x_0"/>
                     <variable symbol="z_0"/>
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                   </formulalist>
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                   <sequent>
                    <formulalist>
                     <constantatomformula symbol="≤" definition="≤">
                      <variable symbol="x_0"/>
                      <variable symbol="y_0"/>
                     </constantatomformula>
                     <constantatomformula symbol="≤" definition="≤">
                      <variable symbol="y_0"/>
                      <variable symbol="z_0"/>
                     </constantatomformula>
                    </formulalist>
                    <formulalist>
                     <constantatomformula symbol="≤" definition="≤">
                      <variable symbol="x_0"/>
                      <variable symbol="z_0"/>
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                    <sequent>
                     <formulalist>
                      <constantatomformula symbol="=">
                       <function symbol="∩">
                        <variable symbol="x_0"/>
                        <variable symbol="y_0"/>
                       </function>
                       <variable symbol="x_0"/>
                      </constantatomformula>
                      <constantatomformula symbol="≤" definition="≤">
                       <variable symbol="y_0"/>
                       <variable symbol="z_0"/>
                      </constantatomformula>
                     </formulalist>
                     <formulalist>
                      <constantatomformula symbol="≤" definition="≤">
                       <variable symbol="x_0"/>
                       <variable symbol="z_0"/>
                      </constantatomformula>
                     </formulalist>
                    </sequent>
                    <rule type="defl" symbol="d:l" param="();[(1)(1 1)(2)][(2)(1 2)];()">
                     <sequent>
                      <formulalist>
                       <constantatomformula symbol="≤" definition="≤">
                        <variable symbol="y_0"/>
                        <variable symbol="z_0"/>
                       </constantatomformula>
                       <constantatomformula symbol="=">
                        <function symbol="∩">
                         <variable symbol="x_0"/>
                         <variable symbol="y_0"/>
                        </function>
                        <variable symbol="x_0"/>
                       </constantatomformula>
                      </formulalist>
                      <formulalist>
                       <constantatomformula symbol="≤" definition="≤">
                        <variable symbol="x_0"/>
                        <variable symbol="z_0"/>
                       </constantatomformula>
                      </formulalist>
                     </sequent>
                     <rule type="perml" symbol="\pi:l" param="(1 2)">
                      <sequent>
                       <formulalist>
                        <constantatomformula symbol="=">
                         <function symbol="∩">
                          <variable symbol="y_0"/>
                          <variable symbol="z_0"/>
                         </function>
                         <variable symbol="y_0"/>
                        </constantatomformula>
                        <constantatomformula symbol="=">
                         <function symbol="∩">
                          <variable symbol="x_0"/>
                          <variable symbol="y_0"/>
                         </function>
                         <variable symbol="x_0"/>
                        </constantatomformula>
                       </formulalist>
                       <formulalist>
                        <constantatomformula symbol="≤" definition="≤">
                         <variable symbol="x_0"/>
                         <variable symbol="z_0"/>
                        </constantatomformula>
                       </formulalist>
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                      <rule type="defr" symbol="d:r" param="();[(1)(1 1)(2)][(2)(1 2)];()">
                       <sequent>
                        <formulalist>
                         <constantatomformula symbol="=">
                          <function symbol="∩">
                           <variable symbol="x_0"/>
                           <variable symbol="y_0"/>
                          </function>
                          <variable symbol="x_0"/>
                         </constantatomformula>
                         <constantatomformula symbol="=">
                          <function symbol="∩">
                           <variable symbol="y_0"/>
                           <variable symbol="z_0"/>
                          </function>
                          <variable symbol="y_0"/>
                         </constantatomformula>
                        </formulalist>
                        <formulalist>
                         <constantatomformula symbol="≤" definition="≤">
                          <variable symbol="x_0"/>
                          <variable symbol="z_0"/>
                         </constantatomformula>
                        </formulalist>
                       </sequent>
                       <rule type="eqr1" symbol="=:r1" param="(2)(1 1)">
                        <sequent>
                         <formulalist>
                          <constantatomformula symbol="=">
                           <function symbol="∩">
                            <variable symbol="x_0"/>
                            <variable symbol="y_0"/>
                           </function>
                           <variable symbol="x_0"/>
                          </constantatomformula>
                          <constantatomformula symbol="=">
                           <function symbol="∩">
                            <variable symbol="y_0"/>
                            <variable symbol="z_0"/>
                           </function>
                           <variable symbol="y_0"/>
                          </constantatomformula>
                         </formulalist>
                         <formulalist>
                          <constantatomformula symbol="=">
                           <function symbol="∩">
                            <variable symbol="x_0"/>
                            <variable symbol="z_0"/>
                           </function>
                           <variable symbol="x_0"/>
                          </constantatomformula>
                         </formulalist>
                        </sequent>
                        <rule type="axiom" symbol="axiom">
                         <sequent>
                          <formulalist>
                           <constantatomformula symbol="=">
                            <function symbol="∩">
                             <variable symbol="x_0"/>
                             <variable symbol="y_0"/>
                            </function>
                            <variable symbol="x_0"/>
                           </constantatomformula>
                          </formulalist>
                          <formulalist>
                           <constantatomformula symbol="=">
                            <function symbol="∩">
                             <variable symbol="x_0"/>
                             <variable symbol="y_0"/>
                            </function>
                            <variable symbol="x_0"/>
                           </constantatomformula>
                          </formulalist>
                         </sequent>
                        </rule>
                        <rule type="eqr1" symbol="=:r1" param="(2 2)">
                         <sequent>
                          <formulalist>
                           <constantatomformula symbol="=">
                            <function symbol="∩">
                             <variable symbol="y_0"/>
                             <variable symbol="z_0"/>
                            </function>
                            <variable symbol="y_0"/>
                           </constantatomformula>
                          </formulalist>
                          <formulalist>
                           <constantatomformula symbol="=">
                            <function symbol="∩">
                             <function symbol="∩">
                              <variable symbol="x_0"/>
                              <variable symbol="y_0"/>
                             </function>
                             <variable symbol="z_0"/>
                            </function>
                            <function symbol="∩">
                             <variable symbol="x_0"/>
                             <variable symbol="y_0"/>
                            </function>
                           </constantatomformula>
                          </formulalist>
                         </sequent>
                         <rule type="axiom" symbol="axiom">
                          <sequent>
                           <formulalist>
                            <constantatomformula symbol="=">
                             <function symbol="∩">
                              <variable symbol="y_0"/>
                              <variable symbol="z_0"/>
                             </function>
                             <variable symbol="y_0"/>
                            </constantatomformula>
                           </formulalist>
                           <formulalist>
                            <constantatomformula symbol="=">
                             <function symbol="∩">
                              <variable symbol="y_0"/>
                              <variable symbol="z_0"/>
                             </function>
                             <variable symbol="y_0"/>
                            </constantatomformula>
                           </formulalist>
                          </sequent>
                         </rule>
                         <rule type="axiom" symbol="axiom">
                          <sequent>
                           <formulalist/>
                           <formulalist>
                            <constantatomformula symbol="=">
                             <function symbol="∩">
                              <function symbol="∩">
                               <variable symbol="x_0"/>
                               <variable symbol="y_0"/>
                              </function>
                              <variable symbol="z_0"/>
                             </function>
                             <function symbol="∩">
                              <variable symbol="x_0"/>
                              <function symbol="∩">
                               <variable symbol="y_0"/>
                               <variable symbol="z_0"/>
                              </function>
                             </function>
                            </constantatomformula>
                           </formulalist>
                          </sequent>
                         </rule>
                        </rule>
                       </rule>
                      </rule>
                     </rule>
                    </rule>
                   </rule>
                  </rule>
                 </rule>
                </rule>
               </rule>
              </rule>
             </rule>
            </rule>
           </rule>
          </rule>
         </rule>
        </rule>
       </rule>
      </rule>
     </rule>
     <rule type="andr" symbol="\and:r">
      <sequent>
       <formulalist>
        <constantatomformula symbol="L1" definition="L1"/>
       </formulalist>
       <formulalist>
        <conjunctiveformula type="and">
         <constantatomformula symbol="GLB" definition="GLB"/>
         <constantatomformula symbol="LUB" definition="LUB"/>
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      </sequent>
      <rule type="defr" symbol="d:r" param="();[()];()">
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        <formulalist/>
        <formulalist>
         <constantatomformula symbol="GLB" definition="GLB"/>
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       </sequent>
       <rule type="forallr" symbol="\all:r">
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         <formulalist/>
         <formulalist>
          <quantifiedformula type="all">
           <variable symbol="x"/>
           <quantifiedformula type="all">
            <variable symbol="y"/>
            <conjunctiveformula type="and">
             <conjunctiveformula type="and">
              <constantatomformula symbol="≤" definition="≤">
               <function symbol="∩">
                <variable symbol="x"/>
                <variable symbol="y"/>
               </function>
               <variable symbol="x"/>
              </constantatomformula>
              <constantatomformula symbol="≤" definition="≤">
               <function symbol="∩">
                <variable symbol="x"/>
                <variable symbol="y"/>
               </function>
               <variable symbol="y"/>
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             <quantifiedformula type="all">
              <variable symbol="z"/>
              <conjunctiveformula type="impl">
               <conjunctiveformula type="and">
                <constantatomformula symbol="≤" definition="≤">
                 <variable symbol="z"/>
                 <variable symbol="x"/>
                </constantatomformula>
                <constantatomformula symbol="≤" definition="≤">
                 <variable symbol="z"/>
                 <variable symbol="y"/>
                </constantatomformula>
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               <constantatomformula symbol="≤" definition="≤">
                <variable symbol="z"/>
                <function symbol="∩">
                 <variable symbol="x"/>
                 <variable symbol="y"/>
                </function>
               </constantatomformula>
              </conjunctiveformula>
             </quantifiedformula>
            </conjunctiveformula>
           </quantifiedformula>
          </quantifiedformula>
         </formulalist>
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        <rule type="forallr" symbol="\all:r">
         <sequent>
          <formulalist/>
          <formulalist>
           <quantifiedformula type="all">
            <variable symbol="y"/>
            <conjunctiveformula type="and">
             <conjunctiveformula type="and">
              <constantatomformula symbol="≤" definition="≤">
               <function symbol="∩">
                <variable symbol="x_0"/>
                <variable symbol="y"/>
               </function>
               <variable symbol="x_0"/>
              </constantatomformula>
              <constantatomformula symbol="≤" definition="≤">
               <function symbol="∩">
                <variable symbol="x_0"/>
                <variable symbol="y"/>
               </function>
               <variable symbol="y"/>
              </constantatomformula>
             </conjunctiveformula>
             <quantifiedformula type="all">
              <variable symbol="z"/>
              <conjunctiveformula type="impl">
               <conjunctiveformula type="and">
                <constantatomformula symbol="≤" definition="≤">
                 <variable symbol="z"/>
                 <variable symbol="x_0"/>
                </constantatomformula>
                <constantatomformula symbol="≤" definition="≤">
                 <variable symbol="z"/>
                 <variable symbol="y"/>
                </constantatomformula>
               </conjunctiveformula>
               <constantatomformula symbol="≤" definition="≤">
                <variable symbol="z"/>
                <function symbol="∩">
                 <variable symbol="x_0"/>
                 <variable symbol="y"/>
                </function>
               </constantatomformula>
              </conjunctiveformula>
             </quantifiedformula>
            </conjunctiveformula>
           </quantifiedformula>
          </formulalist>
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           <formulalist/>
           <formulalist>
            <conjunctiveformula type="and">
             <conjunctiveformula type="and">
              <constantatomformula symbol="≤" definition="≤">
               <function symbol="∩">
                <variable symbol="x_0"/>
                <variable symbol="y_0"/>
               </function>
               <variable symbol="x_0"/>
              </constantatomformula>
              <constantatomformula symbol="≤" definition="≤">
               <function symbol="∩">
                <variable symbol="x_0"/>
                <variable symbol="y_0"/>
               </function>
               <variable symbol="y_0"/>
              </constantatomformula>
             </conjunctiveformula>
             <quantifiedformula type="all">
              <variable symbol="z"/>
              <conjunctiveformula type="impl">
               <conjunctiveformula type="and">
                <constantatomformula symbol="≤" definition="≤">
                 <variable symbol="z"/>
                 <variable symbol="x_0"/>
                </constantatomformula>
                <constantatomformula symbol="≤" definition="≤">
                 <variable symbol="z"/>
                 <variable symbol="y_0"/>
                </constantatomformula>
               </conjunctiveformula>
               <constantatomformula symbol="≤" definition="≤">
                <variable symbol="z"/>
                <function symbol="∩">
                 <variable symbol="x_0"/>
                 <variable symbol="y_0"/>
                </function>
               </constantatomformula>
              </conjunctiveformula>
             </quantifiedformula>
            </conjunctiveformula>
           </formulalist>
          </sequent>
          <rule type="andr" symbol="\and:r">
           <sequent>
            <formulalist/>
            <formulalist>
             <conjunctiveformula type="and">
              <constantatomformula symbol="≤" definition="≤">
               <function symbol="∩">
                <variable symbol="x_0"/>
                <variable symbol="y_0"/>
               </function>
               <variable symbol="x_0"/>
              </constantatomformula>
              <constantatomformula symbol="≤" definition="≤">
               <function symbol="∩">
                <variable symbol="x_0"/>
                <variable symbol="y_0"/>
               </function>
               <variable symbol="y_0"/>
              </constantatomformula>
             </conjunctiveformula>
            </formulalist>
           </sequent>
           <rule type="defr" symbol="d:r" param="();[(1)(1 1)(2)][(2)(1 2)];()">
            <sequent>
             <formulalist/>
             <formulalist>
              <constantatomformula symbol="≤" definition="≤">
               <function symbol="∩">
                <variable symbol="x_0"/>
                <variable symbol="y_0"/>
               </function>
               <variable symbol="x_0"/>
              </constantatomformula>
             </formulalist>
            </sequent>
            <rule type="eqr2" symbol="=:r2" param="(1)">
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              <formulalist/>
              <formulalist>
               <constantatomformula symbol="=">
                <function symbol="∩">
                 <function symbol="∩">
                  <variable symbol="x_0"/>
                  <variable symbol="y_0"/>
                 </function>
                 <variable symbol="x_0"/>
                </function>
                <function symbol="∩">
                 <variable symbol="x_0"/>
                 <variable symbol="y_0"/>
                </function>
               </constantatomformula>
              </formulalist>
             </sequent>
             <rule type="axiom" symbol="axiom">
              <sequent>
               <formulalist/>
               <formulalist>
                <constantatomformula symbol="=">
                 <function symbol="∩">
                  <function symbol="∩">
                   <variable symbol="x_0"/>
                   <variable symbol="y_0"/>
                  </function>
                  <variable symbol="x_0"/>
                 </function>
                 <function symbol="∩">
                  <variable symbol="x_0"/>
                  <function symbol="∩">
                   <variable symbol="x_0"/>
                   <variable symbol="y_0"/>
                  </function>
                 </function>
                </constantatomformula>
               </formulalist>
              </sequent>
             </rule>
             <rule type="eqr1" symbol="=:r1" param="(1)">
              <sequent>
               <formulalist/>
               <formulalist>
                <constantatomformula symbol="=">
                 <function symbol="∩">
                  <variable symbol="x_0"/>
                  <function symbol="∩">
                   <variable symbol="x_0"/>
                   <variable symbol="y_0"/>
                  </function>
                 </function>
                 <function symbol="∩">
                  <variable symbol="x_0"/>
                  <variable symbol="y_0"/>
                 </function>
                </constantatomformula>
               </formulalist>
              </sequent>
              <rule type="axiom" symbol="axiom">
               <sequent>
                <formulalist/>
                <formulalist>
                 <constantatomformula symbol="=">
                  <function symbol="∩">
                   <function symbol="∩">
                    <variable symbol="x_0"/>
                    <variable symbol="x_0"/>
                   </function>
                   <variable symbol="y_0"/>
                  </function>
                  <function symbol="∩">
                   <variable symbol="x_0"/>
                   <function symbol="∩">
                    <variable symbol="x_0"/>
                    <variable symbol="y_0"/>
                   </function>
                  </function>
                 </constantatomformula>
                </formulalist>
               </sequent>
              </rule>
              <rule type="eqr2" symbol="=:r2" param="(1 1)">
               <sequent>
                <formulalist/>
                <formulalist>
                 <constantatomformula symbol="=">
                  <function symbol="∩">
                   <function symbol="∩">
                    <variable symbol="x_0"/>
                    <variable symbol="x_0"/>
                   </function>
                   <variable symbol="y_0"/>
                  </function>
                  <function symbol="∩">
                   <variable symbol="x_0"/>
                   <variable symbol="y_0"/>
                  </function>
                 </constantatomformula>
                </formulalist>
               </sequent>
               <rule type="axiom" symbol="axiom">
                <sequent>
                 <formulalist/>
                 <formulalist>
                  <constantatomformula symbol="=">
                   <function symbol="∩">
                    <variable symbol="x_0"/>
                    <variable symbol="x_0"/>
                   </function>
                   <variable symbol="x_0"/>
                  </constantatomformula>
                 </formulalist>
                </sequent>
               </rule>
               <rule type="axiom" symbol="axiom">
                <sequent>
                 <formulalist/>
                 <formulalist>
                  <constantatomformula symbol="=">
                   <function symbol="∩">
                    <variable symbol="x_0"/>
                    <variable symbol="y_0"/>
                   </function>
                   <function symbol="∩">
                    <variable symbol="x_0"/>
                    <variable symbol="y_0"/>
                   </function>
                  </constantatomformula>
                 </formulalist>
                </sequent>
               </rule>
              </rule>
             </rule>
            </rule>
           </rule>
           <rule type="defr" symbol="d:r" param="();[(1)(1 1)(2)][(2)(1 2)];()">
            <sequent>
             <formulalist/>
             <formulalist>
              <constantatomformula symbol="≤" definition="≤">
               <function symbol="∩">
                <variable symbol="x_0"/>
                <variable symbol="y_0"/>
               </function>
               <variable symbol="y_0"/>
              </constantatomformula>
             </formulalist>
            </sequent>
            <rule type="eqr2" symbol="=:r2" param="(1)">
             <sequent>
              <formulalist/>
              <formulalist>
               <constantatomformula symbol="=">
                <function symbol="∩">
                 <function symbol="∩">
                  <variable symbol="x_0"/>
                  <variable symbol="y_0"/>
                 </function>
                 <variable symbol="y_0"/>
                </function>
                <function symbol="∩">
                 <variable symbol="x_0"/>
                 <variable symbol="y_0"/>
                </function>
               </constantatomformula>
              </formulalist>
             </sequent>
             <rule type="axiom" symbol="axiom">
              <sequent>
               <formulalist/>
               <formulalist>
                <constantatomformula symbol="=">
                 <function symbol="∩">
                  <function symbol="∩">
                   <variable symbol="x_0"/>
                   <variable symbol="y_0"/>
                  </function>
                  <variable symbol="y_0"/>
                 </function>
                 <function symbol="∩">
                  <variable symbol="x_0"/>
                  <function symbol="∩">
                   <variable symbol="y_0"/>
                   <variable symbol="y_0"/>
                  </function>
                 </function>
                </constantatomformula>
               </formulalist>
              </sequent>
             </rule>
             <rule type="eqr2" symbol="=:r2" param="(1 2)">
              <sequent>
               <formulalist/>
               <formulalist>
                <constantatomformula symbol="=">
                 <function symbol="∩">
                  <variable symbol="x_0"/>
                  <function symbol="∩">
                   <variable symbol="y_0"/>
                   <variable symbol="y_0"/>
                  </function>
                 </function>
                 <function symbol="∩">
                  <variable symbol="x_0"/>
                  <variable symbol="y_0"/>
                 </function>
                </constantatomformula>
               </formulalist>
              </sequent>
              <rule type="axiom" symbol="axiom">
               <sequent>
                <formulalist/>
                <formulalist>
                 <constantatomformula symbol="=">
                  <function symbol="∩">
                   <variable symbol="y_0"/>
                   <variable symbol="y_0"/>
                  </function>
                  <variable symbol="y_0"/>
                 </constantatomformula>
                </formulalist>
               </sequent>
              </rule>
              <rule type="axiom" symbol="axiom">
               <sequent>
                <formulalist/>
                <formulalist>
                 <constantatomformula symbol="=">
                  <function symbol="∩">
                   <variable symbol="x_0"/>
                   <variable symbol="y_0"/>
                  </function>
                  <function symbol="∩">
                   <variable symbol="x_0"/>
                   <variable symbol="y_0"/>
                  </function>
                 </constantatomformula>
                </formulalist>
               </sequent>
              </rule>
             </rule>
            </rule>
           </rule>
          </rule>
          <rule type="forallr" symbol="\all:r">
           <sequent>
            <formulalist/>
            <formulalist>
             <quantifiedformula type="all">
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              <conjunctiveformula type="impl">
               <conjunctiveformula type="and">
                <constantatomformula symbol="≤" definition="≤">
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                 <variable symbol="x_0"/>
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                <constantatomformula symbol="≤" definition="≤">
                 <variable symbol="z"/>
                 <variable symbol="y_0"/>
                </constantatomformula>
               </conjunctiveformula>
               <constantatomformula symbol="≤" definition="≤">
                <variable symbol="z"/>
                <function symbol="∩">
                 <variable symbol="x_0"/>
                 <variable symbol="y_0"/>
                </function>
               </constantatomformula>
              </conjunctiveformula>
             </quantifiedformula>
            </formulalist>
           </sequent>
           <rule type="implr" symbol="\impl:r">
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             <formulalist/>
             <formulalist>
              <conjunctiveformula type="impl">
               <conjunctiveformula type="and">
                <constantatomformula symbol="≤" definition="≤">
                 <variable symbol="z_0"/>
                 <variable symbol="x_0"/>
                </constantatomformula>
                <constantatomformula symbol="≤" definition="≤">
                 <variable symbol="z_0"/>
                 <variable symbol="y_0"/>
                </constantatomformula>
               </conjunctiveformula>
               <constantatomformula symbol="≤" definition="≤">
                <variable symbol="z_0"/>
                <function symbol="∩">
                 <variable symbol="x_0"/>
                 <variable symbol="y_0"/>
                </function>
               </constantatomformula>
              </conjunctiveformula>
             </formulalist>
            </sequent>
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              <formulalist>
               <conjunctiveformula type="and">
                <constantatomformula symbol="≤" definition="≤">
                 <variable symbol="z_0"/>
                 <variable symbol="x_0"/>
                </constantatomformula>
                <constantatomformula symbol="≤" definition="≤">
                 <variable symbol="z_0"/>
                 <variable symbol="y_0"/>
                </constantatomformula>
               </conjunctiveformula>
              </formulalist>
              <formulalist>
               <constantatomformula symbol="≤" definition="≤">
                <variable symbol="z_0"/>
                <function symbol="∩">
                 <variable symbol="x_0"/>
                 <variable symbol="y_0"/>
                </function>
               </constantatomformula>
              </formulalist>
             </sequent>
             <rule type="andl1" symbol="\and:l1">
              <sequent>
               <formulalist>
                <conjunctiveformula type="and">
                 <constantatomformula symbol="≤" definition="≤">
                  <variable symbol="z_0"/>
                  <variable symbol="x_0"/>
                 </constantatomformula>
                 <constantatomformula symbol="≤" definition="≤">
                  <variable symbol="z_0"/>
                  <variable symbol="y_0"/>
                 </constantatomformula>
                </conjunctiveformula>
                <conjunctiveformula type="and">
                 <constantatomformula symbol="≤" definition="≤">
                  <variable symbol="z_0"/>
                  <variable symbol="x_0"/>
                 </constantatomformula>
                 <constantatomformula symbol="≤" definition="≤">
                  <variable symbol="z_0"/>
                  <variable symbol="y_0"/>
                 </constantatomformula>
                </conjunctiveformula>
               </formulalist>
               <formulalist>
                <constantatomformula symbol="≤" definition="≤">
                 <variable symbol="z_0"/>
                 <function symbol="∩">
                  <variable symbol="x_0"/>
                  <variable symbol="y_0"/>
                 </function>
                </constantatomformula>
               </formulalist>
              </sequent>
              <rule type="perml" symbol="\pi:l" param="(1 2)">
               <sequent>
                <formulalist>
                 <constantatomformula symbol="≤" definition="≤">
                  <variable symbol="z_0"/>
                  <variable symbol="x_0"/>
                 </constantatomformula>
                 <conjunctiveformula type="and">
                  <constantatomformula symbol="≤" definition="≤">
                   <variable symbol="z_0"/>
                   <variable symbol="x_0"/>
                  </constantatomformula>
                  <constantatomformula symbol="≤" definition="≤">
                   <variable symbol="z_0"/>
                   <variable symbol="y_0"/>
                  </constantatomformula>
                 </conjunctiveformula>
                </formulalist>
                <formulalist>
                 <constantatomformula symbol="≤" definition="≤">
                  <variable symbol="z_0"/>
                  <function symbol="∩">
                   <variable symbol="x_0"/>
                   <variable symbol="y_0"/>
                  </function>
                 </constantatomformula>
                </formulalist>
               </sequent>
               <rule type="andl2" symbol="\and:l2">
                <sequent>
                 <formulalist>
                  <conjunctiveformula type="and">
                   <constantatomformula symbol="≤" definition="≤">
                    <variable symbol="z_0"/>
                    <variable symbol="x_0"/>
                   </constantatomformula>
                   <constantatomformula symbol="≤" definition="≤">
                    <variable symbol="z_0"/>
                    <variable symbol="y_0"/>
                   </constantatomformula>
                  </conjunctiveformula>
                  <constantatomformula symbol="≤" definition="≤">
                   <variable symbol="z_0"/>
                   <variable symbol="x_0"/>
                  </constantatomformula>
                 </formulalist>
                 <formulalist>
                  <constantatomformula symbol="≤" definition="≤">
                   <variable symbol="z_0"/>
                   <function symbol="∩">
                    <variable symbol="x_0"/>
                    <variable symbol="y_0"/>
                   </function>
                  </constantatomformula>
                 </formulalist>
                </sequent>
                <rule type="perml" symbol="\pi:l" param="(1 2)">
                 <sequent>
                  <formulalist>
                   <constantatomformula symbol="≤" definition="≤">
                    <variable symbol="z_0"/>
                    <variable symbol="y_0"/>
                   </constantatomformula>
                   <constantatomformula symbol="≤" definition="≤">
                    <variable symbol="z_0"/>
                    <variable symbol="x_0"/>
                   </constantatomformula>
                  </formulalist>
                  <formulalist>
                   <constantatomformula symbol="≤" definition="≤">
                    <variable symbol="z_0"/>
                    <function symbol="∩">
                     <variable symbol="x_0"/>
                     <variable symbol="y_0"/>
                    </function>
                   </constantatomformula>
                  </formulalist>
                 </sequent>
                 <rule type="defl" symbol="d:l" param="();[(1)(1 1)(2)][(2)(1 2)];()">
                  <sequent>
                   <formulalist>
                    <constantatomformula symbol="≤" definition="≤">
                     <variable symbol="z_0"/>
                     <variable symbol="x_0"/>
                    </constantatomformula>
                    <constantatomformula symbol="≤" definition="≤">
                     <variable symbol="z_0"/>
                     <variable symbol="y_0"/>
                    </constantatomformula>
                   </formulalist>
                   <formulalist>
                    <constantatomformula symbol="≤" definition="≤">
                     <variable symbol="z_0"/>
                     <function symbol="∩">
                      <variable symbol="x_0"/>
                      <variable symbol="y_0"/>
                     </function>
                    </constantatomformula>
                   </formulalist>
                  </sequent>
                  <rule type="perml" symbol="\pi:l" param="(1 2)">
                   <sequent>
                    <formulalist>
                     <constantatomformula symbol="=">
                      <function symbol="∩">
                       <variable symbol="z_0"/>
                       <variable symbol="x_0"/>
                      </function>
                      <variable symbol="z_0"/>
                     </constantatomformula>
                     <constantatomformula symbol="≤" definition="≤">
                      <variable symbol="z_0"/>
                      <variable symbol="y_0"/>
                     </constantatomformula>
                    </formulalist>
                    <formulalist>
                     <constantatomformula symbol="≤" definition="≤">
                      <variable symbol="z_0"/>
                      <function symbol="∩">
                       <variable symbol="x_0"/>
                       <variable symbol="y_0"/>
                      </function>
                     </constantatomformula>
                    </formulalist>
                   </sequent>
                   <rule type="defl" symbol="d:l" param="();[(1)(1 1)(2)][(2)(1 2)];()">
                    <sequent>
                     <formulalist>
                      <constantatomformula symbol="≤" definition="≤">
                       <variable symbol="z_0"/>
                       <variable symbol="y_0"/>
                      </constantatomformula>
                      <constantatomformula symbol="=">
                       <function symbol="∩">
                        <variable symbol="z_0"/>
                        <variable symbol="x_0"/>
                       </function>
                       <variable symbol="z_0"/>
                      </constantatomformula>
                     </formulalist>
                     <formulalist>
                      <constantatomformula symbol="≤" definition="≤">
                       <variable symbol="z_0"/>
                       <function symbol="∩">
                        <variable symbol="x_0"/>
                        <variable symbol="y_0"/>
                       </function>
                      </constantatomformula>
                     </formulalist>
                    </sequent>
                    <rule type="perml" symbol="\pi:l" param="(1 2)">
                     <sequent>
                      <formulalist>
                       <constantatomformula symbol="=">
                        <function symbol="∩">
                         <variable symbol="z_0"/>
                         <variable symbol="y_0"/>
                        </function>
                        <variable symbol="z_0"/>
                       </constantatomformula>
                       <constantatomformula symbol="=">
                        <function symbol="∩">
                         <variable symbol="z_0"/>
                         <variable symbol="x_0"/>
                        </function>
                        <variable symbol="z_0"/>
                       </constantatomformula>
                      </formulalist>
                      <formulalist>
                       <constantatomformula symbol="≤" definition="≤">
                        <variable symbol="z_0"/>
                        <function symbol="∩">
                         <variable symbol="x_0"/>
                         <variable symbol="y_0"/>
                        </function>
                       </constantatomformula>
                      </formulalist>
                     </sequent>
                     <rule type="defr" symbol="d:r" param="();[(1)(1 1)(2)][(2)(1 2)];()">
                      <sequent>
                       <formulalist>
                        <constantatomformula symbol="=">
                         <function symbol="∩">
                          <variable symbol="z_0"/>
                          <variable symbol="x_0"/>
                         </function>
                         <variable symbol="z_0"/>
                        </constantatomformula>
                        <constantatomformula symbol="=">
                         <function symbol="∩">
                          <variable symbol="z_0"/>
                          <variable symbol="y_0"/>
                         </function>
                         <variable symbol="z_0"/>
                        </constantatomformula>
                       </formulalist>
                       <formulalist>
                        <constantatomformula symbol="≤" definition="≤">
                         <variable symbol="z_0"/>
                         <function symbol="∩">
                          <variable symbol="x_0"/>
                          <variable symbol="y_0"/>
                         </function>
                        </constantatomformula>
                       </formulalist>
                      </sequent>
                      <rule type="eqr1" symbol="=:r1" param="(1)">
                       <sequent>
                        <formulalist>
                         <constantatomformula symbol="=">
                          <function symbol="∩">
                           <variable symbol="z_0"/>
                           <variable symbol="x_0"/>
                          </function>
                          <variable symbol="z_0"/>
                         </constantatomformula>
                         <constantatomformula symbol="=">
                          <function symbol="∩">
                           <variable symbol="z_0"/>
                           <variable symbol="y_0"/>
                          </function>
                          <variable symbol="z_0"/>
                         </constantatomformula>
                        </formulalist>
                        <formulalist>
                         <constantatomformula symbol="=">
                          <function symbol="∩">
                           <variable symbol="z_0"/>
                           <function symbol="∩">
                            <variable symbol="x_0"/>
                            <variable symbol="y_0"/>
                           </function>
                          </function>
                          <variable symbol="z_0"/>
                         </constantatomformula>
                        </formulalist>
                       </sequent>
                       <rule type="axiom" symbol="axiom">
                        <sequent>
                         <formulalist/>
                         <formulalist>
                          <constantatomformula symbol="=">
                           <function symbol="∩">
                            <function symbol="∩">
                             <variable symbol="z_0"/>
                             <variable symbol="x_0"/>
                            </function>
                            <variable symbol="y_0"/>
                           </function>
                           <function symbol="∩">
                            <variable symbol="z_0"/>
                            <function symbol="∩">
                             <variable symbol="x_0"/>
                             <variable symbol="y_0"/>
                            </function>
                           </function>
                          </constantatomformula>
                         </formulalist>
                        </sequent>
                       </rule>
                       <rule type="eqr2" symbol="=:r2" param="(1 1)">
                        <sequent>
                         <formulalist>
                          <constantatomformula symbol="=">
                           <function symbol="∩">
                            <variable symbol="z_0"/>
                            <variable symbol="x_0"/>
                           </function>
                           <variable symbol="z_0"/>
                          </constantatomformula>
                          <constantatomformula symbol="=">
                           <function symbol="∩">
                            <variable symbol="z_0"/>
                            <variable symbol="y_0"/>
                           </function>
                           <variable symbol="z_0"/>
                          </constantatomformula>
                         </formulalist>
                         <formulalist>
                          <constantatomformula symbol="=">
                           <function symbol="∩">
                            <function symbol="∩">
                             <variable symbol="z_0"/>
                             <variable symbol="x_0"/>
                            </function>
                            <variable symbol="y_0"/>
                           </function>
                           <variable symbol="z_0"/>
                          </constantatomformula>
                         </formulalist>
                        </sequent>
                        <rule type="axiom" symbol="axiom">
                         <sequent>
                          <formulalist>
                           <constantatomformula symbol="=">
                            <function symbol="∩">
                             <variable symbol="z_0"/>
                             <variable symbol="x_0"/>
                            </function>
                            <variable symbol="z_0"/>
                           </constantatomformula>
                          </formulalist>
                          <formulalist>
                           <constantatomformula symbol="=">
                            <function symbol="∩">
                             <variable symbol="z_0"/>
                             <variable symbol="x_0"/>
                            </function>
                            <variable symbol="z_0"/>
                           </constantatomformula>
                          </formulalist>
                         </sequent>
                        </rule>
                        <rule type="axiom" symbol="axiom">
                         <sequent>
                          <formulalist>
                           <constantatomformula symbol="=">
                            <function symbol="∩">
                             <variable symbol="z_0"/>
                             <variable symbol="y_0"/>
                            </function>
                            <variable symbol="z_0"/>
                           </constantatomformula>
                          </formulalist>
                          <formulalist>
                           <constantatomformula symbol="=">
                            <function symbol="∩">
                             <variable symbol="z_0"/>
                             <variable symbol="y_0"/>
                            </function>
                            <variable symbol="z_0"/>
                           </constantatomformula>
                          </formulalist>
                         </sequent>
                        </rule>
                       </rule>
                      </rule>
                     </rule>
                    </rule>
                   </rule>
                  </rule>
                 </rule>
                </rule>
               </rule>
              </rule>
             </rule>
            </rule>
           </rule>
          </rule>
         </rule>
        </rule>
       </rule>
      </rule>
      <rule type="defr" symbol="d:r" param="();[()];()">
       <sequent>
        <formulalist>
         <constantatomformula symbol="L1" definition="L1"/>
        </formulalist>
        <formulalist>
         <constantatomformula symbol="LUB" definition="LUB"/>
        </formulalist>
       </sequent>
       <rule type="forallr" symbol="\all:r">
        <sequent>
         <formulalist>
          <constantatomformula symbol="L1" definition="L1"/>
         </formulalist>
         <formulalist>
          <quantifiedformula type="all">
           <variable symbol="x"/>
           <quantifiedformula type="all">
            <variable symbol="y"/>
            <conjunctiveformula type="and">
             <conjunctiveformula type="and">
              <constantatomformula symbol="≤" definition="≤">
               <variable symbol="x"/>
               <function symbol="∪">
                <variable symbol="x"/>
                <variable symbol="y"/>
               </function>
              </constantatomformula>
              <constantatomformula symbol="≤" definition="≤">
               <variable symbol="y"/>
               <function symbol="∪">
                <variable symbol="x"/>
                <variable symbol="y"/>
               </function>
              </constantatomformula>
             </conjunctiveformula>
             <quantifiedformula type="all">
              <variable symbol="z"/>
              <conjunctiveformula type="impl">
               <conjunctiveformula type="and">
                <constantatomformula symbol="≤" definition="≤">
                 <variable symbol="x"/>
                 <variable symbol="z"/>
                </constantatomformula>
                <constantatomformula symbol="≤" definition="≤">
                 <variable symbol="y"/>
                 <variable symbol="z"/>
                </constantatomformula>
               </conjunctiveformula>
               <constantatomformula symbol="≤" definition="≤">
                <function symbol="∪">
                 <variable symbol="x"/>
                 <variable symbol="y"/>
                </function>
                <variable symbol="z"/>
               </constantatomformula>
              </conjunctiveformula>
             </quantifiedformula>
            </conjunctiveformula>
           </quantifiedformula>
          </quantifiedformula>
         </formulalist>
        </sequent>
        <rule type="forallr" symbol="\all:r">
         <sequent>
          <formulalist>
           <constantatomformula symbol="L1" definition="L1"/>
          </formulalist>
          <formulalist>
           <quantifiedformula type="all">
            <variable symbol="y"/>
            <conjunctiveformula type="and">
             <conjunctiveformula type="and">
              <constantatomformula symbol="≤" definition="≤">
               <variable symbol="x_0"/>
               <function symbol="∪">
                <variable symbol="x_0"/>
                <variable symbol="y"/>
               </function>
              </constantatomformula>
              <constantatomformula symbol="≤" definition="≤">
               <variable symbol="y"/>
               <function symbol="∪">
                <variable symbol="x_0"/>
                <variable symbol="y"/>
               </function>
              </constantatomformula>
             </conjunctiveformula>
             <quantifiedformula type="all">
              <variable symbol="z"/>
              <conjunctiveformula type="impl">
               <conjunctiveformula type="and">
                <constantatomformula symbol="≤" definition="≤">
                 <variable symbol="x_0"/>
                 <variable symbol="z"/>
                </constantatomformula>
                <constantatomformula symbol="≤" definition="≤">
                 <variable symbol="y"/>
                 <variable symbol="z"/>
                </constantatomformula>
               </conjunctiveformula>
               <constantatomformula symbol="≤" definition="≤">
                <function symbol="∪">
                 <variable symbol="x_0"/>
                 <variable symbol="y"/>
                </function>
                <variable symbol="z"/>
               </constantatomformula>
              </conjunctiveformula>
             </quantifiedformula>
            </conjunctiveformula>
           </quantifiedformula>
          </formulalist>
         </sequent>
         <rule type="contrl" symbol="c:l" param="2">
          <sequent>
           <formulalist>
            <constantatomformula symbol="L1" definition="L1"/>
           </formulalist>
           <formulalist>
            <conjunctiveformula type="and">
             <conjunctiveformula type="and">
              <constantatomformula symbol="≤" definition="≤">
               <variable symbol="x_0"/>
               <function symbol="∪">
                <variable symbol="x_0"/>
                <variable symbol="y_0"/>
               </function>
              </constantatomformula>
              <constantatomformula symbol="≤" definition="≤">
               <variable symbol="y_0"/>
               <function symbol="∪">
                <variable symbol="x_0"/>
                <variable symbol="y_0"/>
               </function>
              </constantatomformula>
             </conjunctiveformula>
             <quantifiedformula type="all">
              <variable symbol="z"/>
              <conjunctiveformula type="impl">
               <conjunctiveformula type="and">
                <constantatomformula symbol="≤" definition="≤">
                 <variable symbol="x_0"/>
                 <variable symbol="z"/>
                </constantatomformula>
                <constantatomformula symbol="≤" definition="≤">
                 <variable symbol="y_0"/>
                 <variable symbol="z"/>
                </constantatomformula>
               </conjunctiveformula>
               <constantatomformula symbol="≤" definition="≤">
                <function symbol="∪">
                 <variable symbol="x_0"/>
                 <variable symbol="y_0"/>
                </function>
                <variable symbol="z"/>
               </constantatomformula>
              </conjunctiveformula>
             </quantifiedformula>
            </conjunctiveformula>
           </formulalist>
          </sequent>
          <rule type="andr" symbol="\and:r">
           <sequent>
            <formulalist>
             <constantatomformula symbol="L1" definition="L1"/>
             <constantatomformula symbol="L1" definition="L1"/>
            </formulalist>
            <formulalist>
             <conjunctiveformula type="and">
              <conjunctiveformula type="and">
               <constantatomformula symbol="≤" definition="≤">
                <variable symbol="x_0"/>
                <function symbol="∪">
                 <variable symbol="x_0"/>
                 <variable symbol="y_0"/>
                </function>
               </constantatomformula>
               <constantatomformula symbol="≤" definition="≤">
                <variable symbol="y_0"/>
                <function symbol="∪">
                 <variable symbol="x_0"/>
                 <variable symbol="y_0"/>
                </function>
               </constantatomformula>
              </conjunctiveformula>
              <quantifiedformula type="all">
               <variable symbol="z"/>
               <conjunctiveformula type="impl">
                <conjunctiveformula type="and">
                 <constantatomformula symbol="≤" definition="≤">
                  <variable symbol="x_0"/>
                  <variable symbol="z"/>
                 </constantatomformula>
                 <constantatomformula symbol="≤" definition="≤">
                  <variable symbol="y_0"/>
                  <variable symbol="z"/>
                 </constantatomformula>
                </conjunctiveformula>
                <constantatomformula symbol="≤" definition="≤">
                 <function symbol="∪">
                  <variable symbol="x_0"/>
                  <variable symbol="y_0"/>
                 </function>
                 <variable symbol="z"/>
                </constantatomformula>
               </conjunctiveformula>
              </quantifiedformula>
             </conjunctiveformula>
            </formulalist>
           </sequent>
           <rule type="contrl" symbol="c:l" param="2">
            <sequent>
             <formulalist>
              <constantatomformula symbol="L1" definition="L1"/>
             </formulalist>
             <formulalist>
              <conjunctiveformula type="and">
               <constantatomformula symbol="≤" definition="≤">
                <variable symbol="x_0"/>
                <function symbol="∪">
                 <variable symbol="x_0"/>
                 <variable symbol="y_0"/>
                </function>
               </constantatomformula>
               <constantatomformula symbol="≤" definition="≤">
                <variable symbol="y_0"/>
                <function symbol="∪">
                 <variable symbol="x_0"/>
                 <variable symbol="y_0"/>
                </function>
               </constantatomformula>
              </conjunctiveformula>
             </formulalist>
            </sequent>
            <rule type="andr" symbol="\and:r">
             <sequent>
              <formulalist>
               <constantatomformula symbol="L1" definition="L1"/>
               <constantatomformula symbol="L1" definition="L1"/>
              </formulalist>
              <formulalist>
               <conjunctiveformula type="and">
                <constantatomformula symbol="≤" definition="≤">
                 <variable symbol="x_0"/>
                 <function symbol="∪">
                  <variable symbol="x_0"/>
                  <variable symbol="y_0"/>
                 </function>
                </constantatomformula>
                <constantatomformula symbol="≤" definition="≤">
                 <variable symbol="y_0"/>
                 <function symbol="∪">
                  <variable symbol="x_0"/>
                  <variable symbol="y_0"/>
                 </function>
                </constantatomformula>
               </conjunctiveformula>
              </formulalist>
             </sequent>
             <rule type="defr" symbol="d:r" param="();[(1)(1 1)(2)][(2)(1 2)];()">
              <sequent>
               <formulalist>
                <constantatomformula symbol="L1" definition="L1"/>
               </formulalist>
               <formulalist>
                <constantatomformula symbol="≤" definition="≤">
                 <variable symbol="x_0"/>
                 <function symbol="∪">
                  <variable symbol="x_0"/>
                  <variable symbol="y_0"/>
                 </function>
                </constantatomformula>
               </formulalist>
              </sequent>
              <rule type="defl" symbol="d:l" param="();[()];()">
               <sequent>
                <formulalist>
                 <constantatomformula symbol="L1" definition="L1"/>
                </formulalist>
                <formulalist>
                 <constantatomformula symbol="=">
                  <function symbol="∩">
                   <variable symbol="x_0"/>
                   <function symbol="∪">
                    <variable symbol="x_0"/>
                    <variable symbol="y_0"/>
                   </function>
                  </function>
                  <variable symbol="x_0"/>
                 </constantatomformula>
                </formulalist>
               </sequent>
               <rule type="foralll" symbol="\all:l">
                <sequent>
                 <formulalist>
                  <quantifiedformula type="all">
                   <variable symbol="x"/>
                   <quantifiedformula type="all">
                    <variable symbol="y"/>
                    <conjunctiveformula type="and">
                     <conjunctiveformula type="impl">
                      <constantatomformula symbol="=">
                       <function symbol="∩">
                        <variable symbol="x"/>
                        <variable symbol="y"/>
                       </function>
                       <variable symbol="x"/>
                      </constantatomformula>
                      <constantatomformula symbol="=">
                       <function symbol="∪">
                        <variable symbol="x"/>
                        <variable symbol="y"/>
                       </function>
                       <variable symbol="y"/>
                      </constantatomformula>
                     </conjunctiveformula>
                     <conjunctiveformula type="impl">
                      <constantatomformula symbol="=">
                       <function symbol="∪">
                        <variable symbol="x"/>
                        <variable symbol="y"/>
                       </function>
                       <variable symbol="y"/>
                      </constantatomformula>
                      <constantatomformula symbol="=">
                       <function symbol="∩">
                        <variable symbol="x"/>
                        <variable symbol="y"/>
                       </function>
                       <variable symbol="x"/>
                      </constantatomformula>
                     </conjunctiveformula>
                    </conjunctiveformula>
                   </quantifiedformula>
                  </quantifiedformula>
                 </formulalist>
                 <formulalist>
                  <constantatomformula symbol="=">
                   <function symbol="∩">
                    <variable symbol="x_0"/>
                    <function symbol="∪">
                     <variable symbol="x_0"/>
                     <variable symbol="y_0"/>
                    </function>
                   </function>
                   <variable symbol="x_0"/>
                  </constantatomformula>
                 </formulalist>
                </sequent>
                <rule type="foralll" symbol="\all:l">
                 <sequent>
                  <formulalist>
                   <quantifiedformula type="all">
                    <variable symbol="y"/>
                    <conjunctiveformula type="and">
                     <conjunctiveformula type="impl">
                      <constantatomformula symbol="=">
                       <function symbol="∩">
                        <variable symbol="x_0"/>
                        <variable symbol="y"/>
                       </function>
                       <variable symbol="x_0"/>
                      </constantatomformula>
                      <constantatomformula symbol="=">
                       <function symbol="∪">
                        <variable symbol="x_0"/>
                        <variable symbol="y"/>
                       </function>
                       <variable symbol="y"/>
                      </constantatomformula>
                     </conjunctiveformula>
                     <conjunctiveformula type="impl">
                      <constantatomformula symbol="=">
                       <function symbol="∪">
                        <variable symbol="x_0"/>
                        <variable symbol="y"/>
                       </function>
                       <variable symbol="y"/>
                      </constantatomformula>
                      <constantatomformula symbol="=">
                       <function symbol="∩">
                        <variable symbol="x_0"/>
                        <variable symbol="y"/>
                       </function>
                       <variable symbol="x_0"/>
                      </constantatomformula>
                     </conjunctiveformula>
                    </conjunctiveformula>
                   </quantifiedformula>
                  </formulalist>
                  <formulalist>
                   <constantatomformula symbol="=">
                    <function symbol="∩">
                     <variable symbol="x_0"/>
                     <function symbol="∪">
                      <variable symbol="x_0"/>
                      <variable symbol="y_0"/>
                     </function>
                    </function>
                    <variable symbol="x_0"/>
                   </constantatomformula>
                  </formulalist>
                 </sequent>
                 <rule type="andl2" symbol="\and:l2">
                  <sequent>
                   <formulalist>
                    <conjunctiveformula type="and">
                     <conjunctiveformula type="impl">
                      <constantatomformula symbol="=">
                       <function symbol="∩">
                        <variable symbol="x_0"/>
                        <function symbol="∪">
                         <variable symbol="x_0"/>
                         <variable symbol="y_0"/>
                        </function>
                       </function>
                       <variable symbol="x_0"/>
                      </constantatomformula>
                      <constantatomformula symbol="=">
                       <function symbol="∪">
                        <variable symbol="x_0"/>
                        <function symbol="∪">
                         <variable symbol="x_0"/>
                         <variable symbol="y_0"/>
                        </function>
                       </function>
                       <function symbol="∪">
                        <variable symbol="x_0"/>
                        <variable symbol="y_0"/>
                       </function>
                      </constantatomformula>
                     </conjunctiveformula>
                     <conjunctiveformula type="impl">
                      <constantatomformula symbol="=">
                       <function symbol="∪">
                        <variable symbol="x_0"/>
                        <function symbol="∪">
                         <variable symbol="x_0"/>
                         <variable symbol="y_0"/>
                        </function>
                       </function>
                       <function symbol="∪">
                        <variable symbol="x_0"/>
                        <variable symbol="y_0"/>
                       </function>
                      </constantatomformula>
                      <constantatomformula symbol="=">
                       <function symbol="∩">
                        <variable symbol="x_0"/>
                        <function symbol="∪">
                         <variable symbol="x_0"/>
                         <variable symbol="y_0"/>
                        </function>
                       </function>
                       <variable symbol="x_0"/>
                      </constantatomformula>
                     </conjunctiveformula>
                    </conjunctiveformula>
                   </formulalist>
                   <formulalist>
                    <constantatomformula symbol="=">
                     <function symbol="∩">
                      <variable symbol="x_0"/>
                      <function symbol="∪">
                       <variable symbol="x_0"/>
                       <variable symbol="y_0"/>
                      </function>
                     </function>
                     <variable symbol="x_0"/>
                    </constantatomformula>
                   </formulalist>
                  </sequent>
                  <rule type="impll" symbol="\impl:l">
                   <sequent>
                    <formulalist>
                     <conjunctiveformula type="impl">
                      <constantatomformula symbol="=">
                       <function symbol="∪">
                        <variable symbol="x_0"/>
                        <function symbol="∪">
                         <variable symbol="x_0"/>
                         <variable symbol="y_0"/>
                        </function>
                       </function>
                       <function symbol="∪">
                        <variable symbol="x_0"/>
                        <variable symbol="y_0"/>
                       </function>
                      </constantatomformula>
                      <constantatomformula symbol="=">
                       <function symbol="∩">
                        <variable symbol="x_0"/>
                        <function symbol="∪">
                         <variable symbol="x_0"/>
                         <variable symbol="y_0"/>
                        </function>
                       </function>
                       <variable symbol="x_0"/>
                      </constantatomformula>
                     </conjunctiveformula>
                    </formulalist>
                    <formulalist>
                     <constantatomformula symbol="=">
                      <function symbol="∩">
                       <variable symbol="x_0"/>
                       <function symbol="∪">
                        <variable symbol="x_0"/>
                        <variable symbol="y_0"/>
                       </function>
                      </function>
                      <variable symbol="x_0"/>
                     </constantatomformula>
                    </formulalist>
                   </sequent>
                   <rule type="eqr1" symbol="=:r1" param="(1)">
                    <sequent>
                     <formulalist/>
                     <formulalist>
                      <constantatomformula symbol="=">
                       <function symbol="∪">
                        <variable symbol="x_0"/>
                        <function symbol="∪">
                         <variable symbol="x_0"/>
                         <variable symbol="y_0"/>
                        </function>
                       </function>
                       <function symbol="∪">
                        <variable symbol="x_0"/>
                        <variable symbol="y_0"/>
                       </function>
                      </constantatomformula>
                     </formulalist>
                    </sequent>
                    <rule type="axiom" symbol="axiom">
                     <sequent>
                      <formulalist/>
                      <formulalist>
                       <constantatomformula symbol="=">
                        <function symbol="∪">
                         <function symbol="∪">
                          <variable symbol="x_0"/>
                          <variable symbol="x_0"/>
                         </function>
                         <variable symbol="y_0"/>
                        </function>
                        <function symbol="∪">
                         <variable symbol="x_0"/>
                         <function symbol="∪">
                          <variable symbol="x_0"/>
                          <variable symbol="y_0"/>
                         </function>
                        </function>
                       </constantatomformula>
                      </formulalist>
                     </sequent>
                    </rule>
                    <rule type="eqr2" symbol="=:r2" param="(1 1)">
                     <sequent>
                      <formulalist/>
                      <formulalist>
                       <constantatomformula symbol="=">
                        <function symbol="∪">
                         <function symbol="∪">
                          <variable symbol="x_0"/>
                          <variable symbol="x_0"/>
                         </function>
                         <variable symbol="y_0"/>
                        </function>
                        <function symbol="∪">
                         <variable symbol="x_0"/>
                         <variable symbol="y_0"/>
                        </function>
                       </constantatomformula>
                      </formulalist>
                     </sequent>
                     <rule type="axiom" symbol="axiom">
                      <sequent>
                       <formulalist/>
                       <formulalist>
                        <constantatomformula symbol="=">
                         <function symbol="∪">
                          <variable symbol="x_0"/>
                          <variable symbol="x_0"/>
                         </function>
                         <variable symbol="x_0"/>
                        </constantatomformula>
                       </formulalist>
                      </sequent>
                     </rule>
                     <rule type="axiom" symbol="axiom">
                      <sequent>
                       <formulalist/>
                       <formulalist>
                        <constantatomformula symbol="=">
                         <function symbol="∪">
                          <variable symbol="x_0"/>
                          <variable symbol="y_0"/>
                         </function>
                         <function symbol="∪">
                          <variable symbol="x_0"/>
                          <variable symbol="y_0"/>
                         </function>
                        </constantatomformula>
                       </formulalist>
                      </sequent>
                     </rule>
                    </rule>
                   </rule>
                   <rule type="axiom" symbol="axiom">
                    <sequent>
                     <formulalist>
                      <constantatomformula symbol="=">
                       <function symbol="∩">
                        <variable symbol="x_0"/>
                        <function symbol="∪">
                         <variable symbol="x_0"/>
                         <variable symbol="y_0"/>
                        </function>
                       </function>
                       <variable symbol="x_0"/>
                      </constantatomformula>
                     </formulalist>
                     <formulalist>
                      <constantatomformula symbol="=">
                       <function symbol="∩">
                        <variable symbol="x_0"/>
                        <function symbol="∪">
                         <variable symbol="x_0"/>
                         <variable symbol="y_0"/>
                        </function>
                       </function>
                       <variable symbol="x_0"/>
                      </constantatomformula>
                     </formulalist>
                    </sequent>
                   </rule>
                  </rule>
                 </rule>
                </rule>
               </rule>
              </rule>
             </rule>
             <rule type="eqr2" symbol="=:r2" param="(2)">
              <sequent>
               <formulalist>
                <constantatomformula symbol="L1" definition="L1"/>
               </formulalist>
               <formulalist>
                <constantatomformula symbol="≤" definition="≤">
                 <variable symbol="y_0"/>
                 <function symbol="∪">
                  <variable symbol="x_0"/>
                  <variable symbol="y_0"/>
                 </function>
                </constantatomformula>
               </formulalist>
              </sequent>
              <rule type="axiom" symbol="axiom">
               <sequent>
                <formulalist/>
                <formulalist>
                 <constantatomformula symbol="=">
                  <function symbol="∪">
                   <variable symbol="x_0"/>
                   <variable symbol="y_0"/>
                  </function>
                  <function symbol="∪">
                   <variable symbol="y_0"/>
                   <variable symbol="x_0"/>
                  </function>
                 </constantatomformula>
                </formulalist>
               </sequent>
              </rule>
              <rule type="defr" symbol="d:r" param="();[(1)(1 1)(2)][(2)(1 2)];()">
               <sequent>
                <formulalist>
                 <constantatomformula symbol="L1" definition="L1"/>
                </formulalist>
                <formulalist>
                 <constantatomformula symbol="≤" definition="≤">
                  <variable symbol="y_0"/>
                  <function symbol="∪">
                   <variable symbol="y_0"/>
                   <variable symbol="x_0"/>
                  </function>
                 </constantatomformula>
                </formulalist>
               </sequent>
               <rule type="defl" symbol="d:l" param="();[()];()">
                <sequent>
                 <formulalist>
                  <constantatomformula symbol="L1" definition="L1"/>
                 </formulalist>
                 <formulalist>
                  <constantatomformula symbol="=">
                   <function symbol="∩">
                    <variable symbol="y_0"/>
                    <function symbol="∪">
                     <variable symbol="y_0"/>
                     <variable symbol="x_0"/>
                    </function>
                   </function>
                   <variable symbol="y_0"/>
                  </constantatomformula>
                 </formulalist>
                </sequent>
                <rule type="foralll" symbol="\all:l">
                 <sequent>
                  <formulalist>
                   <quantifiedformula type="all">
                    <variable symbol="x"/>
                    <quantifiedformula type="all">
                     <variable symbol="y"/>
                     <conjunctiveformula type="and">
                      <conjunctiveformula type="impl">
                       <constantatomformula symbol="=">
                        <function symbol="∩">
                         <variable symbol="x"/>
                         <variable symbol="y"/>
                        </function>
                        <variable symbol="x"/>
                       </constantatomformula>
                       <constantatomformula symbol="=">
                        <function symbol="∪">
                         <variable symbol="x"/>
                         <variable symbol="y"/>
                        </function>
                        <variable symbol="y"/>
                       </constantatomformula>
                      </conjunctiveformula>
                      <conjunctiveformula type="impl">
                       <constantatomformula symbol="=">
                        <function symbol="∪">
                         <variable symbol="x"/>
                         <variable symbol="y"/>
                        </function>
                        <variable symbol="y"/>
                       </constantatomformula>
                       <constantatomformula symbol="=">
                        <function symbol="∩">
                         <variable symbol="x"/>
                         <variable symbol="y"/>
                        </function>
                        <variable symbol="x"/>
                       </constantatomformula>
                      </conjunctiveformula>
                     </conjunctiveformula>
                    </quantifiedformula>
                   </quantifiedformula>
                  </formulalist>
                  <formulalist>
                   <constantatomformula symbol="=">
                    <function symbol="∩">
                     <variable symbol="y_0"/>
                     <function symbol="∪">
                      <variable symbol="y_0"/>
                      <variable symbol="x_0"/>
                     </function>
                    </function>
                    <variable symbol="y_0"/>
                   </constantatomformula>
                  </formulalist>
                 </sequent>
                 <rule type="foralll" symbol="\all:l">
                  <sequent>
                   <formulalist>
                    <quantifiedformula type="all">
                     <variable symbol="y"/>
                     <conjunctiveformula type="and">
                      <conjunctiveformula type="impl">
                       <constantatomformula symbol="=">
                        <function symbol="∩">
                         <variable symbol="y_0"/>
                         <variable symbol="y"/>
                        </function>
                        <variable symbol="y_0"/>
                       </constantatomformula>
                       <constantatomformula symbol="=">
                        <function symbol="∪">
                         <variable symbol="y_0"/>
                         <variable symbol="y"/>
                        </function>
                        <variable symbol="y"/>
                       </constantatomformula>
                      </conjunctiveformula>
                      <conjunctiveformula type="impl">
                       <constantatomformula symbol="=">
                        <function symbol="∪">
                         <variable symbol="y_0"/>
                         <variable symbol="y"/>
                        </function>
                        <variable symbol="y"/>
                       </constantatomformula>
                       <constantatomformula symbol="=">
                        <function symbol="∩">
                         <variable symbol="y_0"/>
                         <variable symbol="y"/>
                        </function>
                        <variable symbol="y_0"/>
                       </constantatomformula>
                      </conjunctiveformula>
                     </conjunctiveformula>
                    </quantifiedformula>
                   </formulalist>
                   <formulalist>
                    <constantatomformula symbol="=">
                     <function symbol="∩">
                      <variable symbol="y_0"/>
                      <function symbol="∪">
                       <variable symbol="y_0"/>
                       <variable symbol="x_0"/>
                      </function>
                     </function>
                     <variable symbol="y_0"/>
                    </constantatomformula>
                   </formulalist>
                  </sequent>
                  <rule type="andl2" symbol="\and:l2">
                   <sequent>
                    <formulalist>
                     <conjunctiveformula type="and">
                      <conjunctiveformula type="impl">
                       <constantatomformula symbol="=">
                        <function symbol="∩">
                         <variable symbol="y_0"/>
                         <function symbol="∪">
                          <variable symbol="y_0"/>
                          <variable symbol="x_0"/>
                         </function>
                        </function>
                        <variable symbol="y_0"/>
                       </constantatomformula>
                       <constantatomformula symbol="=">
                        <function symbol="∪">
                         <variable symbol="y_0"/>
                         <function symbol="∪">
                          <variable symbol="y_0"/>
                          <variable symbol="x_0"/>
                         </function>
                        </function>
                        <function symbol="∪">
                         <variable symbol="y_0"/>
                         <variable symbol="x_0"/>
                        </function>
                       </constantatomformula>
                      </conjunctiveformula>
                      <conjunctiveformula type="impl">
                       <constantatomformula symbol="=">
                        <function symbol="∪">
                         <variable symbol="y_0"/>
                         <function symbol="∪">
                          <variable symbol="y_0"/>
                          <variable symbol="x_0"/>
                         </function>
                        </function>
                        <function symbol="∪">
                         <variable symbol="y_0"/>
                         <variable symbol="x_0"/>
                        </function>
                       </constantatomformula>
                       <constantatomformula symbol="=">
                        <function symbol="∩">
                         <variable symbol="y_0"/>
                         <function symbol="∪">
                          <variable symbol="y_0"/>
                          <variable symbol="x_0"/>
                         </function>
                        </function>
                        <variable symbol="y_0"/>
                       </constantatomformula>
                      </conjunctiveformula>
                     </conjunctiveformula>
                    </formulalist>
                    <formulalist>
                     <constantatomformula symbol="=">
                      <function symbol="∩">
                       <variable symbol="y_0"/>
                       <function symbol="∪">
                        <variable symbol="y_0"/>
                        <variable symbol="x_0"/>
                       </function>
                      </function>
                      <variable symbol="y_0"/>
                     </constantatomformula>
                    </formulalist>
                   </sequent>
                   <rule type="impll" symbol="\impl:l">
                    <sequent>
                     <formulalist>
                      <conjunctiveformula type="impl">
                       <constantatomformula symbol="=">
                        <function symbol="∪">
                         <variable symbol="y_0"/>
                         <function symbol="∪">
                          <variable symbol="y_0"/>
                          <variable symbol="x_0"/>
                         </function>
                        </function>
                        <function symbol="∪">
                         <variable symbol="y_0"/>
                         <variable symbol="x_0"/>
                        </function>
                       </constantatomformula>
                       <constantatomformula symbol="=">
                        <function symbol="∩">
                         <variable symbol="y_0"/>
                         <function symbol="∪">
                          <variable symbol="y_0"/>
                          <variable symbol="x_0"/>
                         </function>
                        </function>
                        <variable symbol="y_0"/>
                       </constantatomformula>
                      </conjunctiveformula>
                     </formulalist>
                     <formulalist>
                      <constantatomformula symbol="=">
                       <function symbol="∩">
                        <variable symbol="y_0"/>
                        <function symbol="∪">
                         <variable symbol="y_0"/>
                         <variable symbol="x_0"/>
                        </function>
                       </function>
                       <variable symbol="y_0"/>
                      </constantatomformula>
                     </formulalist>
                    </sequent>
                    <rule type="eqr1" symbol="=:r1" param="(1)">
                     <sequent>
                      <formulalist/>
                      <formulalist>
                       <constantatomformula symbol="=">
                        <function symbol="∪">
                         <variable symbol="y_0"/>
                         <function symbol="∪">
                          <variable symbol="y_0"/>
                          <variable symbol="x_0"/>
                         </function>
                        </function>
                        <function symbol="∪">
                         <variable symbol="y_0"/>
                         <variable symbol="x_0"/>
                        </function>
                       </constantatomformula>
                      </formulalist>
                     </sequent>
                     <rule type="axiom" symbol="axiom">
                      <sequent>
                       <formulalist/>
                       <formulalist>
                        <constantatomformula symbol="=">
                         <function symbol="∪">
                          <function symbol="∪">
                           <variable symbol="y_0"/>
                           <variable symbol="y_0"/>
                          </function>
                          <variable symbol="x_0"/>
                         </function>
                         <function symbol="∪">
                          <variable symbol="y_0"/>
                          <function symbol="∪">
                           <variable symbol="y_0"/>
                           <variable symbol="x_0"/>
                          </function>
                         </function>
                        </constantatomformula>
                       </formulalist>
                      </sequent>
                     </rule>
                     <rule type="eqr2" symbol="=:r2" param="(1 1)">
                      <sequent>
                       <formulalist/>
                       <formulalist>
                        <constantatomformula symbol="=">
                         <function symbol="∪">
                          <function symbol="∪">
                           <variable symbol="y_0"/>
                           <variable symbol="y_0"/>
                          </function>
                          <variable symbol="x_0"/>
                         </function>
                         <function symbol="∪">
                          <variable symbol="y_0"/>
                          <variable symbol="x_0"/>
                         </function>
                        </constantatomformula>
                       </formulalist>
                      </sequent>
                      <rule type="axiom" symbol="axiom">
                       <sequent>
                        <formulalist/>
                        <formulalist>
                         <constantatomformula symbol="=">
                          <function symbol="∪">
                           <variable symbol="y_0"/>
                           <variable symbol="y_0"/>
                          </function>
                          <variable symbol="y_0"/>
                         </constantatomformula>
                        </formulalist>
                       </sequent>
                      </rule>
                      <rule type="axiom" symbol="axiom">
                       <sequent>
                        <formulalist/>
                        <formulalist>
                         <constantatomformula symbol="=">
                          <function symbol="∪">
                           <variable symbol="y_0"/>
                           <variable symbol="x_0"/>
                          </function>
                          <function symbol="∪">
                           <variable symbol="y_0"/>
                           <variable symbol="x_0"/>
                          </function>
                         </constantatomformula>
                        </formulalist>
                       </sequent>
                      </rule>
                     </rule>
                    </rule>
                    <rule type="axiom" symbol="axiom">
                     <sequent>
                      <formulalist>
                       <constantatomformula symbol="=">
                        <function symbol="∩">
                         <variable symbol="y_0"/>
                         <function symbol="∪">
                          <variable symbol="y_0"/>
                          <variable symbol="x_0"/>
                         </function>
                        </function>
                        <variable symbol="y_0"/>
                       </constantatomformula>
                      </formulalist>
                      <formulalist>
                       <constantatomformula symbol="=">
                        <function symbol="∩">
                         <variable symbol="y_0"/>
                         <function symbol="∪">
                          <variable symbol="y_0"/>
                          <variable symbol="x_0"/>
                         </function>
                        </function>
                        <variable symbol="y_0"/>
                       </constantatomformula>
                      </formulalist>
                     </sequent>
                    </rule>
                   </rule>
                  </rule>
                 </rule>
                </rule>
               </rule>
              </rule>
             </rule>
            </rule>
           </rule>
           <rule type="forallr" symbol="\all:r">
            <sequent>
             <formulalist>
              <constantatomformula symbol="L1" definition="L1"/>
             </formulalist>
             <formulalist>
              <quantifiedformula type="all">
               <variable symbol="z"/>
               <conjunctiveformula type="impl">
                <conjunctiveformula type="and">
                 <constantatomformula symbol="≤" definition="≤">
                  <variable symbol="x_0"/>
                  <variable symbol="z"/>
                 </constantatomformula>
                 <constantatomformula symbol="≤" definition="≤">
                  <variable symbol="y_0"/>
                  <variable symbol="z"/>
                 </constantatomformula>
                </conjunctiveformula>
                <constantatomformula symbol="≤" definition="≤">
                 <function symbol="∪">
                  <variable symbol="x_0"/>
                  <variable symbol="y_0"/>
                 </function>
                 <variable symbol="z"/>
                </constantatomformula>
               </conjunctiveformula>
              </quantifiedformula>
             </formulalist>
            </sequent>
            <rule type="implr" symbol="\impl:r">
             <sequent>
              <formulalist>
               <constantatomformula symbol="L1" definition="L1"/>
              </formulalist>
              <formulalist>
               <conjunctiveformula type="impl">
                <conjunctiveformula type="and">
                 <constantatomformula symbol="≤" definition="≤">
                  <variable symbol="x_0"/>
                  <variable symbol="z_0"/>
                 </constantatomformula>
                 <constantatomformula symbol="≤" definition="≤">
                  <variable symbol="y_0"/>
                  <variable symbol="z_0"/>
                 </constantatomformula>
                </conjunctiveformula>
                <constantatomformula symbol="≤" definition="≤">
                 <function symbol="∪">
                  <variable symbol="x_0"/>
                  <variable symbol="y_0"/>
                 </function>
                 <variable symbol="z_0"/>
                </constantatomformula>
               </conjunctiveformula>
              </formulalist>
             </sequent>
             <rule type="contrl" symbol="c:l" param="2,1">
              <sequent>
               <formulalist>
                <conjunctiveformula type="and">
                 <constantatomformula symbol="≤" definition="≤">
                  <variable symbol="x_0"/>
                  <variable symbol="z_0"/>
                 </constantatomformula>
                 <constantatomformula symbol="≤" definition="≤">
                  <variable symbol="y_0"/>
                  <variable symbol="z_0"/>
                 </constantatomformula>
                </conjunctiveformula>
                <constantatomformula symbol="L1" definition="L1"/>
               </formulalist>
               <formulalist>
                <constantatomformula symbol="≤" definition="≤">
                 <function symbol="∪">
                  <variable symbol="x_0"/>
                  <variable symbol="y_0"/>
                 </function>
                 <variable symbol="z_0"/>
                </constantatomformula>
               </formulalist>
              </sequent>
              <rule type="perml" symbol="\pi:l" param="(2 3)">
               <sequent>
                <formulalist>
                 <conjunctiveformula type="and">
                  <constantatomformula symbol="≤" definition="≤">
                   <variable symbol="x_0"/>
                   <variable symbol="z_0"/>
                  </constantatomformula>
                  <constantatomformula symbol="≤" definition="≤">
                   <variable symbol="y_0"/>
                   <variable symbol="z_0"/>
                  </constantatomformula>
                 </conjunctiveformula>
                 <conjunctiveformula type="and">
                  <constantatomformula symbol="≤" definition="≤">
                   <variable symbol="x_0"/>
                   <variable symbol="z_0"/>
                  </constantatomformula>
                  <constantatomformula symbol="≤" definition="≤">
                   <variable symbol="y_0"/>
                   <variable symbol="z_0"/>
                  </constantatomformula>
                 </conjunctiveformula>
                 <constantatomformula symbol="L1" definition="L1"/>
                </formulalist>
                <formulalist>
                 <constantatomformula symbol="≤" definition="≤">
                  <function symbol="∪">
                   <variable symbol="x_0"/>
                   <variable symbol="y_0"/>
                  </function>
                  <variable symbol="z_0"/>
                 </constantatomformula>
                </formulalist>
               </sequent>
               <rule type="andl1" symbol="\and:l1">
                <sequent>
                 <formulalist>
                  <conjunctiveformula type="and">
                   <constantatomformula symbol="≤" definition="≤">
                    <variable symbol="x_0"/>
                    <variable symbol="z_0"/>
                   </constantatomformula>
                   <constantatomformula symbol="≤" definition="≤">
                    <variable symbol="y_0"/>
                    <variable symbol="z_0"/>
                   </constantatomformula>
                  </conjunctiveformula>
                  <constantatomformula symbol="L1" definition="L1"/>
                  <conjunctiveformula type="and">
                   <constantatomformula symbol="≤" definition="≤">
                    <variable symbol="x_0"/>
                    <variable symbol="z_0"/>
                   </constantatomformula>
                   <constantatomformula symbol="≤" definition="≤">
                    <variable symbol="y_0"/>
                    <variable symbol="z_0"/>
                   </constantatomformula>
                  </conjunctiveformula>
                 </formulalist>
                 <formulalist>
                  <constantatomformula symbol="≤" definition="≤">
                   <function symbol="∪">
                    <variable symbol="x_0"/>
                    <variable symbol="y_0"/>
                   </function>
                   <variable symbol="z_0"/>
                  </constantatomformula>
                 </formulalist>
                </sequent>
                <rule type="perml" symbol="\pi:l" param="(1 3)">
                 <sequent>
                  <formulalist>
                   <constantatomformula symbol="≤" definition="≤">
                    <variable symbol="x_0"/>
                    <variable symbol="z_0"/>
                   </constantatomformula>
                   <constantatomformula symbol="L1" definition="L1"/>
                   <conjunctiveformula type="and">
                    <constantatomformula symbol="≤" definition="≤">
                     <variable symbol="x_0"/>
                     <variable symbol="z_0"/>
                    </constantatomformula>
                    <constantatomformula symbol="≤" definition="≤">
                     <variable symbol="y_0"/>
                     <variable symbol="z_0"/>
                    </constantatomformula>
                   </conjunctiveformula>
                  </formulalist>
                  <formulalist>
                   <constantatomformula symbol="≤" definition="≤">
                    <function symbol="∪">
                     <variable symbol="x_0"/>
                     <variable symbol="y_0"/>
                    </function>
                    <variable symbol="z_0"/>
                   </constantatomformula>
                  </formulalist>
                 </sequent>
                 <rule type="andl2" symbol="\and:l2">
                  <sequent>
                   <formulalist>
                    <conjunctiveformula type="and">
                     <constantatomformula symbol="≤" definition="≤">
                      <variable symbol="x_0"/>
                      <variable symbol="z_0"/>
                     </constantatomformula>
                     <constantatomformula symbol="≤" definition="≤">
                      <variable symbol="y_0"/>
                      <variable symbol="z_0"/>
                     </constantatomformula>
                    </conjunctiveformula>
                    <constantatomformula symbol="L1" definition="L1"/>
                    <constantatomformula symbol="≤" definition="≤">
                     <variable symbol="x_0"/>
                     <variable symbol="z_0"/>
                    </constantatomformula>
                   </formulalist>
                   <formulalist>
                    <constantatomformula symbol="≤" definition="≤">
                     <function symbol="∪">
                      <variable symbol="x_0"/>
                      <variable symbol="y_0"/>
                     </function>
                     <variable symbol="z_0"/>
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                                            </conjunctiveformula>
                                            <conjunctiveformula type="impl">
                                             <constantatomformula symbol="=">
                                              <function symbol="∪">
                                               <variable symbol="x"/>
                                               <variable symbol="y"/>
                                              </function>
                                              <variable symbol="y"/>
                                             </constantatomformula>
                                             <constantatomformula symbol="=">
                                              <function symbol="∩">
                                               <variable symbol="x"/>
                                               <variable symbol="y"/>
                                              </function>
                                              <variable symbol="x"/>
                                             </constantatomformula>
                                            </conjunctiveformula>
                                           </conjunctiveformula>
                                          </quantifiedformula>
                                         </quantifiedformula>
                                         <constantatomformula symbol="=">
                                          <function symbol="∪">
                                           <variable symbol="x_0"/>
                                           <variable symbol="z_0"/>
                                          </function>
                                          <variable symbol="z_0"/>
                                         </constantatomformula>
                                        </formulalist>
                                        <formulalist>
                                         <constantatomformula symbol="=">
                                          <function symbol="∩">
                                           <function symbol="∪">
                                            <variable symbol="x_0"/>
                                            <variable symbol="y_0"/>
                                           </function>
                                           <variable symbol="z_0"/>
                                          </function>
                                          <function symbol="∪">
                                           <variable symbol="x_0"/>
                                           <variable symbol="y_0"/>
                                          </function>
                                         </constantatomformula>
                                        </formulalist>
                                       </sequent>
                                       <rule type="foralll" symbol="\all:l">
                                        <sequent>
                                         <formulalist>
                                          <quantifiedformula type="all">
                                           <variable symbol="x"/>
                                           <quantifiedformula type="all">
                                            <variable symbol="y"/>
                                            <conjunctiveformula type="and">
                                             <conjunctiveformula type="impl">
                                              <constantatomformula symbol="=">
                                               <function symbol="∩">
                                                <variable symbol="x"/>
                                                <variable symbol="y"/>
                                               </function>
                                               <variable symbol="x"/>
                                              </constantatomformula>
                                              <constantatomformula symbol="=">
                                               <function symbol="∪">
                                                <variable symbol="x"/>
                                                <variable symbol="y"/>
                                               </function>
                                               <variable symbol="y"/>
                                              </constantatomformula>
                                             </conjunctiveformula>
                                             <conjunctiveformula type="impl">
                                              <constantatomformula symbol="=">
                                               <function symbol="∪">
                                                <variable symbol="x"/>
                                                <variable symbol="y"/>
                                               </function>
                                               <variable symbol="y"/>
                                              </constantatomformula>
                                              <constantatomformula symbol="=">
                                               <function symbol="∩">
                                                <variable symbol="x"/>
                                                <variable symbol="y"/>
                                               </function>
                                               <variable symbol="x"/>
                                              </constantatomformula>
                                             </conjunctiveformula>
                                            </conjunctiveformula>
                                           </quantifiedformula>
                                          </quantifiedformula>
                                          <constantatomformula symbol="=">
                                           <function symbol="∪">
                                            <variable symbol="y_0"/>
                                            <variable symbol="z_0"/>
                                           </function>
                                           <variable symbol="z_0"/>
                                          </constantatomformula>
                                          <constantatomformula symbol="=">
                                           <function symbol="∪">
                                            <variable symbol="x_0"/>
                                            <variable symbol="z_0"/>
                                           </function>
                                           <variable symbol="z_0"/>
                                          </constantatomformula>
                                         </formulalist>
                                         <formulalist>
                                          <constantatomformula symbol="=">
                                           <function symbol="∩">
                                            <function symbol="∪">
                                             <variable symbol="x_0"/>
                                             <variable symbol="y_0"/>
                                            </function>
                                            <variable symbol="z_0"/>
                                           </function>
                                           <function symbol="∪">
                                            <variable symbol="x_0"/>
                                            <variable symbol="y_0"/>
                                           </function>
                                          </constantatomformula>
                                         </formulalist>
                                        </sequent>
                                        <rule type="foralll" symbol="\all:l">
                                         <sequent>
                                          <formulalist>
                                           <quantifiedformula type="all">
                                            <variable symbol="y"/>
                                            <conjunctiveformula type="and">
                                             <conjunctiveformula type="impl">
                                              <constantatomformula symbol="=">
                                               <function symbol="∩">
                                                <function symbol="∪">
                                                 <variable symbol="x_0"/>
                                                 <variable symbol="y_0"/>
                                                </function>
                                                <variable symbol="y"/>
                                               </function>
                                               <function symbol="∪">
                                                <variable symbol="x_0"/>
                                                <variable symbol="y_0"/>
                                               </function>
                                              </constantatomformula>
                                              <constantatomformula symbol="=">
                                               <function symbol="∪">
                                                <function symbol="∪">
                                                 <variable symbol="x_0"/>
                                                 <variable symbol="y_0"/>
                                                </function>
                                                <variable symbol="y"/>
                                               </function>
                                               <variable symbol="y"/>
                                              </constantatomformula>
                                             </conjunctiveformula>
                                             <conjunctiveformula type="impl">
                                              <constantatomformula symbol="=">
                                               <function symbol="∪">
                                                <function symbol="∪">
                                                 <variable symbol="x_0"/>
                                                 <variable symbol="y_0"/>
                                                </function>
                                                <variable symbol="y"/>
                                               </function>
                                               <variable symbol="y"/>
                                              </constantatomformula>
                                              <constantatomformula symbol="=">
                                               <function symbol="∩">
                                                <function symbol="∪">
                                                 <variable symbol="x_0"/>
                                                 <variable symbol="y_0"/>
                                                </function>
                                                <variable symbol="y"/>
                                               </function>
                                               <function symbol="∪">
                                                <variable symbol="x_0"/>
                                                <variable symbol="y_0"/>
                                               </function>
                                              </constantatomformula>
                                             </conjunctiveformula>
                                            </conjunctiveformula>
                                           </quantifiedformula>
                                           <constantatomformula symbol="=">
                                            <function symbol="∪">
                                             <variable symbol="y_0"/>
                                             <variable symbol="z_0"/>
                                            </function>
                                            <variable symbol="z_0"/>
                                           </constantatomformula>
                                           <constantatomformula symbol="=">
                                            <function symbol="∪">
                                             <variable symbol="x_0"/>
                                             <variable symbol="z_0"/>
                                            </function>
                                            <variable symbol="z_0"/>
                                           </constantatomformula>
                                          </formulalist>
                                          <formulalist>
                                           <constantatomformula symbol="=">
                                            <function symbol="∩">
                                             <function symbol="∪">
                                              <variable symbol="x_0"/>
                                              <variable symbol="y_0"/>
                                             </function>
                                             <variable symbol="z_0"/>
                                            </function>
                                            <function symbol="∪">
                                             <variable symbol="x_0"/>
                                             <variable symbol="y_0"/>
                                            </function>
                                           </constantatomformula>
                                          </formulalist>
                                         </sequent>
                                         <rule type="andl2" symbol="\and:l2">
                                          <sequent>
                                           <formulalist>
                                            <conjunctiveformula type="and">
                                             <conjunctiveformula type="impl">
                                              <constantatomformula symbol="=">
                                               <function symbol="∩">
                                                <function symbol="∪">
                                                 <variable symbol="x_0"/>
                                                 <variable symbol="y_0"/>
                                                </function>
                                                <variable symbol="z_0"/>
                                               </function>
                                               <function symbol="∪">
                                                <variable symbol="x_0"/>
                                                <variable symbol="y_0"/>
                                               </function>
                                              </constantatomformula>
                                              <constantatomformula symbol="=">
                                               <function symbol="∪">
                                                <function symbol="∪">
                                                 <variable symbol="x_0"/>
                                                 <variable symbol="y_0"/>
                                                </function>
                                                <variable symbol="z_0"/>
                                               </function>
                                               <variable symbol="z_0"/>
                                              </constantatomformula>
                                             </conjunctiveformula>
                                             <conjunctiveformula type="impl">
                                              <constantatomformula symbol="=">
                                               <function symbol="∪">
                                                <function symbol="∪">
                                                 <variable symbol="x_0"/>
                                                 <variable symbol="y_0"/>
                                                </function>
                                                <variable symbol="z_0"/>
                                               </function>
                                               <variable symbol="z_0"/>
                                              </constantatomformula>
                                              <constantatomformula symbol="=">
                                               <function symbol="∩">
                                                <function symbol="∪">
                                                 <variable symbol="x_0"/>
                                                 <variable symbol="y_0"/>
                                                </function>
                                                <variable symbol="z_0"/>
                                               </function>
                                               <function symbol="∪">
                                                <variable symbol="x_0"/>
                                                <variable symbol="y_0"/>
                                               </function>
                                              </constantatomformula>
                                             </conjunctiveformula>
                                            </conjunctiveformula>
                                            <constantatomformula symbol="=">
                                             <function symbol="∪">
                                              <variable symbol="y_0"/>
                                              <variable symbol="z_0"/>
                                             </function>
                                             <variable symbol="z_0"/>
                                            </constantatomformula>
                                            <constantatomformula symbol="=">
                                             <function symbol="∪">
                                              <variable symbol="x_0"/>
                                              <variable symbol="z_0"/>
                                             </function>
                                             <variable symbol="z_0"/>
                                            </constantatomformula>
                                           </formulalist>
                                           <formulalist>
                                            <constantatomformula symbol="=">
                                             <function symbol="∩">
                                              <function symbol="∪">
                                               <variable symbol="x_0"/>
                                               <variable symbol="y_0"/>
                                              </function>
                                              <variable symbol="z_0"/>
                                             </function>
                                             <function symbol="∪">
                                              <variable symbol="x_0"/>
                                              <variable symbol="y_0"/>
                                             </function>
                                            </constantatomformula>
                                           </formulalist>
                                          </sequent>
                                          <rule type="impll" symbol="\impl:l">
                                           <sequent>
                                            <formulalist>
                                             <conjunctiveformula type="impl">
                                              <constantatomformula symbol="=">
                                               <function symbol="∪">
                                                <function symbol="∪">
                                                 <variable symbol="x_0"/>
                                                 <variable symbol="y_0"/>
                                                </function>
                                                <variable symbol="z_0"/>
                                               </function>
                                               <variable symbol="z_0"/>
                                              </constantatomformula>
                                              <constantatomformula symbol="=">
                                               <function symbol="∩">
                                                <function symbol="∪">
                                                 <variable symbol="x_0"/>
                                                 <variable symbol="y_0"/>
                                                </function>
                                                <variable symbol="z_0"/>
                                               </function>
                                               <function symbol="∪">
                                                <variable symbol="x_0"/>
                                                <variable symbol="y_0"/>
                                               </function>
                                              </constantatomformula>
                                             </conjunctiveformula>
                                             <constantatomformula symbol="=">
                                              <function symbol="∪">
                                               <variable symbol="y_0"/>
                                               <variable symbol="z_0"/>
                                              </function>
                                              <variable symbol="z_0"/>
                                             </constantatomformula>
                                             <constantatomformula symbol="=">
                                              <function symbol="∪">
                                               <variable symbol="x_0"/>
                                               <variable symbol="z_0"/>
                                              </function>
                                              <variable symbol="z_0"/>
                                             </constantatomformula>
                                            </formulalist>
                                            <formulalist>
                                             <constantatomformula symbol="=">
                                              <function symbol="∩">
                                               <function symbol="∪">
                                                <variable symbol="x_0"/>
                                                <variable symbol="y_0"/>
                                               </function>
                                               <variable symbol="z_0"/>
                                              </function>
                                              <function symbol="∪">
                                               <variable symbol="x_0"/>
                                               <variable symbol="y_0"/>
                                              </function>
                                             </constantatomformula>
                                            </formulalist>
                                           </sequent>
                                           <rule type="eqr2" symbol="=:r2" param="(1)">
                                            <sequent>
                                             <formulalist>
                                              <constantatomformula symbol="=">
                                               <function symbol="∪">
                                                <variable symbol="y_0"/>
                                                <variable symbol="z_0"/>
                                               </function>
                                               <variable symbol="z_0"/>
                                              </constantatomformula>
                                              <constantatomformula symbol="=">
                                               <function symbol="∪">
                                                <variable symbol="x_0"/>
                                                <variable symbol="z_0"/>
                                               </function>
                                               <variable symbol="z_0"/>
                                              </constantatomformula>
                                             </formulalist>
                                             <formulalist>
                                              <constantatomformula symbol="=">
                                               <function symbol="∪">
                                                <function symbol="∪">
                                                 <variable symbol="x_0"/>
                                                 <variable symbol="y_0"/>
                                                </function>
                                                <variable symbol="z_0"/>
                                               </function>
                                               <variable symbol="z_0"/>
                                              </constantatomformula>
                                             </formulalist>
                                            </sequent>
                                            <rule type="axiom" symbol="axiom">
                                             <sequent>
                                              <formulalist/>
                                              <formulalist>
                                               <constantatomformula symbol="=">
                                                <function symbol="∪">
                                                 <function symbol="∪">
                                                  <variable symbol="x_0"/>
                                                  <variable symbol="y_0"/>
                                                 </function>
                                                 <variable symbol="z_0"/>
                                                </function>
                                                <function symbol="∪">
                                                 <variable symbol="x_0"/>
                                                 <function symbol="∪">
                                                  <variable symbol="y_0"/>
                                                  <variable symbol="z_0"/>
                                                 </function>
                                                </function>
                                               </constantatomformula>
                                              </formulalist>
                                             </sequent>
                                            </rule>
                                            <rule type="eqr2" symbol="=:r2" param="(1 2)">
                                             <sequent>
                                              <formulalist>
                                               <constantatomformula symbol="=">
                                                <function symbol="∪">
                                                 <variable symbol="y_0"/>
                                                 <variable symbol="z_0"/>
                                                </function>
                                                <variable symbol="z_0"/>
                                               </constantatomformula>
                                               <constantatomformula symbol="=">
                                                <function symbol="∪">
                                                 <variable symbol="x_0"/>
                                                 <variable symbol="z_0"/>
                                                </function>
                                                <variable symbol="z_0"/>
                                               </constantatomformula>
                                              </formulalist>
                                              <formulalist>
                                               <constantatomformula symbol="=">
                                                <function symbol="∪">
                                                 <variable symbol="x_0"/>
                                                 <function symbol="∪">
                                                  <variable symbol="y_0"/>
                                                  <variable symbol="z_0"/>
                                                 </function>
                                                </function>
                                                <variable symbol="z_0"/>
                                               </constantatomformula>
                                              </formulalist>
                                             </sequent>
                                             <rule type="axiom" symbol="axiom">
                                              <sequent>
                                               <formulalist>
                                                <constantatomformula symbol="=">
                                                 <function symbol="∪">
                                                  <variable symbol="y_0"/>
                                                  <variable symbol="z_0"/>
                                                 </function>
                                                 <variable symbol="z_0"/>
                                                </constantatomformula>
                                               </formulalist>
                                               <formulalist>
                                                <constantatomformula symbol="=">
                                                 <function symbol="∪">
                                                  <variable symbol="y_0"/>
                                                  <variable symbol="z_0"/>
                                                 </function>
                                                 <variable symbol="z_0"/>
                                                </constantatomformula>
                                               </formulalist>
                                              </sequent>
                                             </rule>
                                             <rule type="contrr" symbol="c:r" param="2">
                                              <sequent>
                                               <formulalist>
                                                <constantatomformula symbol="=">
                                                 <function symbol="∪">
                                                  <variable symbol="x_0"/>
                                                  <variable symbol="z_0"/>
                                                 </function>
                                                 <variable symbol="z_0"/>
                                                </constantatomformula>
                                               </formulalist>
                                               <formulalist>
                                                <constantatomformula symbol="=">
                                                 <function symbol="∪">
                                                  <variable symbol="x_0"/>
                                                  <variable symbol="z_0"/>
                                                 </function>
                                                 <variable symbol="z_0"/>
                                                </constantatomformula>
                                               </formulalist>
                                              </sequent>
                                              <rule type="contrl" symbol="c:l" param="2">
                                               <sequent>
                                                <formulalist>
                                                 <constantatomformula symbol="=">
                                                  <function symbol="∪">
                                                   <variable symbol="x_0"/>
                                                   <variable symbol="z_0"/>
                                                  </function>
                                                  <variable symbol="z_0"/>
                                                 </constantatomformula>
                                                </formulalist>
                                                <formulalist>
                                                 <constantatomformula symbol="=">
                                                  <function symbol="∪">
                                                   <variable symbol="x_0"/>
                                                   <variable symbol="z_0"/>
                                                  </function>
                                                  <variable symbol="z_0"/>
                                                 </constantatomformula>
                                                 <constantatomformula symbol="=">
                                                  <function symbol="∪">
                                                   <variable symbol="x_0"/>
                                                   <variable symbol="z_0"/>
                                                  </function>
                                                  <variable symbol="z_0"/>
                                                 </constantatomformula>
                                                </formulalist>
                                               </sequent>
                                               <rule type="eqr2" symbol="=:r2" param="(1)">
                                                <sequent>
                                                 <formulalist>
                                                  <constantatomformula symbol="=">
                                                   <function symbol="∪">
                                                    <variable symbol="x_0"/>
                                                    <variable symbol="z_0"/>
                                                   </function>
                                                   <variable symbol="z_0"/>
                                                  </constantatomformula>
                                                  <constantatomformula symbol="=">
                                                   <function symbol="∪">
                                                    <variable symbol="x_0"/>
                                                    <variable symbol="z_0"/>
                                                   </function>
                                                   <variable symbol="z_0"/>
                                                  </constantatomformula>
                                                 </formulalist>
                                                 <formulalist>
                                                  <constantatomformula symbol="=">
                                                   <function symbol="∪">
                                                    <variable symbol="x_0"/>
                                                    <variable symbol="z_0"/>
                                                   </function>
                                                   <variable symbol="z_0"/>
                                                  </constantatomformula>
                                                  <constantatomformula symbol="=">
                                                   <function symbol="∪">
                                                    <variable symbol="x_0"/>
                                                    <variable symbol="z_0"/>
                                                   </function>
                                                   <variable symbol="z_0"/>
                                                  </constantatomformula>
                                                 </formulalist>
                                                </sequent>
                                                <rule type="axiom" symbol="axiom">
                                                 <sequent>
                                                  <formulalist>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="∪">
                                                     <variable symbol="x_0"/>
                                                     <variable symbol="z_0"/>
                                                    </function>
                                                    <variable symbol="z_0"/>
                                                   </constantatomformula>
                                                  </formulalist>
                                                  <formulalist>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="∪">
                                                     <variable symbol="x_0"/>
                                                     <variable symbol="z_0"/>
                                                    </function>
                                                    <variable symbol="z_0"/>
                                                   </constantatomformula>
                                                  </formulalist>
                                                 </sequent>
                                                </rule>
                                                <rule type="weakr" symbol="w:r">
                                                 <sequent>
                                                  <formulalist>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="∪">
                                                     <variable symbol="x_0"/>
                                                     <variable symbol="z_0"/>
                                                    </function>
                                                    <variable symbol="z_0"/>
                                                   </constantatomformula>
                                                  </formulalist>
                                                  <formulalist>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="∪">
                                                     <variable symbol="x_0"/>
                                                     <variable symbol="z_0"/>
                                                    </function>
                                                    <variable symbol="z_0"/>
                                                   </constantatomformula>
                                                   <constantatomformula symbol="=">
                                                    <variable symbol="z_0"/>
                                                    <variable symbol="z_0"/>
                                                   </constantatomformula>
                                                  </formulalist>
                                                 </sequent>
                                                 <rule type="axiom" symbol="axiom">
                                                  <sequent>
                                                   <formulalist>
                                                    <constantatomformula symbol="=">
                                                     <function symbol="∪">
                                                      <variable symbol="x_0"/>
                                                      <variable symbol="z_0"/>
                                                     </function>
                                                     <variable symbol="z_0"/>
                                                    </constantatomformula>
                                                   </formulalist>
                                                   <formulalist>
                                                    <constantatomformula symbol="=">
                                                     <function symbol="∪">
                                                      <variable symbol="x_0"/>
                                                      <variable symbol="z_0"/>
                                                     </function>
                                                     <variable symbol="z_0"/>
                                                    </constantatomformula>
                                                   </formulalist>
                                                  </sequent>
                                                 </rule>
                                                </rule>
                                               </rule>
                                              </rule>
                                             </rule>
                                            </rule>
                                           </rule>
                                           <rule type="axiom" symbol="axiom">
                                            <sequent>
                                             <formulalist>
                                              <constantatomformula symbol="=">
                                               <function symbol="∩">
                                                <function symbol="∪">
                                                 <variable symbol="x_0"/>
                                                 <variable symbol="y_0"/>
                                                </function>
                                                <variable symbol="z_0"/>
                                               </function>
                                               <function symbol="∪">
                                                <variable symbol="x_0"/>
                                                <variable symbol="y_0"/>
                                               </function>
                                              </constantatomformula>
                                             </formulalist>
                                             <formulalist>
                                              <constantatomformula symbol="=">
                                               <function symbol="∩">
                                                <function symbol="∪">
                                                 <variable symbol="x_0"/>
                                                 <variable symbol="y_0"/>
                                                </function>
                                                <variable symbol="z_0"/>
                                               </function>
                                               <function symbol="∪">
                                                <variable symbol="x_0"/>
                                                <variable symbol="y_0"/>
                                               </function>
                                              </constantatomformula>
                                             </formulalist>
                                            </sequent>
                                           </rule>
                                          </rule>
                                         </rule>
                                        </rule>
                                       </rule>
                                      </rule>
                                     </rule>
                                    </rule>
                                   </rule>
                                  </rule>
                                 </rule>
                                </rule>
                               </rule>
                              </rule>
                             </rule>
                            </rule>
                           </rule>
                          </rule>
                         </rule>
                        </rule>
                       </rule>
                      </rule>
                     </rule>
                    </rule>
                   </rule>
                  </rule>
                 </rule>
                </rule>
               </rule>
              </rule>
             </rule>
            </rule>
           </rule>
          </rule>
         </rule>
        </rule>
       </rule>
      </rule>
     </rule>
    </rule>
   </rule>
   <rule type="defl" symbol="d:l" param="();[()];()">
    <sequent>
     <formulalist>
      <constantatomformula symbol="L3" definition="L3"/>
     </formulalist>
     <formulalist>
      <constantatomformula symbol="L2" definition="L2"/>
     </formulalist>
    </sequent>
    <rule type="contrl" symbol="c:l" param="2">
     <sequent>
      <formulalist>
       <conjunctiveformula type="and">
        <constantatomformula symbol="POSET" definition="POSET"/>
        <conjunctiveformula type="and">
         <constantatomformula symbol="GLB" definition="GLB"/>
         <constantatomformula symbol="LUB" definition="LUB"/>
        </conjunctiveformula>
       </conjunctiveformula>
      </formulalist>
      <formulalist>
       <constantatomformula symbol="L2" definition="L2"/>
      </formulalist>
     </sequent>
     <rule type="andl2" symbol="\and:l2">
      <sequent>
       <formulalist>
        <conjunctiveformula type="and">
         <constantatomformula symbol="POSET" definition="POSET"/>
         <conjunctiveformula type="and">
          <constantatomformula symbol="GLB" definition="GLB"/>
          <constantatomformula symbol="LUB" definition="LUB"/>
         </conjunctiveformula>
        </conjunctiveformula>
        <conjunctiveformula type="and">
         <constantatomformula symbol="POSET" definition="POSET"/>
         <conjunctiveformula type="and">
          <constantatomformula symbol="GLB" definition="GLB"/>
          <constantatomformula symbol="LUB" definition="LUB"/>
         </conjunctiveformula>
        </conjunctiveformula>
       </formulalist>
       <formulalist>
        <constantatomformula symbol="L2" definition="L2"/>
       </formulalist>
      </sequent>
      <rule type="contrl" symbol="c:l" param="2,1">
       <sequent>
        <formulalist>
         <conjunctiveformula type="and">
          <constantatomformula symbol="GLB" definition="GLB"/>
          <constantatomformula symbol="LUB" definition="LUB"/>
         </conjunctiveformula>
         <conjunctiveformula type="and">
          <constantatomformula symbol="POSET" definition="POSET"/>
          <conjunctiveformula type="and">
           <constantatomformula symbol="GLB" definition="GLB"/>
           <constantatomformula symbol="LUB" definition="LUB"/>
          </conjunctiveformula>
         </conjunctiveformula>
        </formulalist>
        <formulalist>
         <constantatomformula symbol="L2" definition="L2"/>
        </formulalist>
       </sequent>
       <rule type="perml" symbol="\pi:l" param="(2 3)">
        <sequent>
         <formulalist>
          <conjunctiveformula type="and">
           <constantatomformula symbol="GLB" definition="GLB"/>
           <constantatomformula symbol="LUB" definition="LUB"/>
          </conjunctiveformula>
          <conjunctiveformula type="and">
           <constantatomformula symbol="GLB" definition="GLB"/>
           <constantatomformula symbol="LUB" definition="LUB"/>
          </conjunctiveformula>
          <conjunctiveformula type="and">
           <constantatomformula symbol="POSET" definition="POSET"/>
           <conjunctiveformula type="and">
            <constantatomformula symbol="GLB" definition="GLB"/>
            <constantatomformula symbol="LUB" definition="LUB"/>
           </conjunctiveformula>
          </conjunctiveformula>
         </formulalist>
         <formulalist>
          <constantatomformula symbol="L2" definition="L2"/>
         </formulalist>
        </sequent>
        <rule type="andl1" symbol="\and:l1">
         <sequent>
          <formulalist>
           <conjunctiveformula type="and">
            <constantatomformula symbol="GLB" definition="GLB"/>
            <constantatomformula symbol="LUB" definition="LUB"/>
           </conjunctiveformula>
           <conjunctiveformula type="and">
            <constantatomformula symbol="POSET" definition="POSET"/>
            <conjunctiveformula type="and">
             <constantatomformula symbol="GLB" definition="GLB"/>
             <constantatomformula symbol="LUB" definition="LUB"/>
            </conjunctiveformula>
           </conjunctiveformula>
           <conjunctiveformula type="and">
            <constantatomformula symbol="GLB" definition="GLB"/>
            <constantatomformula symbol="LUB" definition="LUB"/>
           </conjunctiveformula>
          </formulalist>
          <formulalist>
           <constantatomformula symbol="L2" definition="L2"/>
          </formulalist>
         </sequent>
         <rule type="perml" symbol="\pi:l" param="(1 3)">
          <sequent>
           <formulalist>
            <constantatomformula symbol="GLB" definition="GLB"/>
            <conjunctiveformula type="and">
             <constantatomformula symbol="POSET" definition="POSET"/>
             <conjunctiveformula type="and">
              <constantatomformula symbol="GLB" definition="GLB"/>
              <constantatomformula symbol="LUB" definition="LUB"/>
             </conjunctiveformula>
            </conjunctiveformula>
            <conjunctiveformula type="and">
             <constantatomformula symbol="GLB" definition="GLB"/>
             <constantatomformula symbol="LUB" definition="LUB"/>
            </conjunctiveformula>
           </formulalist>
           <formulalist>
            <constantatomformula symbol="L2" definition="L2"/>
           </formulalist>
          </sequent>
          <rule type="andl2" symbol="\and:l2">
           <sequent>
            <formulalist>
             <conjunctiveformula type="and">
              <constantatomformula symbol="GLB" definition="GLB"/>
              <constantatomformula symbol="LUB" definition="LUB"/>
             </conjunctiveformula>
             <conjunctiveformula type="and">
              <constantatomformula symbol="POSET" definition="POSET"/>
              <conjunctiveformula type="and">
               <constantatomformula symbol="GLB" definition="GLB"/>
               <constantatomformula symbol="LUB" definition="LUB"/>
              </conjunctiveformula>
             </conjunctiveformula>
             <constantatomformula symbol="GLB" definition="GLB"/>
            </formulalist>
            <formulalist>
             <constantatomformula symbol="L2" definition="L2"/>
            </formulalist>
           </sequent>
           <rule type="perml" symbol="\pi:l" param="(1 2)">
            <sequent>
             <formulalist>
              <constantatomformula symbol="LUB" definition="LUB"/>
              <conjunctiveformula type="and">
               <constantatomformula symbol="POSET" definition="POSET"/>
               <conjunctiveformula type="and">
                <constantatomformula symbol="GLB" definition="GLB"/>
                <constantatomformula symbol="LUB" definition="LUB"/>
               </conjunctiveformula>
              </conjunctiveformula>
              <constantatomformula symbol="GLB" definition="GLB"/>
             </formulalist>
             <formulalist>
              <constantatomformula symbol="L2" definition="L2"/>
             </formulalist>
            </sequent>
            <rule type="andl1" symbol="\and:l1">
             <sequent>
              <formulalist>
               <conjunctiveformula type="and">
                <constantatomformula symbol="POSET" definition="POSET"/>
                <conjunctiveformula type="and">
                 <constantatomformula symbol="GLB" definition="GLB"/>
                 <constantatomformula symbol="LUB" definition="LUB"/>
                </conjunctiveformula>
               </conjunctiveformula>
               <constantatomformula symbol="LUB" definition="LUB"/>
               <constantatomformula symbol="GLB" definition="GLB"/>
              </formulalist>
              <formulalist>
               <constantatomformula symbol="L2" definition="L2"/>
              </formulalist>
             </sequent>
             <rule type="defl" symbol="d:l" param="();[()];()">
              <sequent>
               <formulalist>
                <constantatomformula symbol="POSET" definition="POSET"/>
                <constantatomformula symbol="LUB" definition="LUB"/>
                <constantatomformula symbol="GLB" definition="GLB"/>
               </formulalist>
               <formulalist>
                <constantatomformula symbol="L2" definition="L2"/>
               </formulalist>
              </sequent>
              <rule type="contrl" symbol="c:l" param="2,1,1">
               <sequent>
                <formulalist>
                 <conjunctiveformula type="and">
                  <constantatomformula symbol="R" definition="R"/>
                  <conjunctiveformula type="and">
                   <constantatomformula symbol="AS" definition="AS"/>
                   <constantatomformula symbol="T" definition="T"/>
                  </conjunctiveformula>
                 </conjunctiveformula>
                 <constantatomformula symbol="LUB" definition="LUB"/>
                 <constantatomformula symbol="GLB" definition="GLB"/>
                </formulalist>
                <formulalist>
                 <constantatomformula symbol="L2" definition="L2"/>
                </formulalist>
               </sequent>
               <rule type="andl1" symbol="\and:l1">
                <sequent>
                 <formulalist>
                  <conjunctiveformula type="and">
                   <constantatomformula symbol="R" definition="R"/>
                   <conjunctiveformula type="and">
                    <constantatomformula symbol="AS" definition="AS"/>
                    <constantatomformula symbol="T" definition="T"/>
                   </conjunctiveformula>
                  </conjunctiveformula>
                  <conjunctiveformula type="and">
                   <constantatomformula symbol="R" definition="R"/>
                   <conjunctiveformula type="and">
                    <constantatomformula symbol="AS" definition="AS"/>
                    <constantatomformula symbol="T" definition="T"/>
                   </conjunctiveformula>
                  </conjunctiveformula>
                  <constantatomformula symbol="LUB" definition="LUB"/>
                  <constantatomformula symbol="GLB" definition="GLB"/>
                 </formulalist>
                 <formulalist>
                  <constantatomformula symbol="L2" definition="L2"/>
                 </formulalist>
                </sequent>
                <rule type="perml" symbol="\pi:l" param="(1 2)">
                 <sequent>
                  <formulalist>
                   <constantatomformula symbol="R" definition="R"/>
                   <conjunctiveformula type="and">
                    <constantatomformula symbol="R" definition="R"/>
                    <conjunctiveformula type="and">
                     <constantatomformula symbol="AS" definition="AS"/>
                     <constantatomformula symbol="T" definition="T"/>
                    </conjunctiveformula>
                   </conjunctiveformula>
                   <constantatomformula symbol="LUB" definition="LUB"/>
                   <constantatomformula symbol="GLB" definition="GLB"/>
                  </formulalist>
                  <formulalist>
                   <constantatomformula symbol="L2" definition="L2"/>
                  </formulalist>
                 </sequent>
                 <rule type="andl2" symbol="\and:l2">
                  <sequent>
                   <formulalist>
                    <conjunctiveformula type="and">
                     <constantatomformula symbol="R" definition="R"/>
                     <conjunctiveformula type="and">
                      <constantatomformula symbol="AS" definition="AS"/>
                      <constantatomformula symbol="T" definition="T"/>
                     </conjunctiveformula>
                    </conjunctiveformula>
                    <constantatomformula symbol="R" definition="R"/>
                    <constantatomformula symbol="LUB" definition="LUB"/>
                    <constantatomformula symbol="GLB" definition="GLB"/>
                   </formulalist>
                   <formulalist>
                    <constantatomformula symbol="L2" definition="L2"/>
                   </formulalist>
                  </sequent>
                  <rule type="andl1" symbol="\and:l1">
                   <sequent>
                    <formulalist>
                     <conjunctiveformula type="and">
                      <constantatomformula symbol="AS" definition="AS"/>
                      <constantatomformula symbol="T" definition="T"/>
                     </conjunctiveformula>
                     <constantatomformula symbol="R" definition="R"/>
                     <constantatomformula symbol="LUB" definition="LUB"/>
                     <constantatomformula symbol="GLB" definition="GLB"/>
                    </formulalist>
                    <formulalist>
                     <constantatomformula symbol="L2" definition="L2"/>
                    </formulalist>
                   </sequent>
                   <rule type="defr" symbol="d:r" param="();[()];()">
                    <sequent>
                     <formulalist>
                      <constantatomformula symbol="AS" definition="AS"/>
                      <constantatomformula symbol="R" definition="R"/>
                      <constantatomformula symbol="LUB" definition="LUB"/>
                      <constantatomformula symbol="GLB" definition="GLB"/>
                     </formulalist>
                     <formulalist>
                      <constantatomformula symbol="L2" definition="L2"/>
                     </formulalist>
                    </sequent>
                    <rule type="contrl" symbol="c:l" param="2,2,2,2">
                     <sequent>
                      <formulalist>
                       <constantatomformula symbol="AS" definition="AS"/>
                       <constantatomformula symbol="R" definition="R"/>
                       <constantatomformula symbol="LUB" definition="LUB"/>
                       <constantatomformula symbol="GLB" definition="GLB"/>
                      </formulalist>
                      <formulalist>
                       <conjunctiveformula type="and">
                        <quantifiedformula type="all">
                         <variable symbol="x"/>
                         <quantifiedformula type="all">
                          <variable symbol="y"/>
                          <constantatomformula symbol="=">
                           <function symbol="∪">
                            <function symbol="∩">
                             <variable symbol="x"/>
                             <variable symbol="y"/>
                            </function>
                            <variable symbol="x"/>
                           </function>
                           <variable symbol="x"/>
                          </constantatomformula>
                         </quantifiedformula>
                        </quantifiedformula>
                        <quantifiedformula type="all">
                         <variable symbol="x"/>
                         <quantifiedformula type="all">
                          <variable symbol="y"/>
                          <constantatomformula symbol="=">
                           <function symbol="∩">
                            <function symbol="∪">
                             <variable symbol="x"/>
                             <variable symbol="y"/>
                            </function>
                            <variable symbol="x"/>
                           </function>
                           <variable symbol="x"/>
                          </constantatomformula>
                         </quantifiedformula>
                        </quantifiedformula>
                       </conjunctiveformula>
                      </formulalist>
                     </sequent>
                     <rule type="perml" symbol="\pi:l" param="(2 5)(4 8)(6 7)">
                      <sequent>
                       <formulalist>
                        <constantatomformula symbol="AS" definition="AS"/>
                        <constantatomformula symbol="AS" definition="AS"/>
                        <constantatomformula symbol="R" definition="R"/>
                        <constantatomformula symbol="R" definition="R"/>
                        <constantatomformula symbol="LUB" definition="LUB"/>
                        <constantatomformula symbol="LUB" definition="LUB"/>
                        <constantatomformula symbol="GLB" definition="GLB"/>
                        <constantatomformula symbol="GLB" definition="GLB"/>
                       </formulalist>
                       <formulalist>
                        <conjunctiveformula type="and">
                         <quantifiedformula type="all">
                          <variable symbol="x"/>
                          <quantifiedformula type="all">
                           <variable symbol="y"/>
                           <constantatomformula symbol="=">
                            <function symbol="∪">
                             <function symbol="∩">
                              <variable symbol="x"/>
                              <variable symbol="y"/>
                             </function>
                             <variable symbol="x"/>
                            </function>
                            <variable symbol="x"/>
                           </constantatomformula>
                          </quantifiedformula>
                         </quantifiedformula>
                         <quantifiedformula type="all">
                          <variable symbol="x"/>
                          <quantifiedformula type="all">
                           <variable symbol="y"/>
                           <constantatomformula symbol="=">
                            <function symbol="∩">
                             <function symbol="∪">
                              <variable symbol="x"/>
                              <variable symbol="y"/>
                             </function>
                             <variable symbol="x"/>
                            </function>
                            <variable symbol="x"/>
                           </constantatomformula>
                          </quantifiedformula>
                         </quantifiedformula>
                        </conjunctiveformula>
                       </formulalist>
                      </sequent>
                      <rule type="andr" symbol="\and:r">
                       <sequent>
                        <formulalist>
                         <constantatomformula symbol="AS" definition="AS"/>
                         <constantatomformula symbol="LUB" definition="LUB"/>
                         <constantatomformula symbol="R" definition="R"/>
                         <constantatomformula symbol="GLB" definition="GLB"/>
                         <constantatomformula symbol="AS" definition="AS"/>
                         <constantatomformula symbol="GLB" definition="GLB"/>
                         <constantatomformula symbol="LUB" definition="LUB"/>
                         <constantatomformula symbol="R" definition="R"/>
                        </formulalist>
                        <formulalist>
                         <conjunctiveformula type="and">
                          <quantifiedformula type="all">
                           <variable symbol="x"/>
                           <quantifiedformula type="all">
                            <variable symbol="y"/>
                            <constantatomformula symbol="=">
                             <function symbol="∪">
                              <function symbol="∩">
                               <variable symbol="x"/>
                               <variable symbol="y"/>
                              </function>
                              <variable symbol="x"/>
                             </function>
                             <variable symbol="x"/>
                            </constantatomformula>
                           </quantifiedformula>
                          </quantifiedformula>
                          <quantifiedformula type="all">
                           <variable symbol="x"/>
                           <quantifiedformula type="all">
                            <variable symbol="y"/>
                            <constantatomformula symbol="=">
                             <function symbol="∩">
                              <function symbol="∪">
                               <variable symbol="x"/>
                               <variable symbol="y"/>
                              </function>
                              <variable symbol="x"/>
                             </function>
                             <variable symbol="x"/>
                            </constantatomformula>
                           </quantifiedformula>
                          </quantifiedformula>
                         </conjunctiveformula>
                        </formulalist>
                       </sequent>
                       <rule type="forallr" symbol="\all:r">
                        <sequent>
                         <formulalist>
                          <constantatomformula symbol="AS" definition="AS"/>
                          <constantatomformula symbol="LUB" definition="LUB"/>
                          <constantatomformula symbol="R" definition="R"/>
                          <constantatomformula symbol="GLB" definition="GLB"/>
                         </formulalist>
                         <formulalist>
                          <quantifiedformula type="all">
                           <variable symbol="x"/>
                           <quantifiedformula type="all">
                            <variable symbol="y"/>
                            <constantatomformula symbol="=">
                             <function symbol="∪">
                              <function symbol="∩">
                               <variable symbol="x"/>
                               <variable symbol="y"/>
                              </function>
                              <variable symbol="x"/>
                             </function>
                             <variable symbol="x"/>
                            </constantatomformula>
                           </quantifiedformula>
                          </quantifiedformula>
                         </formulalist>
                        </sequent>
                        <rule type="forallr" symbol="\all:r">
                         <sequent>
                          <formulalist>
                           <constantatomformula symbol="AS" definition="AS"/>
                           <constantatomformula symbol="LUB" definition="LUB"/>
                           <constantatomformula symbol="R" definition="R"/>
                           <constantatomformula symbol="GLB" definition="GLB"/>
                          </formulalist>
                          <formulalist>
                           <quantifiedformula type="all">
                            <variable symbol="y"/>
                            <constantatomformula symbol="=">
                             <function symbol="∪">
                              <function symbol="∩">
                               <variable symbol="x_0"/>
                               <variable symbol="y"/>
                              </function>
                              <variable symbol="x_0"/>
                             </function>
                             <variable symbol="x_0"/>
                            </constantatomformula>
                           </quantifiedformula>
                          </formulalist>
                         </sequent>
                         <rule type="defl" symbol="d:l" param="();[()];()">
                          <sequent>
                           <formulalist>
                            <constantatomformula symbol="AS" definition="AS"/>
                            <constantatomformula symbol="LUB" definition="LUB"/>
                            <constantatomformula symbol="R" definition="R"/>
                            <constantatomformula symbol="GLB" definition="GLB"/>
                           </formulalist>
                           <formulalist>
                            <constantatomformula symbol="=">
                             <function symbol="∪">
                              <function symbol="∩">
                               <variable symbol="x_0"/>
                               <variable symbol="y_0"/>
                              </function>
                              <variable symbol="x_0"/>
                             </function>
                             <variable symbol="x_0"/>
                            </constantatomformula>
                           </formulalist>
                          </sequent>
                          <rule type="foralll" symbol="\all:l">
                           <sequent>
                            <formulalist>
                             <quantifiedformula type="all">
                              <variable symbol="x"/>
                              <quantifiedformula type="all">
                               <variable symbol="y"/>
                               <conjunctiveformula type="impl">
                                <conjunctiveformula type="and">
                                 <constantatomformula symbol="≤" definition="≤">
                                  <variable symbol="x"/>
                                  <variable symbol="y"/>
                                 </constantatomformula>
                                 <constantatomformula symbol="≤" definition="≤">
                                  <variable symbol="y"/>
                                  <variable symbol="x"/>
                                 </constantatomformula>
                                </conjunctiveformula>
                                <constantatomformula symbol="=">
                                 <variable symbol="x"/>
                                 <variable symbol="y"/>
                                </constantatomformula>
                               </conjunctiveformula>
                              </quantifiedformula>
                             </quantifiedformula>
                             <constantatomformula symbol="LUB" definition="LUB"/>
                             <constantatomformula symbol="R" definition="R"/>
                             <constantatomformula symbol="GLB" definition="GLB"/>
                            </formulalist>
                            <formulalist>
                             <constantatomformula symbol="=">
                              <function symbol="∪">
                               <function symbol="∩">
                                <variable symbol="x_0"/>
                                <variable symbol="y_0"/>
                               </function>
                               <variable symbol="x_0"/>
                              </function>
                              <variable symbol="x_0"/>
                             </constantatomformula>
                            </formulalist>
                           </sequent>
                           <rule type="foralll" symbol="\all:l">
                            <sequent>
                             <formulalist>
                              <quantifiedformula type="all">
                               <variable symbol="y"/>
                               <conjunctiveformula type="impl">
                                <conjunctiveformula type="and">
                                 <constantatomformula symbol="≤" definition="≤">
                                  <function symbol="∪">
                                   <function symbol="∩">
                                    <variable symbol="x_0"/>
                                    <variable symbol="y_0"/>
                                   </function>
                                   <variable symbol="x_0"/>
                                  </function>
                                  <variable symbol="y"/>
                                 </constantatomformula>
                                 <constantatomformula symbol="≤" definition="≤">
                                  <variable symbol="y"/>
                                  <function symbol="∪">
                                   <function symbol="∩">
                                    <variable symbol="x_0"/>
                                    <variable symbol="y_0"/>
                                   </function>
                                   <variable symbol="x_0"/>
                                  </function>
                                 </constantatomformula>
                                </conjunctiveformula>
                                <constantatomformula symbol="=">
                                 <function symbol="∪">
                                  <function symbol="∩">
                                   <variable symbol="x_0"/>
                                   <variable symbol="y_0"/>
                                  </function>
                                  <variable symbol="x_0"/>
                                 </function>
                                 <variable symbol="y"/>
                                </constantatomformula>
                               </conjunctiveformula>
                              </quantifiedformula>
                              <constantatomformula symbol="LUB" definition="LUB"/>
                              <constantatomformula symbol="R" definition="R"/>
                              <constantatomformula symbol="GLB" definition="GLB"/>
                             </formulalist>
                             <formulalist>
                              <constantatomformula symbol="=">
                               <function symbol="∪">
                                <function symbol="∩">
                                 <variable symbol="x_0"/>
                                 <variable symbol="y_0"/>
                                </function>
                                <variable symbol="x_0"/>
                               </function>
                               <variable symbol="x_0"/>
                              </constantatomformula>
                             </formulalist>
                            </sequent>
                            <rule type="impll" symbol="\impl:l">
                             <sequent>
                              <formulalist>
                               <conjunctiveformula type="impl">
                                <conjunctiveformula type="and">
                                 <constantatomformula symbol="≤" definition="≤">
                                  <function symbol="∪">
                                   <function symbol="∩">
                                    <variable symbol="x_0"/>
                                    <variable symbol="y_0"/>
                                   </function>
                                   <variable symbol="x_0"/>
                                  </function>
                                  <variable symbol="x_0"/>
                                 </constantatomformula>
                                 <constantatomformula symbol="≤" definition="≤">
                                  <variable symbol="x_0"/>
                                  <function symbol="∪">
                                   <function symbol="∩">
                                    <variable symbol="x_0"/>
                                    <variable symbol="y_0"/>
                                   </function>
                                   <variable symbol="x_0"/>
                                  </function>
                                 </constantatomformula>
                                </conjunctiveformula>
                                <constantatomformula symbol="=">
                                 <function symbol="∪">
                                  <function symbol="∩">
                                   <variable symbol="x_0"/>
                                   <variable symbol="y_0"/>
                                  </function>
                                  <variable symbol="x_0"/>
                                 </function>
                                 <variable symbol="x_0"/>
                                </constantatomformula>
                               </conjunctiveformula>
                               <constantatomformula symbol="LUB" definition="LUB"/>
                               <constantatomformula symbol="R" definition="R"/>
                               <constantatomformula symbol="GLB" definition="GLB"/>
                              </formulalist>
                              <formulalist>
                               <constantatomformula symbol="=">
                                <function symbol="∪">
                                 <function symbol="∩">
                                  <variable symbol="x_0"/>
                                  <variable symbol="y_0"/>
                                 </function>
                                 <variable symbol="x_0"/>
                                </function>
                                <variable symbol="x_0"/>
                               </constantatomformula>
                              </formulalist>
                             </sequent>
                             <rule type="defl" symbol="d:l" param="();[()];()">
                              <sequent>
                               <formulalist>
                                <constantatomformula symbol="LUB" definition="LUB"/>
                                <constantatomformula symbol="R" definition="R"/>
                                <constantatomformula symbol="GLB" definition="GLB"/>
                               </formulalist>
                               <formulalist>
                                <conjunctiveformula type="and">
                                 <constantatomformula symbol="≤" definition="≤">
                                  <function symbol="∪">
                                   <function symbol="∩">
                                    <variable symbol="x_0"/>
                                    <variable symbol="y_0"/>
                                   </function>
                                   <variable symbol="x_0"/>
                                  </function>
                                  <variable symbol="x_0"/>
                                 </constantatomformula>
                                 <constantatomformula symbol="≤" definition="≤">
                                  <variable symbol="x_0"/>
                                  <function symbol="∪">
                                   <function symbol="∩">
                                    <variable symbol="x_0"/>
                                    <variable symbol="y_0"/>
                                   </function>
                                   <variable symbol="x_0"/>
                                  </function>
                                 </constantatomformula>
                                </conjunctiveformula>
                               </formulalist>
                              </sequent>
                              <rule type="foralll" symbol="\all:l">
                               <sequent>
                                <formulalist>
                                 <quantifiedformula type="all">
                                  <variable symbol="x"/>
                                  <quantifiedformula type="all">
                                   <variable symbol="y"/>
                                   <conjunctiveformula type="and">
                                    <conjunctiveformula type="and">
                                     <constantatomformula symbol="≤" definition="≤">
                                      <variable symbol="x"/>
                                      <function symbol="∪">
                                       <variable symbol="x"/>
                                       <variable symbol="y"/>
                                      </function>
                                     </constantatomformula>
                                     <constantatomformula symbol="≤" definition="≤">
                                      <variable symbol="y"/>
                                      <function symbol="∪">
                                       <variable symbol="x"/>
                                       <variable symbol="y"/>
                                      </function>
                                     </constantatomformula>
                                    </conjunctiveformula>
                                    <quantifiedformula type="all">
                                     <variable symbol="z"/>
                                     <conjunctiveformula type="impl">
                                      <conjunctiveformula type="and">
                                       <constantatomformula symbol="≤" definition="≤">
                                        <variable symbol="x"/>
                                        <variable symbol="z"/>
                                       </constantatomformula>
                                       <constantatomformula symbol="≤" definition="≤">
                                        <variable symbol="y"/>
                                        <variable symbol="z"/>
                                       </constantatomformula>
                                      </conjunctiveformula>
                                      <constantatomformula symbol="≤" definition="≤">
                                       <function symbol="∪">
                                        <variable symbol="x"/>
                                        <variable symbol="y"/>
                                       </function>
                                       <variable symbol="z"/>
                                      </constantatomformula>
                                     </conjunctiveformula>
                                    </quantifiedformula>
                                   </conjunctiveformula>
                                  </quantifiedformula>
                                 </quantifiedformula>
                                 <constantatomformula symbol="R" definition="R"/>
                                 <constantatomformula symbol="GLB" definition="GLB"/>
                                </formulalist>
                                <formulalist>
                                 <conjunctiveformula type="and">
                                  <constantatomformula symbol="≤" definition="≤">
                                   <function symbol="∪">
                                    <function symbol="∩">
                                     <variable symbol="x_0"/>
                                     <variable symbol="y_0"/>
                                    </function>
                                    <variable symbol="x_0"/>
                                   </function>
                                   <variable symbol="x_0"/>
                                  </constantatomformula>
                                  <constantatomformula symbol="≤" definition="≤">
                                   <variable symbol="x_0"/>
                                   <function symbol="∪">
                                    <function symbol="∩">
                                     <variable symbol="x_0"/>
                                     <variable symbol="y_0"/>
                                    </function>
                                    <variable symbol="x_0"/>
                                   </function>
                                  </constantatomformula>
                                 </conjunctiveformula>
                                </formulalist>
                               </sequent>
                               <rule type="foralll" symbol="\all:l">
                                <sequent>
                                 <formulalist>
                                  <quantifiedformula type="all">
                                   <variable symbol="y"/>
                                   <conjunctiveformula type="and">
                                    <conjunctiveformula type="and">
                                     <constantatomformula symbol="≤" definition="≤">
                                      <function symbol="∩">
                                       <variable symbol="x_0"/>
                                       <variable symbol="y_0"/>
                                      </function>
                                      <function symbol="∪">
                                       <function symbol="∩">
                                        <variable symbol="x_0"/>
                                        <variable symbol="y_0"/>
                                       </function>
                                       <variable symbol="y"/>
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                                     <constantatomformula symbol="≤" definition="≤">
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                                      <function symbol="∪">
                                       <function symbol="∩">
                                        <variable symbol="x_0"/>
                                        <variable symbol="y_0"/>
                                       </function>
                                       <variable symbol="y"/>
                                      </function>
                                     </constantatomformula>
                                    </conjunctiveformula>
                                    <quantifiedformula type="all">
                                     <variable symbol="z"/>
                                     <conjunctiveformula type="impl">
                                      <conjunctiveformula type="and">
                                       <constantatomformula symbol="≤" definition="≤">
                                        <function symbol="∩">
                                         <variable symbol="x_0"/>
                                         <variable symbol="y_0"/>
                                        </function>
                                        <variable symbol="z"/>
                                       </constantatomformula>
                                       <constantatomformula symbol="≤" definition="≤">
                                        <variable symbol="y"/>
                                        <variable symbol="z"/>
                                       </constantatomformula>
                                      </conjunctiveformula>
                                      <constantatomformula symbol="≤" definition="≤">
                                       <function symbol="∪">
                                        <function symbol="∩">
                                         <variable symbol="x_0"/>
                                         <variable symbol="y_0"/>
                                        </function>
                                        <variable symbol="y"/>
                                       </function>
                                       <variable symbol="z"/>
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                                     </conjunctiveformula>
                                    </quantifiedformula>
                                   </conjunctiveformula>
                                  </quantifiedformula>
                                  <constantatomformula symbol="R" definition="R"/>
                                  <constantatomformula symbol="GLB" definition="GLB"/>
                                 </formulalist>
                                 <formulalist>
                                  <conjunctiveformula type="and">
                                   <constantatomformula symbol="≤" definition="≤">
                                    <function symbol="∪">
                                     <function symbol="∩">
                                      <variable symbol="x_0"/>
                                      <variable symbol="y_0"/>
                                     </function>
                                     <variable symbol="x_0"/>
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                                    <variable symbol="x_0"/>
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                                   <constantatomformula symbol="≤" definition="≤">
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                                    <function symbol="∪">
                                     <function symbol="∩">
                                      <variable symbol="x_0"/>
                                      <variable symbol="y_0"/>
                                     </function>
                                     <variable symbol="x_0"/>
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                                   </constantatomformula>
                                  </conjunctiveformula>
                                 </formulalist>
                                </sequent>
                                <rule type="contrl" symbol="c:l" param="2,1,1">
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                                  <formulalist>
                                   <conjunctiveformula type="and">
                                    <conjunctiveformula type="and">
                                     <constantatomformula symbol="≤" definition="≤">
                                      <function symbol="∩">
                                       <variable symbol="x_0"/>
                                       <variable symbol="y_0"/>
                                      </function>
                                      <function symbol="∪">
                                       <function symbol="∩">
                                        <variable symbol="x_0"/>
                                        <variable symbol="y_0"/>
                                       </function>
                                       <variable symbol="x_0"/>
                                      </function>
                                     </constantatomformula>
                                     <constantatomformula symbol="≤" definition="≤">
                                      <variable symbol="x_0"/>
                                      <function symbol="∪">
                                       <function symbol="∩">
                                        <variable symbol="x_0"/>
                                        <variable symbol="y_0"/>
                                       </function>
                                       <variable symbol="x_0"/>
                                      </function>
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                                    <quantifiedformula type="all">
                                     <variable symbol="z"/>
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                                      <conjunctiveformula type="and">
                                       <constantatomformula symbol="≤" definition="≤">
                                        <function symbol="∩">
                                         <variable symbol="x_0"/>
                                         <variable symbol="y_0"/>
                                        </function>
                                        <variable symbol="z"/>
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                                       <constantatomformula symbol="≤" definition="≤">
                                        <variable symbol="x_0"/>
                                        <variable symbol="z"/>
                                       </constantatomformula>
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                                      <constantatomformula symbol="≤" definition="≤">
                                       <function symbol="∪">
                                        <function symbol="∩">
                                         <variable symbol="x_0"/>
                                         <variable symbol="y_0"/>
                                        </function>
                                        <variable symbol="x_0"/>
                                       </function>
                                       <variable symbol="z"/>
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                                     </conjunctiveformula>
                                    </quantifiedformula>
                                   </conjunctiveformula>
                                   <constantatomformula symbol="R" definition="R"/>
                                   <constantatomformula symbol="GLB" definition="GLB"/>
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                                  <formulalist>
                                   <conjunctiveformula type="and">
                                    <constantatomformula symbol="≤" definition="≤">
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                                      <function symbol="∩">
                                       <variable symbol="x_0"/>
                                       <variable symbol="y_0"/>
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                                      <variable symbol="x_0"/>
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                                     <variable symbol="x_0"/>
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                                    <constantatomformula symbol="≤" definition="≤">
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                                     <function symbol="∪">
                                      <function symbol="∩">
                                       <variable symbol="x_0"/>
                                       <variable symbol="y_0"/>
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                                      <variable symbol="x_0"/>
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                                    </constantatomformula>
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                                 </sequent>
                                 <rule type="perml" symbol="\pi:l" param="(2 4)">
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                                   <formulalist>
                                    <conjunctiveformula type="and">
                                     <conjunctiveformula type="and">
                                      <constantatomformula symbol="≤" definition="≤">
                                       <function symbol="∩">
                                        <variable symbol="x_0"/>
                                        <variable symbol="y_0"/>
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                                        <function symbol="∩">
                                         <variable symbol="x_0"/>
                                         <variable symbol="y_0"/>
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                                        <variable symbol="x_0"/>
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                                      <constantatomformula symbol="≤" definition="≤">
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                                       <function symbol="∪">
                                        <function symbol="∩">
                                         <variable symbol="x_0"/>
                                         <variable symbol="y_0"/>
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                                        <variable symbol="x_0"/>
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                                      </constantatomformula>
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                                     <quantifiedformula type="all">
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                                       <conjunctiveformula type="and">
                                        <constantatomformula symbol="≤" definition="≤">
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                                          <variable symbol="x_0"/>
                                          <variable symbol="y_0"/>
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                                        <constantatomformula symbol="≤" definition="≤">
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                                       <constantatomformula symbol="≤" definition="≤">
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                                         <function symbol="∩">
                                          <variable symbol="x_0"/>
                                          <variable symbol="y_0"/>
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                                         <variable symbol="x_0"/>
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                                    <conjunctiveformula type="and">
                                     <conjunctiveformula type="and">
                                      <constantatomformula symbol="≤" definition="≤">
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                                        <variable symbol="x_0"/>
                                        <variable symbol="y_0"/>
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                                        <function symbol="∩">
                                         <variable symbol="x_0"/>
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                                        <variable symbol="x_0"/>
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                                      <constantatomformula symbol="≤" definition="≤">
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                                        <function symbol="∩">
                                         <variable symbol="x_0"/>
                                         <variable symbol="y_0"/>
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                                        <variable symbol="x_0"/>
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                                     <quantifiedformula type="all">
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                                       <conjunctiveformula type="and">
                                        <constantatomformula symbol="≤" definition="≤">
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                                        <constantatomformula symbol="≤" definition="≤">
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                                       <constantatomformula symbol="≤" definition="≤">
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                                         <function symbol="∩">
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                                          <variable symbol="y_0"/>
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                                        <variable symbol="z"/>
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                                    <conjunctiveformula type="and">
                                     <constantatomformula symbol="≤" definition="≤">
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                                     <constantatomformula symbol="≤" definition="≤">
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                                       <constantatomformula symbol="≤" definition="≤">
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                                         <constantatomformula symbol="≤" definition="≤">
                                          <variable symbol="x_0"/>
                                          <function symbol="∪">
                                           <function symbol="∩">
                                            <variable symbol="x_0"/>
                                            <variable symbol="y_0"/>
                                           </function>
                                           <variable symbol="x_0"/>
                                          </function>
                                         </constantatomformula>
                                        </conjunctiveformula>
                                       </formulalist>
                                      </sequent>
                                      <rule type="perml" symbol="\pi:l" param="(1 4)">
                                       <sequent>
                                        <formulalist>
                                         <constantatomformula symbol="≤" definition="≤">
                                          <variable symbol="x_0"/>
                                          <function symbol="∪">
                                           <function symbol="∩">
                                            <variable symbol="x_0"/>
                                            <variable symbol="y_0"/>
                                           </function>
                                           <variable symbol="x_0"/>
                                          </function>
                                         </constantatomformula>
                                         <constantatomformula symbol="GLB" definition="GLB"/>
                                         <constantatomformula symbol="R" definition="R"/>
                                         <quantifiedformula type="all">
                                          <variable symbol="z"/>
                                          <conjunctiveformula type="impl">
                                           <conjunctiveformula type="and">
                                            <constantatomformula symbol="≤" definition="≤">
                                             <function symbol="∩">
                                              <variable symbol="x_0"/>
                                              <variable symbol="y_0"/>
                                             </function>
                                             <variable symbol="z"/>
                                            </constantatomformula>
                                            <constantatomformula symbol="≤" definition="≤">
                                             <variable symbol="x_0"/>
                                             <variable symbol="z"/>
                                            </constantatomformula>
                                           </conjunctiveformula>
                                           <constantatomformula symbol="≤" definition="≤">
                                            <function symbol="∪">
                                             <function symbol="∩">
                                              <variable symbol="x_0"/>
                                              <variable symbol="y_0"/>
                                             </function>
                                             <variable symbol="x_0"/>
                                            </function>
                                            <variable symbol="z"/>
                                           </constantatomformula>
                                          </conjunctiveformula>
                                         </quantifiedformula>
                                        </formulalist>
                                        <formulalist>
                                         <conjunctiveformula type="and">
                                          <constantatomformula symbol="≤" definition="≤">
                                           <function symbol="∪">
                                            <function symbol="∩">
                                             <variable symbol="x_0"/>
                                             <variable symbol="y_0"/>
                                            </function>
                                            <variable symbol="x_0"/>
                                           </function>
                                           <variable symbol="x_0"/>
                                          </constantatomformula>
                                          <constantatomformula symbol="≤" definition="≤">
                                           <variable symbol="x_0"/>
                                           <function symbol="∪">
                                            <function symbol="∩">
                                             <variable symbol="x_0"/>
                                             <variable symbol="y_0"/>
                                            </function>
                                            <variable symbol="x_0"/>
                                           </function>
                                          </constantatomformula>
                                         </conjunctiveformula>
                                        </formulalist>
                                       </sequent>
                                       <rule type="andr" symbol="\and:r">
                                        <sequent>
                                         <formulalist>
                                          <quantifiedformula type="all">
                                           <variable symbol="z"/>
                                           <conjunctiveformula type="impl">
                                            <conjunctiveformula type="and">
                                             <constantatomformula symbol="≤" definition="≤">
                                              <function symbol="∩">
                                               <variable symbol="x_0"/>
                                               <variable symbol="y_0"/>
                                              </function>
                                              <variable symbol="z"/>
                                             </constantatomformula>
                                             <constantatomformula symbol="≤" definition="≤">
                                              <variable symbol="x_0"/>
                                              <variable symbol="z"/>
                                             </constantatomformula>
                                            </conjunctiveformula>
                                            <constantatomformula symbol="≤" definition="≤">
                                             <function symbol="∪">
                                              <function symbol="∩">
                                               <variable symbol="x_0"/>
                                               <variable symbol="y_0"/>
                                              </function>
                                              <variable symbol="x_0"/>
                                             </function>
                                             <variable symbol="z"/>
                                            </constantatomformula>
                                           </conjunctiveformula>
                                          </quantifiedformula>
                                          <constantatomformula symbol="GLB" definition="GLB"/>
                                          <constantatomformula symbol="R" definition="R"/>
                                          <constantatomformula symbol="≤" definition="≤">
                                           <variable symbol="x_0"/>
                                           <function symbol="∪">
                                            <function symbol="∩">
                                             <variable symbol="x_0"/>
                                             <variable symbol="y_0"/>
                                            </function>
                                            <variable symbol="x_0"/>
                                           </function>
                                          </constantatomformula>
                                         </formulalist>
                                         <formulalist>
                                          <conjunctiveformula type="and">
                                           <constantatomformula symbol="≤" definition="≤">
                                            <function symbol="∪">
                                             <function symbol="∩">
                                              <variable symbol="x_0"/>
                                              <variable symbol="y_0"/>
                                             </function>
                                             <variable symbol="x_0"/>
                                            </function>
                                            <variable symbol="x_0"/>
                                           </constantatomformula>
                                           <constantatomformula symbol="≤" definition="≤">
                                            <variable symbol="x_0"/>
                                            <function symbol="∪">
                                             <function symbol="∩">
                                              <variable symbol="x_0"/>
                                              <variable symbol="y_0"/>
                                             </function>
                                             <variable symbol="x_0"/>
                                            </function>
                                           </constantatomformula>
                                          </conjunctiveformula>
                                         </formulalist>
                                        </sequent>
                                        <rule type="foralll" symbol="\all:l">
                                         <sequent>
                                          <formulalist>
                                           <quantifiedformula type="all">
                                            <variable symbol="z"/>
                                            <conjunctiveformula type="impl">
                                             <conjunctiveformula type="and">
                                              <constantatomformula symbol="≤" definition="≤">
                                               <function symbol="∩">
                                                <variable symbol="x_0"/>
                                                <variable symbol="y_0"/>
                                               </function>
                                               <variable symbol="z"/>
                                              </constantatomformula>
                                              <constantatomformula symbol="≤" definition="≤">
                                               <variable symbol="x_0"/>
                                               <variable symbol="z"/>
                                              </constantatomformula>
                                             </conjunctiveformula>
                                             <constantatomformula symbol="≤" definition="≤">
                                              <function symbol="∪">
                                               <function symbol="∩">
                                                <variable symbol="x_0"/>
                                                <variable symbol="y_0"/>
                                               </function>
                                               <variable symbol="x_0"/>
                                              </function>
                                              <variable symbol="z"/>
                                             </constantatomformula>
                                            </conjunctiveformula>
                                           </quantifiedformula>
                                           <constantatomformula symbol="GLB" definition="GLB"/>
                                           <constantatomformula symbol="R" definition="R"/>
                                          </formulalist>
                                          <formulalist>
                                           <constantatomformula symbol="≤" definition="≤">
                                            <function symbol="∪">
                                             <function symbol="∩">
                                              <variable symbol="x_0"/>
                                              <variable symbol="y_0"/>
                                             </function>
                                             <variable symbol="x_0"/>
                                            </function>
                                            <variable symbol="x_0"/>
                                           </constantatomformula>
                                          </formulalist>
                                         </sequent>
                                         <rule type="impll" symbol="\impl:l">
                                          <sequent>
                                           <formulalist>
                                            <conjunctiveformula type="impl">
                                             <conjunctiveformula type="and">
                                              <constantatomformula symbol="≤" definition="≤">
                                               <function symbol="∩">
                                                <variable symbol="x_0"/>
                                                <variable symbol="y_0"/>
                                               </function>
                                               <variable symbol="x_0"/>
                                              </constantatomformula>
                                              <constantatomformula symbol="≤" definition="≤">
                                               <variable symbol="x_0"/>
                                               <variable symbol="x_0"/>
                                              </constantatomformula>
                                             </conjunctiveformula>
                                             <constantatomformula symbol="≤" definition="≤">
                                              <function symbol="∪">
                                               <function symbol="∩">
                                                <variable symbol="x_0"/>
                                                <variable symbol="y_0"/>
                                               </function>
                                               <variable symbol="x_0"/>
                                              </function>
                                              <variable symbol="x_0"/>
                                             </constantatomformula>
                                            </conjunctiveformula>
                                            <constantatomformula symbol="GLB" definition="GLB"/>
                                            <constantatomformula symbol="R" definition="R"/>
                                           </formulalist>
                                           <formulalist>
                                            <constantatomformula symbol="≤" definition="≤">
                                             <function symbol="∪">
                                              <function symbol="∩">
                                               <variable symbol="x_0"/>
                                               <variable symbol="y_0"/>
                                              </function>
                                              <variable symbol="x_0"/>
                                             </function>
                                             <variable symbol="x_0"/>
                                            </constantatomformula>
                                           </formulalist>
                                          </sequent>
                                          <rule type="andr" symbol="\and:r">
                                           <sequent>
                                            <formulalist>
                                             <constantatomformula symbol="GLB" definition="GLB"/>
                                             <constantatomformula symbol="R" definition="R"/>
                                            </formulalist>
                                            <formulalist>
                                             <conjunctiveformula type="and">
                                              <constantatomformula symbol="≤" definition="≤">
                                               <function symbol="∩">
                                                <variable symbol="x_0"/>
                                                <variable symbol="y_0"/>
                                               </function>
                                               <variable symbol="x_0"/>
                                              </constantatomformula>
                                              <constantatomformula symbol="≤" definition="≤">
                                               <variable symbol="x_0"/>
                                               <variable symbol="x_0"/>
                                              </constantatomformula>
                                             </conjunctiveformula>
                                            </formulalist>
                                           </sequent>
                                           <rule type="defl" symbol="d:l" param="();[()];()">
                                            <sequent>
                                             <formulalist>
                                              <constantatomformula symbol="GLB" definition="GLB"/>
                                             </formulalist>
                                             <formulalist>
                                              <constantatomformula symbol="≤" definition="≤">
                                               <function symbol="∩">
                                                <variable symbol="x_0"/>
                                                <variable symbol="y_0"/>
                                               </function>
                                               <variable symbol="x_0"/>
                                              </constantatomformula>
                                             </formulalist>
                                            </sequent>
                                            <rule type="foralll" symbol="\all:l">
                                             <sequent>
                                              <formulalist>
                                               <quantifiedformula type="all">
                                                <variable symbol="x"/>
                                                <quantifiedformula type="all">
                                                 <variable symbol="y"/>
                                                 <conjunctiveformula type="and">
                                                  <conjunctiveformula type="and">
                                                   <constantatomformula symbol="≤" definition="≤">
                                                    <function symbol="∩">
                                                     <variable symbol="x"/>
                                                     <variable symbol="y"/>
                                                    </function>
                                                    <variable symbol="x"/>
                                                   </constantatomformula>
                                                   <constantatomformula symbol="≤" definition="≤">
                                                    <function symbol="∩">
                                                     <variable symbol="x"/>
                                                     <variable symbol="y"/>
                                                    </function>
                                                    <variable symbol="y"/>
                                                   </constantatomformula>
                                                  </conjunctiveformula>
                                                  <quantifiedformula type="all">
                                                   <variable symbol="z"/>
                                                   <conjunctiveformula type="impl">
                                                    <conjunctiveformula type="and">
                                                     <constantatomformula symbol="≤" definition="≤">
                                                      <variable symbol="z"/>
                                                      <variable symbol="x"/>
                                                     </constantatomformula>
                                                     <constantatomformula symbol="≤" definition="≤">
                                                      <variable symbol="z"/>
                                                      <variable symbol="y"/>
                                                     </constantatomformula>
                                                    </conjunctiveformula>
                                                    <constantatomformula symbol="≤" definition="≤">
                                                     <variable symbol="z"/>
                                                     <function symbol="∩">
                                                      <variable symbol="x"/>
                                                      <variable symbol="y"/>
                                                     </function>
                                                    </constantatomformula>
                                                   </conjunctiveformula>
                                                  </quantifiedformula>
                                                 </conjunctiveformula>
                                                </quantifiedformula>
                                               </quantifiedformula>
                                              </formulalist>
                                              <formulalist>
                                               <constantatomformula symbol="≤" definition="≤">
                                                <function symbol="∩">
                                                 <variable symbol="x_0"/>
                                                 <variable symbol="y_0"/>
                                                </function>
                                                <variable symbol="x_0"/>
                                               </constantatomformula>
                                              </formulalist>
                                             </sequent>
                                             <rule type="foralll" symbol="\all:l">
                                              <sequent>
                                               <formulalist>
                                                <quantifiedformula type="all">
                                                 <variable symbol="y"/>
                                                 <conjunctiveformula type="and">
                                                  <conjunctiveformula type="and">
                                                   <constantatomformula symbol="≤" definition="≤">
                                                    <function symbol="∩">
                                                     <variable symbol="x_0"/>
                                                     <variable symbol="y"/>
                                                    </function>
                                                    <variable symbol="x_0"/>
                                                   </constantatomformula>
                                                   <constantatomformula symbol="≤" definition="≤">
                                                    <function symbol="∩">
                                                     <variable symbol="x_0"/>
                                                     <variable symbol="y"/>
                                                    </function>
                                                    <variable symbol="y"/>
                                                   </constantatomformula>
                                                  </conjunctiveformula>
                                                  <quantifiedformula type="all">
                                                   <variable symbol="z"/>
                                                   <conjunctiveformula type="impl">
                                                    <conjunctiveformula type="and">
                                                     <constantatomformula symbol="≤" definition="≤">
                                                      <variable symbol="z"/>
                                                      <variable symbol="x_0"/>
                                                     </constantatomformula>
                                                     <constantatomformula symbol="≤" definition="≤">
                                                      <variable symbol="z"/>
                                                      <variable symbol="y"/>
                                                     </constantatomformula>
                                                    </conjunctiveformula>
                                                    <constantatomformula symbol="≤" definition="≤">
                                                     <variable symbol="z"/>
                                                     <function symbol="∩">
                                                      <variable symbol="x_0"/>
                                                      <variable symbol="y"/>
                                                     </function>
                                                    </constantatomformula>
                                                   </conjunctiveformula>
                                                  </quantifiedformula>
                                                 </conjunctiveformula>
                                                </quantifiedformula>
                                               </formulalist>
                                               <formulalist>
                                                <constantatomformula symbol="≤" definition="≤">
                                                 <function symbol="∩">
                                                  <variable symbol="x_0"/>
                                                  <variable symbol="y_0"/>
                                                 </function>
                                                 <variable symbol="x_0"/>
                                                </constantatomformula>
                                               </formulalist>
                                              </sequent>
                                              <rule type="andl1" symbol="\and:l1">
                                               <sequent>
                                                <formulalist>
                                                 <conjunctiveformula type="and">
                                                  <conjunctiveformula type="and">
                                                   <constantatomformula symbol="≤" definition="≤">
                                                    <function symbol="∩">
                                                     <variable symbol="x_0"/>
                                                     <variable symbol="y_0"/>
                                                    </function>
                                                    <variable symbol="x_0"/>
                                                   </constantatomformula>
                                                   <constantatomformula symbol="≤" definition="≤">
                                                    <function symbol="∩">
                                                     <variable symbol="x_0"/>
                                                     <variable symbol="y_0"/>
                                                    </function>
                                                    <variable symbol="y_0"/>
                                                   </constantatomformula>
                                                  </conjunctiveformula>
                                                  <quantifiedformula type="all">
                                                   <variable symbol="z"/>
                                                   <conjunctiveformula type="impl">
                                                    <conjunctiveformula type="and">
                                                     <constantatomformula symbol="≤" definition="≤">
                                                      <variable symbol="z"/>
                                                      <variable symbol="x_0"/>
                                                     </constantatomformula>
                                                     <constantatomformula symbol="≤" definition="≤">
                                                      <variable symbol="z"/>
                                                      <variable symbol="y_0"/>
                                                     </constantatomformula>
                                                    </conjunctiveformula>
                                                    <constantatomformula symbol="≤" definition="≤">
                                                     <variable symbol="z"/>
                                                     <function symbol="∩">
                                                      <variable symbol="x_0"/>
                                                      <variable symbol="y_0"/>
                                                     </function>
                                                    </constantatomformula>
                                                   </conjunctiveformula>
                                                  </quantifiedformula>
                                                 </conjunctiveformula>
                                                </formulalist>
                                                <formulalist>
                                                 <constantatomformula symbol="≤" definition="≤">
                                                  <function symbol="∩">
                                                   <variable symbol="x_0"/>
                                                   <variable symbol="y_0"/>
                                                  </function>
                                                  <variable symbol="x_0"/>
                                                 </constantatomformula>
                                                </formulalist>
                                               </sequent>
                                               <rule type="andl1" symbol="\and:l1">
                                                <sequent>
                                                 <formulalist>
                                                  <conjunctiveformula type="and">
                                                   <constantatomformula symbol="≤" definition="≤">
                                                    <function symbol="∩">
                                                     <variable symbol="x_0"/>
                                                     <variable symbol="y_0"/>
                                                    </function>
                                                    <variable symbol="x_0"/>
                                                   </constantatomformula>
                                                   <constantatomformula symbol="≤" definition="≤">
                                                    <function symbol="∩">
                                                     <variable symbol="x_0"/>
                                                     <variable symbol="y_0"/>
                                                    </function>
                                                    <variable symbol="y_0"/>
                                                   </constantatomformula>
                                                  </conjunctiveformula>
                                                 </formulalist>
                                                 <formulalist>
                                                  <constantatomformula symbol="≤" definition="≤">
                                                   <function symbol="∩">
                                                    <variable symbol="x_0"/>
                                                    <variable symbol="y_0"/>
                                                   </function>
                                                   <variable symbol="x_0"/>
                                                  </constantatomformula>
                                                 </formulalist>
                                                </sequent>
                                                <rule type="defl" symbol="d:l" param="();[(1)(1 1)(2)][(2)(1 2)];()">
                                                 <sequent>
                                                  <formulalist>
                                                   <constantatomformula symbol="≤" definition="≤">
                                                    <function symbol="∩">
                                                     <variable symbol="x_0"/>
                                                     <variable symbol="y_0"/>
                                                    </function>
                                                    <variable symbol="x_0"/>
                                                   </constantatomformula>
                                                  </formulalist>
                                                  <formulalist>
                                                   <constantatomformula symbol="≤" definition="≤">
                                                    <function symbol="∩">
                                                     <variable symbol="x_0"/>
                                                     <variable symbol="y_0"/>
                                                    </function>
                                                    <variable symbol="x_0"/>
                                                   </constantatomformula>
                                                  </formulalist>
                                                 </sequent>
                                                 <rule type="defr" symbol="d:r" param="();[(1)(1 1)(2)][(2)(1 2)];()">
                                                  <sequent>
                                                   <formulalist>
                                                    <constantatomformula symbol="=">
                                                     <function symbol="∩">
                                                      <function symbol="∩">
                                                       <variable symbol="x_0"/>
                                                       <variable symbol="y_0"/>
                                                      </function>
                                                      <variable symbol="x_0"/>
                                                     </function>
                                                     <function symbol="∩">
                                                      <variable symbol="x_0"/>
                                                      <variable symbol="y_0"/>
                                                     </function>
                                                    </constantatomformula>
                                                   </formulalist>
                                                   <formulalist>
                                                    <constantatomformula symbol="≤" definition="≤">
                                                     <function symbol="∩">
                                                      <variable symbol="x_0"/>
                                                      <variable symbol="y_0"/>
                                                     </function>
                                                     <variable symbol="x_0"/>
                                                    </constantatomformula>
                                                   </formulalist>
                                                  </sequent>
                                                  <rule type="axiom" symbol="axiom">
                                                   <sequent>
                                                    <formulalist>
                                                     <constantatomformula symbol="=">
                                                      <function symbol="∩">
                                                       <function symbol="∩">
                                                        <variable symbol="x_0"/>
                                                        <variable symbol="y_0"/>
                                                       </function>
                                                       <variable symbol="x_0"/>
                                                      </function>
                                                      <function symbol="∩">
                                                       <variable symbol="x_0"/>
                                                       <variable symbol="y_0"/>
                                                      </function>
                                                     </constantatomformula>
                                                    </formulalist>
                                                    <formulalist>
                                                     <constantatomformula symbol="=">
                                                      <function symbol="∩">
                                                       <function symbol="∩">
                                                        <variable symbol="x_0"/>
                                                        <variable symbol="y_0"/>
                                                       </function>
                                                       <variable symbol="x_0"/>
                                                      </function>
                                                      <function symbol="∩">
                                                       <variable symbol="x_0"/>
                                                       <variable symbol="y_0"/>
                                                      </function>
                                                     </constantatomformula>
                                                    </formulalist>
                                                   </sequent>
                                                  </rule>
                                                 </rule>
                                                </rule>
                                               </rule>
                                              </rule>
                                             </rule>
                                            </rule>
                                           </rule>
                                           <rule type="defl" symbol="d:l" param="();[()];()">
                                            <sequent>
                                             <formulalist>
                                              <constantatomformula symbol="R" definition="R"/>
                                             </formulalist>
                                             <formulalist>
                                              <constantatomformula symbol="≤" definition="≤">
                                               <variable symbol="x_0"/>
                                               <variable symbol="x_0"/>
                                              </constantatomformula>
                                             </formulalist>
                                            </sequent>
                                            <rule type="foralll" symbol="\all:l">
                                             <sequent>
                                              <formulalist>
                                               <quantifiedformula type="all">
                                                <variable symbol="x"/>
                                                <constantatomformula symbol="≤" definition="≤">
                                                 <variable symbol="x"/>
                                                 <variable symbol="x"/>
                                                </constantatomformula>
                                               </quantifiedformula>
                                              </formulalist>
                                              <formulalist>
                                               <constantatomformula symbol="≤" definition="≤">
                                                <variable symbol="x_0"/>
                                                <variable symbol="x_0"/>
                                               </constantatomformula>
                                              </formulalist>
                                             </sequent>
                                             <rule type="defl" symbol="d:l" param="();[(1)(1 1)(2)][(2)(1 2)];()">
                                              <sequent>
                                               <formulalist>
                                                <constantatomformula symbol="≤" definition="≤">
                                                 <variable symbol="x_0"/>
                                                 <variable symbol="x_0"/>
                                                </constantatomformula>
                                               </formulalist>
                                               <formulalist>
                                                <constantatomformula symbol="≤" definition="≤">
                                                 <variable symbol="x_0"/>
                                                 <variable symbol="x_0"/>
                                                </constantatomformula>
                                               </formulalist>
                                              </sequent>
                                              <rule type="defr" symbol="d:r" param="();[(1)(1 1)(2)][(2)(1 2)];()">
                                               <sequent>
                                                <formulalist>
                                                 <constantatomformula symbol="=">
                                                  <function symbol="∩">
                                                   <variable symbol="x_0"/>
                                                   <variable symbol="x_0"/>
                                                  </function>
                                                  <variable symbol="x_0"/>
                                                 </constantatomformula>
                                                </formulalist>
                                                <formulalist>
                                                 <constantatomformula symbol="≤" definition="≤">
                                                  <variable symbol="x_0"/>
                                                  <variable symbol="x_0"/>
                                                 </constantatomformula>
                                                </formulalist>
                                               </sequent>
                                               <rule type="axiom" symbol="axiom">
                                                <sequent>
                                                 <formulalist>
                                                  <constantatomformula symbol="=">
                                                   <function symbol="∩">
                                                    <variable symbol="x_0"/>
                                                    <variable symbol="x_0"/>
                                                   </function>
                                                   <variable symbol="x_0"/>
                                                  </constantatomformula>
                                                 </formulalist>
                                                 <formulalist>
                                                  <constantatomformula symbol="=">
                                                   <function symbol="∩">
                                                    <variable symbol="x_0"/>
                                                    <variable symbol="x_0"/>
                                                   </function>
                                                   <variable symbol="x_0"/>
                                                  </constantatomformula>
                                                 </formulalist>
                                                </sequent>
                                               </rule>
                                              </rule>
                                             </rule>
                                            </rule>
                                           </rule>
                                          </rule>
                                          <rule type="defl" symbol="d:l" param="();[(1)(1 1)(2)][(2)(1 2)];()">
                                           <sequent>
                                            <formulalist>
                                             <constantatomformula symbol="≤" definition="≤">
                                              <function symbol="∪">
                                               <function symbol="∩">
                                                <variable symbol="x_0"/>
                                                <variable symbol="y_0"/>
                                               </function>
                                               <variable symbol="x_0"/>
                                              </function>
                                              <variable symbol="x_0"/>
                                             </constantatomformula>
                                            </formulalist>
                                            <formulalist>
                                             <constantatomformula symbol="≤" definition="≤">
                                              <function symbol="∪">
                                               <function symbol="∩">
                                                <variable symbol="x_0"/>
                                                <variable symbol="y_0"/>
                                               </function>
                                               <variable symbol="x_0"/>
                                              </function>
                                              <variable symbol="x_0"/>
                                             </constantatomformula>
                                            </formulalist>
                                           </sequent>
                                           <rule type="defr" symbol="d:r" param="();[(1)(1 1)(2)][(2)(1 2)];()">
                                            <sequent>
                                             <formulalist>
                                              <constantatomformula symbol="=">
                                               <function symbol="∩">
                                                <function symbol="∪">
                                                 <function symbol="∩">
                                                  <variable symbol="x_0"/>
                                                  <variable symbol="y_0"/>
                                                 </function>
                                                 <variable symbol="x_0"/>
                                                </function>
                                                <variable symbol="x_0"/>
                                               </function>
                                               <function symbol="∪">
                                                <function symbol="∩">
                                                 <variable symbol="x_0"/>
                                                 <variable symbol="y_0"/>
                                                </function>
                                                <variable symbol="x_0"/>
                                               </function>
                                              </constantatomformula>
                                             </formulalist>
                                             <formulalist>
                                              <constantatomformula symbol="≤" definition="≤">
                                               <function symbol="∪">
                                                <function symbol="∩">
                                                 <variable symbol="x_0"/>
                                                 <variable symbol="y_0"/>
                                                </function>
                                                <variable symbol="x_0"/>
                                               </function>
                                               <variable symbol="x_0"/>
                                              </constantatomformula>
                                             </formulalist>
                                            </sequent>
                                            <rule type="axiom" symbol="axiom">
                                             <sequent>
                                              <formulalist>
                                               <constantatomformula symbol="=">
                                                <function symbol="∩">
                                                 <function symbol="∪">
                                                  <function symbol="∩">
                                                   <variable symbol="x_0"/>
                                                   <variable symbol="y_0"/>
                                                  </function>
                                                  <variable symbol="x_0"/>
                                                 </function>
                                                 <variable symbol="x_0"/>
                                                </function>
                                                <function symbol="∪">
                                                 <function symbol="∩">
                                                  <variable symbol="x_0"/>
                                                  <variable symbol="y_0"/>
                                                 </function>
                                                 <variable symbol="x_0"/>
                                                </function>
                                               </constantatomformula>
                                              </formulalist>
                                              <formulalist>
                                               <constantatomformula symbol="=">
                                                <function symbol="∩">
                                                 <function symbol="∪">
                                                  <function symbol="∩">
                                                   <variable symbol="x_0"/>
                                                   <variable symbol="y_0"/>
                                                  </function>
                                                  <variable symbol="x_0"/>
                                                 </function>
                                                 <variable symbol="x_0"/>
                                                </function>
                                                <function symbol="∪">
                                                 <function symbol="∩">
                                                  <variable symbol="x_0"/>
                                                  <variable symbol="y_0"/>
                                                 </function>
                                                 <variable symbol="x_0"/>
                                                </function>
                                               </constantatomformula>
                                              </formulalist>
                                             </sequent>
                                            </rule>
                                           </rule>
                                          </rule>
                                         </rule>
                                        </rule>
                                        <rule type="defl" symbol="d:l" param="();[(1)(1 1)(2)][(2)(1 2)];()">
                                         <sequent>
                                          <formulalist>
                                           <constantatomformula symbol="≤" definition="≤">
                                            <variable symbol="x_0"/>
                                            <function symbol="∪">
                                             <function symbol="∩">
                                              <variable symbol="x_0"/>
                                              <variable symbol="y_0"/>
                                             </function>
                                             <variable symbol="x_0"/>
                                            </function>
                                           </constantatomformula>
                                          </formulalist>
                                          <formulalist>
                                           <constantatomformula symbol="≤" definition="≤">
                                            <variable symbol="x_0"/>
                                            <function symbol="∪">
                                             <function symbol="∩">
                                              <variable symbol="x_0"/>
                                              <variable symbol="y_0"/>
                                             </function>
                                             <variable symbol="x_0"/>
                                            </function>
                                           </constantatomformula>
                                          </formulalist>
                                         </sequent>
                                         <rule type="defr" symbol="d:r" param="();[(1)(1 1)(2)][(2)(1 2)];()">
                                          <sequent>
                                           <formulalist>
                                            <constantatomformula symbol="=">
                                             <function symbol="∩">
                                              <variable symbol="x_0"/>
                                              <function symbol="∪">
                                               <function symbol="∩">
                                                <variable symbol="x_0"/>
                                                <variable symbol="y_0"/>
                                               </function>
                                               <variable symbol="x_0"/>
                                              </function>
                                             </function>
                                             <variable symbol="x_0"/>
                                            </constantatomformula>
                                           </formulalist>
                                           <formulalist>
                                            <constantatomformula symbol="≤" definition="≤">
                                             <variable symbol="x_0"/>
                                             <function symbol="∪">
                                              <function symbol="∩">
                                               <variable symbol="x_0"/>
                                               <variable symbol="y_0"/>
                                              </function>
                                              <variable symbol="x_0"/>
                                             </function>
                                            </constantatomformula>
                                           </formulalist>
                                          </sequent>
                                          <rule type="axiom" symbol="axiom">
                                           <sequent>
                                            <formulalist>
                                             <constantatomformula symbol="=">
                                              <function symbol="∩">
                                               <variable symbol="x_0"/>
                                               <function symbol="∪">
                                                <function symbol="∩">
                                                 <variable symbol="x_0"/>
                                                 <variable symbol="y_0"/>
                                                </function>
                                                <variable symbol="x_0"/>
                                               </function>
                                              </function>
                                              <variable symbol="x_0"/>
                                             </constantatomformula>
                                            </formulalist>
                                            <formulalist>
                                             <constantatomformula symbol="=">
                                              <function symbol="∩">
                                               <variable symbol="x_0"/>
                                               <function symbol="∪">
                                                <function symbol="∩">
                                                 <variable symbol="x_0"/>
                                                 <variable symbol="y_0"/>
                                                </function>
                                                <variable symbol="x_0"/>
                                               </function>
                                              </function>
                                              <variable symbol="x_0"/>
                                             </constantatomformula>
                                            </formulalist>
                                           </sequent>
                                          </rule>
                                         </rule>
                                        </rule>
                                       </rule>
                                      </rule>
                                     </rule>
                                    </rule>
                                   </rule>
                                  </rule>
                                 </rule>
                                </rule>
                               </rule>
                              </rule>
                             </rule>
                             <rule type="axiom" symbol="axiom">
                              <sequent>
                               <formulalist>
                                <constantatomformula symbol="=">
                                 <function symbol="∪">
                                  <function symbol="∩">
                                   <variable symbol="x_0"/>
                                   <variable symbol="y_0"/>
                                  </function>
                                  <variable symbol="x_0"/>
                                 </function>
                                 <variable symbol="x_0"/>
                                </constantatomformula>
                               </formulalist>
                               <formulalist>
                                <constantatomformula symbol="=">
                                 <function symbol="∪">
                                  <function symbol="∩">
                                   <variable symbol="x_0"/>
                                   <variable symbol="y_0"/>
                                  </function>
                                  <variable symbol="x_0"/>
                                 </function>
                                 <variable symbol="x_0"/>
                                </constantatomformula>
                               </formulalist>
                              </sequent>
                             </rule>
                            </rule>
                           </rule>
                          </rule>
                         </rule>
                        </rule>
                       </rule>
                       <rule type="forallr" symbol="\all:r">
                        <sequent>
                         <formulalist>
                          <constantatomformula symbol="AS" definition="AS"/>
                          <constantatomformula symbol="GLB" definition="GLB"/>
                          <constantatomformula symbol="LUB" definition="LUB"/>
                          <constantatomformula symbol="R" definition="R"/>
                         </formulalist>
                         <formulalist>
                          <quantifiedformula type="all">
                           <variable symbol="x"/>
                           <quantifiedformula type="all">
                            <variable symbol="y"/>
                            <constantatomformula symbol="=">
                             <function symbol="∩">
                              <function symbol="∪">
                               <variable symbol="x"/>
                               <variable symbol="y"/>
                              </function>
                              <variable symbol="x"/>
                             </function>
                             <variable symbol="x"/>
                            </constantatomformula>
                           </quantifiedformula>
                          </quantifiedformula>
                         </formulalist>
                        </sequent>
                        <rule type="forallr" symbol="\all:r">
                         <sequent>
                          <formulalist>
                           <constantatomformula symbol="AS" definition="AS"/>
                           <constantatomformula symbol="GLB" definition="GLB"/>
                           <constantatomformula symbol="LUB" definition="LUB"/>
                           <constantatomformula symbol="R" definition="R"/>
                          </formulalist>
                          <formulalist>
                           <quantifiedformula type="all">
                            <variable symbol="y"/>
                            <constantatomformula symbol="=">
                             <function symbol="∩">
                              <function symbol="∪">
                               <variable symbol="x_0"/>
                               <variable symbol="y"/>
                              </function>
                              <variable symbol="x_0"/>
                             </function>
                             <variable symbol="x_0"/>
                            </constantatomformula>
                           </quantifiedformula>
                          </formulalist>
                         </sequent>
                         <rule type="defl" symbol="d:l" param="();[()];()">
                          <sequent>
                           <formulalist>
                            <constantatomformula symbol="AS" definition="AS"/>
                            <constantatomformula symbol="GLB" definition="GLB"/>
                            <constantatomformula symbol="LUB" definition="LUB"/>
                            <constantatomformula symbol="R" definition="R"/>
                           </formulalist>
                           <formulalist>
                            <constantatomformula symbol="=">
                             <function symbol="∩">
                              <function symbol="∪">
                               <variable symbol="x_0"/>
                               <variable symbol="y_0"/>
                              </function>
                              <variable symbol="x_0"/>
                             </function>
                             <variable symbol="x_0"/>
                            </constantatomformula>
                           </formulalist>
                          </sequent>
                          <rule type="foralll" symbol="\all:l">
                           <sequent>
                            <formulalist>
                             <quantifiedformula type="all">
                              <variable symbol="x"/>
                              <quantifiedformula type="all">
                               <variable symbol="y"/>
                               <conjunctiveformula type="impl">
                                <conjunctiveformula type="and">
                                 <constantatomformula symbol="≤" definition="≤">
                                  <variable symbol="x"/>
                                  <variable symbol="y"/>
                                 </constantatomformula>
                                 <constantatomformula symbol="≤" definition="≤">
                                  <variable symbol="y"/>
                                  <variable symbol="x"/>
                                 </constantatomformula>
                                </conjunctiveformula>
                                <constantatomformula symbol="=">
                                 <variable symbol="x"/>
                                 <variable symbol="y"/>
                                </constantatomformula>
                               </conjunctiveformula>
                              </quantifiedformula>
                             </quantifiedformula>
                             <constantatomformula symbol="GLB" definition="GLB"/>
                             <constantatomformula symbol="LUB" definition="LUB"/>
                             <constantatomformula symbol="R" definition="R"/>
                            </formulalist>
                            <formulalist>
                             <constantatomformula symbol="=">
                              <function symbol="∩">
                               <function symbol="∪">
                                <variable symbol="x_0"/>
                                <variable symbol="y_0"/>
                               </function>
                               <variable symbol="x_0"/>
                              </function>
                              <variable symbol="x_0"/>
                             </constantatomformula>
                            </formulalist>
                           </sequent>
                           <rule type="foralll" symbol="\all:l">
                            <sequent>
                             <formulalist>
                              <quantifiedformula type="all">
                               <variable symbol="y"/>
                               <conjunctiveformula type="impl">
                                <conjunctiveformula type="and">
                                 <constantatomformula symbol="≤" definition="≤">
                                  <function symbol="∩">
                                   <function symbol="∪">
                                    <variable symbol="x_0"/>
                                    <variable symbol="y_0"/>
                                   </function>
                                   <variable symbol="x_0"/>
                                  </function>
                                  <variable symbol="y"/>
                                 </constantatomformula>
                                 <constantatomformula symbol="≤" definition="≤">
                                  <variable symbol="y"/>
                                  <function symbol="∩">
                                   <function symbol="∪">
                                    <variable symbol="x_0"/>
                                    <variable symbol="y_0"/>
                                   </function>
                                   <variable symbol="x_0"/>
                                  </function>
                                 </constantatomformula>
                                </conjunctiveformula>
                                <constantatomformula symbol="=">
                                 <function symbol="∩">
                                  <function symbol="∪">
                                   <variable symbol="x_0"/>
                                   <variable symbol="y_0"/>
                                  </function>
                                  <variable symbol="x_0"/>
                                 </function>
                                 <variable symbol="y"/>
                                </constantatomformula>
                               </conjunctiveformula>
                              </quantifiedformula>
                              <constantatomformula symbol="GLB" definition="GLB"/>
                              <constantatomformula symbol="LUB" definition="LUB"/>
                              <constantatomformula symbol="R" definition="R"/>
                             </formulalist>
                             <formulalist>
                              <constantatomformula symbol="=">
                               <function symbol="∩">
                                <function symbol="∪">
                                 <variable symbol="x_0"/>
                                 <variable symbol="y_0"/>
                                </function>
                                <variable symbol="x_0"/>
                               </function>
                               <variable symbol="x_0"/>
                              </constantatomformula>
                             </formulalist>
                            </sequent>
                            <rule type="impll" symbol="\impl:l">
                             <sequent>
                              <formulalist>
                               <conjunctiveformula type="impl">
                                <conjunctiveformula type="and">
                                 <constantatomformula symbol="≤" definition="≤">
                                  <function symbol="∩">
                                   <function symbol="∪">
                                    <variable symbol="x_0"/>
                                    <variable symbol="y_0"/>
                                   </function>
                                   <variable symbol="x_0"/>
                                  </function>
                                  <variable symbol="x_0"/>
                                 </constantatomformula>
                                 <constantatomformula symbol="≤" definition="≤">
                                  <variable symbol="x_0"/>
                                  <function symbol="∩">
                                   <function symbol="∪">
                                    <variable symbol="x_0"/>
                                    <variable symbol="y_0"/>
                                   </function>
                                   <variable symbol="x_0"/>
                                  </function>
                                 </constantatomformula>
                                </conjunctiveformula>
                                <constantatomformula symbol="=">
                                 <function symbol="∩">
                                  <function symbol="∪">
                                   <variable symbol="x_0"/>
                                   <variable symbol="y_0"/>
                                  </function>
                                  <variable symbol="x_0"/>
                                 </function>
                                 <variable symbol="x_0"/>
                                </constantatomformula>
                               </conjunctiveformula>
                               <constantatomformula symbol="GLB" definition="GLB"/>
                               <constantatomformula symbol="LUB" definition="LUB"/>
                               <constantatomformula symbol="R" definition="R"/>
                              </formulalist>
                              <formulalist>
                               <constantatomformula symbol="=">
                                <function symbol="∩">
                                 <function symbol="∪">
                                  <variable symbol="x_0"/>
                                  <variable symbol="y_0"/>
                                 </function>
                                 <variable symbol="x_0"/>
                                </function>
                                <variable symbol="x_0"/>
                               </constantatomformula>
                              </formulalist>
                             </sequent>
                             <rule type="defl" symbol="d:l" param="();[()];()">
                              <sequent>
                               <formulalist>
                                <constantatomformula symbol="GLB" definition="GLB"/>
                                <constantatomformula symbol="LUB" definition="LUB"/>
                                <constantatomformula symbol="R" definition="R"/>
                               </formulalist>
                               <formulalist>
                                <conjunctiveformula type="and">
                                 <constantatomformula symbol="≤" definition="≤">
                                  <function symbol="∩">
                                   <function symbol="∪">
                                    <variable symbol="x_0"/>
                                    <variable symbol="y_0"/>
                                   </function>
                                   <variable symbol="x_0"/>
                                  </function>
                                  <variable symbol="x_0"/>
                                 </constantatomformula>
                                 <constantatomformula symbol="≤" definition="≤">
                                  <variable symbol="x_0"/>
                                  <function symbol="∩">
                                   <function symbol="∪">
                                    <variable symbol="x_0"/>
                                    <variable symbol="y_0"/>
                                   </function>
                                   <variable symbol="x_0"/>
                                  </function>
                                 </constantatomformula>
                                </conjunctiveformula>
                               </formulalist>
                              </sequent>
                              <rule type="foralll" symbol="\all:l">
                               <sequent>
                                <formulalist>
                                 <quantifiedformula type="all">
                                  <variable symbol="x"/>
                                  <quantifiedformula type="all">
                                   <variable symbol="y"/>
                                   <conjunctiveformula type="and">
                                    <conjunctiveformula type="and">
                                     <constantatomformula symbol="≤" definition="≤">
                                      <function symbol="∩">
                                       <variable symbol="x"/>
                                       <variable symbol="y"/>
                                      </function>
                                      <variable symbol="x"/>
                                     </constantatomformula>
                                     <constantatomformula symbol="≤" definition="≤">
                                      <function symbol="∩">
                                       <variable symbol="x"/>
                                       <variable symbol="y"/>
                                      </function>
                                      <variable symbol="y"/>
                                     </constantatomformula>
                                    </conjunctiveformula>
                                    <quantifiedformula type="all">
                                     <variable symbol="z"/>
                                     <conjunctiveformula type="impl">
                                      <conjunctiveformula type="and">
                                       <constantatomformula symbol="≤" definition="≤">
                                        <variable symbol="z"/>
                                        <variable symbol="x"/>
                                       </constantatomformula>
                                       <constantatomformula symbol="≤" definition="≤">
                                        <variable symbol="z"/>
                                        <variable symbol="y"/>
                                       </constantatomformula>
                                      </conjunctiveformula>
                                      <constantatomformula symbol="≤" definition="≤">
                                       <variable symbol="z"/>
                                       <function symbol="∩">
                                        <variable symbol="x"/>
                                        <variable symbol="y"/>
                                       </function>
                                      </constantatomformula>
                                     </conjunctiveformula>
                                    </quantifiedformula>
                                   </conjunctiveformula>
                                  </quantifiedformula>
                                 </quantifiedformula>
                                 <constantatomformula symbol="LUB" definition="LUB"/>
                                 <constantatomformula symbol="R" definition="R"/>
                                </formulalist>
                                <formulalist>
                                 <conjunctiveformula type="and">
                                  <constantatomformula symbol="≤" definition="≤">
                                   <function symbol="∩">
                                    <function symbol="∪">
                                     <variable symbol="x_0"/>
                                     <variable symbol="y_0"/>
                                    </function>
                                    <variable symbol="x_0"/>
                                   </function>
                                   <variable symbol="x_0"/>
                                  </constantatomformula>
                                  <constantatomformula symbol="≤" definition="≤">
                                   <variable symbol="x_0"/>
                                   <function symbol="∩">
                                    <function symbol="∪">
                                     <variable symbol="x_0"/>
                                     <variable symbol="y_0"/>
                                    </function>
                                    <variable symbol="x_0"/>
                                   </function>
                                  </constantatomformula>
                                 </conjunctiveformula>
                                </formulalist>
                               </sequent>
                               <rule type="foralll" symbol="\all:l">
                                <sequent>
                                 <formulalist>
                                  <quantifiedformula type="all">
                                   <variable symbol="y"/>
                                   <conjunctiveformula type="and">
                                    <conjunctiveformula type="and">
                                     <constantatomformula symbol="≤" definition="≤">
                                      <function symbol="∩">
                                       <function symbol="∪">
                                        <variable symbol="x_0"/>
                                        <variable symbol="y_0"/>
                                       </function>
                                       <variable symbol="y"/>
                                      </function>
                                      <function symbol="∪">
                                       <variable symbol="x_0"/>
                                       <variable symbol="y_0"/>
                                      </function>
                                     </constantatomformula>
                                     <constantatomformula symbol="≤" definition="≤">
                                      <function symbol="∩">
                                       <function symbol="∪">
                                        <variable symbol="x_0"/>
                                        <variable symbol="y_0"/>
                                       </function>
                                       <variable symbol="y"/>
                                      </function>
                                      <variable symbol="y"/>
                                     </constantatomformula>
                                    </conjunctiveformula>
                                    <quantifiedformula type="all">
                                     <variable symbol="z"/>
                                     <conjunctiveformula type="impl">
                                      <conjunctiveformula type="and">
                                       <constantatomformula symbol="≤" definition="≤">
                                        <variable symbol="z"/>
                                        <function symbol="∪">
                                         <variable symbol="x_0"/>
                                         <variable symbol="y_0"/>
                                        </function>
                                       </constantatomformula>
                                       <constantatomformula symbol="≤" definition="≤">
                                        <variable symbol="z"/>
                                        <variable symbol="y"/>
                                       </constantatomformula>
                                      </conjunctiveformula>
                                      <constantatomformula symbol="≤" definition="≤">
                                       <variable symbol="z"/>
                                       <function symbol="∩">
                                        <function symbol="∪">
                                         <variable symbol="x_0"/>
                                         <variable symbol="y_0"/>
                                        </function>
                                        <variable symbol="y"/>
                                       </function>
                                      </constantatomformula>
                                     </conjunctiveformula>
                                    </quantifiedformula>
                                   </conjunctiveformula>
                                  </quantifiedformula>
                                  <constantatomformula symbol="LUB" definition="LUB"/>
                                  <constantatomformula symbol="R" definition="R"/>
                                 </formulalist>
                                 <formulalist>
                                  <conjunctiveformula type="and">
                                   <constantatomformula symbol="≤" definition="≤">
                                    <function symbol="∩">
                                     <function symbol="∪">
                                      <variable symbol="x_0"/>
                                      <variable symbol="y_0"/>
                                     </function>
                                     <variable symbol="x_0"/>
                                    </function>
                                    <variable symbol="x_0"/>
                                   </constantatomformula>
                                   <constantatomformula symbol="≤" definition="≤">
                                    <variable symbol="x_0"/>
                                    <function symbol="∩">
                                     <function symbol="∪">
                                      <variable symbol="x_0"/>
                                      <variable symbol="y_0"/>
                                     </function>
                                     <variable symbol="x_0"/>
                                    </function>
                                   </constantatomformula>
                                  </conjunctiveformula>
                                 </formulalist>
                                </sequent>
                                <rule type="contrl" symbol="c:l" param="2,1,1">
                                 <sequent>
                                  <formulalist>
                                   <conjunctiveformula type="and">
                                    <conjunctiveformula type="and">
                                     <constantatomformula symbol="≤" definition="≤">
                                      <function symbol="∩">
                                       <function symbol="∪">
                                        <variable symbol="x_0"/>
                                        <variable symbol="y_0"/>
                                       </function>
                                       <variable symbol="x_0"/>
                                      </function>
                                      <function symbol="∪">
                                       <variable symbol="x_0"/>
                                       <variable symbol="y_0"/>
                                      </function>
                                     </constantatomformula>
                                     <constantatomformula symbol="≤" definition="≤">
                                      <function symbol="∩">
                                       <function symbol="∪">
                                        <variable symbol="x_0"/>
                                        <variable symbol="y_0"/>
                                       </function>
                                       <variable symbol="x_0"/>
                                      </function>
                                      <variable symbol="x_0"/>
                                     </constantatomformula>
                                    </conjunctiveformula>
                                    <quantifiedformula type="all">
                                     <variable symbol="z"/>
                                     <conjunctiveformula type="impl">
                                      <conjunctiveformula type="and">
                                       <constantatomformula symbol="≤" definition="≤">
                                        <variable symbol="z"/>
                                        <function symbol="∪">
                                         <variable symbol="x_0"/>
                                         <variable symbol="y_0"/>
                                        </function>
                                       </constantatomformula>
                                       <constantatomformula symbol="≤" definition="≤">
                                        <variable symbol="z"/>
                                        <variable symbol="x_0"/>
                                       </constantatomformula>
                                      </conjunctiveformula>
                                      <constantatomformula symbol="≤" definition="≤">
                                       <variable symbol="z"/>
                                       <function symbol="∩">
                                        <function symbol="∪">
                                         <variable symbol="x_0"/>
                                         <variable symbol="y_0"/>
                                        </function>
                                        <variable symbol="x_0"/>
                                       </function>
                                      </constantatomformula>
                                     </conjunctiveformula>
                                    </quantifiedformula>
                                   </conjunctiveformula>
                                   <constantatomformula symbol="LUB" definition="LUB"/>
                                   <constantatomformula symbol="R" definition="R"/>
                                  </formulalist>
                                  <formulalist>
                                   <conjunctiveformula type="and">
                                    <constantatomformula symbol="≤" definition="≤">
                                     <function symbol="∩">
                                      <function symbol="∪">
                                       <variable symbol="x_0"/>
                                       <variable symbol="y_0"/>
                                      </function>
                                      <variable symbol="x_0"/>
                                     </function>
                                     <variable symbol="x_0"/>
                                    </constantatomformula>
                                    <constantatomformula symbol="≤" definition="≤">
                                     <variable symbol="x_0"/>
                                     <function symbol="∩">
                                      <function symbol="∪">
                                       <variable symbol="x_0"/>
                                       <variable symbol="y_0"/>
                                      </function>
                                      <variable symbol="x_0"/>
                                     </function>
                                    </constantatomformula>
                                   </conjunctiveformula>
                                  </formulalist>
                                 </sequent>
                                 <rule type="andl2" symbol="\and:l2">
                                  <sequent>
                                   <formulalist>
                                    <conjunctiveformula type="and">
                                     <conjunctiveformula type="and">
                                      <constantatomformula symbol="≤" definition="≤">
                                       <function symbol="∩">
                                        <function symbol="∪">
                                         <variable symbol="x_0"/>
                                         <variable symbol="y_0"/>
                                        </function>
                                        <variable symbol="x_0"/>
                                       </function>
                                       <function symbol="∪">
                                        <variable symbol="x_0"/>
                                        <variable symbol="y_0"/>
                                       </function>
                                      </constantatomformula>
                                      <constantatomformula symbol="≤" definition="≤">
                                       <function symbol="∩">
                                        <function symbol="∪">
                                         <variable symbol="x_0"/>
                                         <variable symbol="y_0"/>
                                        </function>
                                        <variable symbol="x_0"/>
                                       </function>
                                       <variable symbol="x_0"/>
                                      </constantatomformula>
                                     </conjunctiveformula>
                                     <quantifiedformula type="all">
                                      <variable symbol="z"/>
                                      <conjunctiveformula type="impl">
                                       <conjunctiveformula type="and">
                                        <constantatomformula symbol="≤" definition="≤">
                                         <variable symbol="z"/>
                                         <function symbol="∪">
                                          <variable symbol="x_0"/>
                                          <variable symbol="y_0"/>
                                         </function>
                                        </constantatomformula>
                                        <constantatomformula symbol="≤" definition="≤">
                                         <variable symbol="z"/>
                                         <variable symbol="x_0"/>
                                        </constantatomformula>
                                       </conjunctiveformula>
                                       <constantatomformula symbol="≤" definition="≤">
                                        <variable symbol="z"/>
                                        <function symbol="∩">
                                         <function symbol="∪">
                                          <variable symbol="x_0"/>
                                          <variable symbol="y_0"/>
                                         </function>
                                         <variable symbol="x_0"/>
                                        </function>
                                       </constantatomformula>
                                      </conjunctiveformula>
                                     </quantifiedformula>
                                    </conjunctiveformula>
                                    <conjunctiveformula type="and">
                                     <conjunctiveformula type="and">
                                      <constantatomformula symbol="≤" definition="≤">
                                       <function symbol="∩">
                                        <function symbol="∪">
                                         <variable symbol="x_0"/>
                                         <variable symbol="y_0"/>
                                        </function>
                                        <variable symbol="x_0"/>
                                       </function>
                                       <function symbol="∪">
                                        <variable symbol="x_0"/>
                                        <variable symbol="y_0"/>
                                       </function>
                                      </constantatomformula>
                                      <constantatomformula symbol="≤" definition="≤">
                                       <function symbol="∩">
                                        <function symbol="∪">
                                         <variable symbol="x_0"/>
                                         <variable symbol="y_0"/>
                                        </function>
                                        <variable symbol="x_0"/>
                                       </function>
                                       <variable symbol="x_0"/>
                                      </constantatomformula>
                                     </conjunctiveformula>
                                     <quantifiedformula type="all">
                                      <variable symbol="z"/>
                                      <conjunctiveformula type="impl">
                                       <conjunctiveformula type="and">
                                        <constantatomformula symbol="≤" definition="≤">
                                         <variable symbol="z"/>
                                         <function symbol="∪">
                                          <variable symbol="x_0"/>
                                          <variable symbol="y_0"/>
                                         </function>
                                        </constantatomformula>
                                        <constantatomformula symbol="≤" definition="≤">
                                         <variable symbol="z"/>
                                         <variable symbol="x_0"/>
                                        </constantatomformula>
                                       </conjunctiveformula>
                                       <constantatomformula symbol="≤" definition="≤">
                                        <variable symbol="z"/>
                                        <function symbol="∩">
                                         <function symbol="∪">
                                          <variable symbol="x_0"/>
                                          <variable symbol="y_0"/>
                                         </function>
                                         <variable symbol="x_0"/>
                                        </function>
                                       </constantatomformula>
                                      </conjunctiveformula>
                                     </quantifiedformula>
                                    </conjunctiveformula>
                                    <constantatomformula symbol="LUB" definition="LUB"/>
                                    <constantatomformula symbol="R" definition="R"/>
                                   </formulalist>
                                   <formulalist>
                                    <conjunctiveformula type="and">
                                     <constantatomformula symbol="≤" definition="≤">
                                      <function symbol="∩">
                                       <function symbol="∪">
                                        <variable symbol="x_0"/>
                                        <variable symbol="y_0"/>
                                       </function>
                                       <variable symbol="x_0"/>
                                      </function>
                                      <variable symbol="x_0"/>
                                     </constantatomformula>
                                     <constantatomformula symbol="≤" definition="≤">
                                      <variable symbol="x_0"/>
                                      <function symbol="∩">
                                       <function symbol="∪">
                                        <variable symbol="x_0"/>
                                        <variable symbol="y_0"/>
                                       </function>
                                       <variable symbol="x_0"/>
                                      </function>
                                     </constantatomformula>
                                    </conjunctiveformula>
                                   </formulalist>
                                  </sequent>
                                  <rule type="perml" symbol="\pi:l" param="(1 2)">
                                   <sequent>
                                    <formulalist>
                                     <quantifiedformula type="all">
                                      <variable symbol="z"/>
                                      <conjunctiveformula type="impl">
                                       <conjunctiveformula type="and">
                                        <constantatomformula symbol="≤" definition="≤">
                                         <variable symbol="z"/>
                                         <function symbol="∪">
                                          <variable symbol="x_0"/>
                                          <variable symbol="y_0"/>
                                         </function>
                                        </constantatomformula>
                                        <constantatomformula symbol="≤" definition="≤">
                                         <variable symbol="z"/>
                                         <variable symbol="x_0"/>
                                        </constantatomformula>
                                       </conjunctiveformula>
                                       <constantatomformula symbol="≤" definition="≤">
                                        <variable symbol="z"/>
                                        <function symbol="∩">
                                         <function symbol="∪">
                                          <variable symbol="x_0"/>
                                          <variable symbol="y_0"/>
                                         </function>
                                         <variable symbol="x_0"/>
                                        </function>
                                       </constantatomformula>
                                      </conjunctiveformula>
                                     </quantifiedformula>
                                     <conjunctiveformula type="and">
                                      <conjunctiveformula type="and">
                                       <constantatomformula symbol="≤" definition="≤">
                                        <function symbol="∩">
                                         <function symbol="∪">
                                          <variable symbol="x_0"/>
                                          <variable symbol="y_0"/>
                                         </function>
                                         <variable symbol="x_0"/>
                                        </function>
                                        <function symbol="∪">
                                         <variable symbol="x_0"/>
                                         <variable symbol="y_0"/>
                                        </function>
                                       </constantatomformula>
                                       <constantatomformula symbol="≤" definition="≤">
                                        <function symbol="∩">
                                         <function symbol="∪">
                                          <variable symbol="x_0"/>
                                          <variable symbol="y_0"/>
                                         </function>
                                         <variable symbol="x_0"/>
                                        </function>
                                        <variable symbol="x_0"/>
                                       </constantatomformula>
                                      </conjunctiveformula>
                                      <quantifiedformula type="all">
                                       <variable symbol="z"/>
                                       <conjunctiveformula type="impl">
                                        <conjunctiveformula type="and">
                                         <constantatomformula symbol="≤" definition="≤">
                                          <variable symbol="z"/>
                                          <function symbol="∪">
                                           <variable symbol="x_0"/>
                                           <variable symbol="y_0"/>
                                          </function>
                                         </constantatomformula>
                                         <constantatomformula symbol="≤" definition="≤">
                                          <variable symbol="z"/>
                                          <variable symbol="x_0"/>
                                         </constantatomformula>
                                        </conjunctiveformula>
                                        <constantatomformula symbol="≤" definition="≤">
                                         <variable symbol="z"/>
                                         <function symbol="∩">
                                          <function symbol="∪">
                                           <variable symbol="x_0"/>
                                           <variable symbol="y_0"/>
                                          </function>
                                          <variable symbol="x_0"/>
                                         </function>
                                        </constantatomformula>
                                       </conjunctiveformula>
                                      </quantifiedformula>
                                     </conjunctiveformula>
                                     <constantatomformula symbol="LUB" definition="LUB"/>
                                     <constantatomformula symbol="R" definition="R"/>
                                    </formulalist>
                                    <formulalist>
                                     <conjunctiveformula type="and">
                                      <constantatomformula symbol="≤" definition="≤">
                                       <function symbol="∩">
                                        <function symbol="∪">
                                         <variable symbol="x_0"/>
                                         <variable symbol="y_0"/>
                                        </function>
                                        <variable symbol="x_0"/>
                                       </function>
                                       <variable symbol="x_0"/>
                                      </constantatomformula>
                                      <constantatomformula symbol="≤" definition="≤">
                                       <variable symbol="x_0"/>
                                       <function symbol="∩">
                                        <function symbol="∪">
                                         <variable symbol="x_0"/>
                                         <variable symbol="y_0"/>
                                        </function>
                                        <variable symbol="x_0"/>
                                       </function>
                                      </constantatomformula>
                                     </conjunctiveformula>
                                    </formulalist>
                                   </sequent>
                                   <rule type="andl1" symbol="\and:l1">
                                    <sequent>
                                     <formulalist>
                                      <conjunctiveformula type="and">
                                       <conjunctiveformula type="and">
                                        <constantatomformula symbol="≤" definition="≤">
                                         <function symbol="∩">
                                          <function symbol="∪">
                                           <variable symbol="x_0"/>
                                           <variable symbol="y_0"/>
                                          </function>
                                          <variable symbol="x_0"/>
                                         </function>
                                         <function symbol="∪">
                                          <variable symbol="x_0"/>
                                          <variable symbol="y_0"/>
                                         </function>
                                        </constantatomformula>
                                        <constantatomformula symbol="≤" definition="≤">
                                         <function symbol="∩">
                                          <function symbol="∪">
                                           <variable symbol="x_0"/>
                                           <variable symbol="y_0"/>
                                          </function>
                                          <variable symbol="x_0"/>
                                         </function>
                                         <variable symbol="x_0"/>
                                        </constantatomformula>
                                       </conjunctiveformula>
                                       <quantifiedformula type="all">
                                        <variable symbol="z"/>
                                        <conjunctiveformula type="impl">
                                         <conjunctiveformula type="and">
                                          <constantatomformula symbol="≤" definition="≤">
                                           <variable symbol="z"/>
                                           <function symbol="∪">
                                            <variable symbol="x_0"/>
                                            <variable symbol="y_0"/>
                                           </function>
                                          </constantatomformula>
                                          <constantatomformula symbol="≤" definition="≤">
                                           <variable symbol="z"/>
                                           <variable symbol="x_0"/>
                                          </constantatomformula>
                                         </conjunctiveformula>
                                         <constantatomformula symbol="≤" definition="≤">
                                          <variable symbol="z"/>
                                          <function symbol="∩">
                                           <function symbol="∪">
                                            <variable symbol="x_0"/>
                                            <variable symbol="y_0"/>
                                           </function>
                                           <variable symbol="x_0"/>
                                          </function>
                                         </constantatomformula>
                                        </conjunctiveformula>
                                       </quantifiedformula>
                                      </conjunctiveformula>
                                      <quantifiedformula type="all">
                                       <variable symbol="z"/>
                                       <conjunctiveformula type="impl">
                                        <conjunctiveformula type="and">
                                         <constantatomformula symbol="≤" definition="≤">
                                          <variable symbol="z"/>
                                          <function symbol="∪">
                                           <variable symbol="x_0"/>
                                           <variable symbol="y_0"/>
                                          </function>
                                         </constantatomformula>
                                         <constantatomformula symbol="≤" definition="≤">
                                          <variable symbol="z"/>
                                          <variable symbol="x_0"/>
                                         </constantatomformula>
                                        </conjunctiveformula>
                                        <constantatomformula symbol="≤" definition="≤">
                                         <variable symbol="z"/>
                                         <function symbol="∩">
                                          <function symbol="∪">
                                           <variable symbol="x_0"/>
                                           <variable symbol="y_0"/>
                                          </function>
                                          <variable symbol="x_0"/>
                                         </function>
                                        </constantatomformula>
                                       </conjunctiveformula>
                                      </quantifiedformula>
                                      <constantatomformula symbol="LUB" definition="LUB"/>
                                      <constantatomformula symbol="R" definition="R"/>
                                     </formulalist>
                                     <formulalist>
                                      <conjunctiveformula type="and">
                                       <constantatomformula symbol="≤" definition="≤">
                                        <function symbol="∩">
                                         <function symbol="∪">
                                          <variable symbol="x_0"/>
                                          <variable symbol="y_0"/>
                                         </function>
                                         <variable symbol="x_0"/>
                                        </function>
                                        <variable symbol="x_0"/>
                                       </constantatomformula>
                                       <constantatomformula symbol="≤" definition="≤">
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                                        <function symbol="∩">
                                         <function symbol="∪">
                                          <variable symbol="x_0"/>
                                          <variable symbol="y_0"/>
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                                         <variable symbol="x_0"/>
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                                       </constantatomformula>
                                      </conjunctiveformula>
                                     </formulalist>
                                    </sequent>
                                    <rule type="andl2" symbol="\and:l2">
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                                      <formulalist>
                                       <conjunctiveformula type="and">
                                        <constantatomformula symbol="≤" definition="≤">
                                         <function symbol="∩">
                                          <function symbol="∪">
                                           <variable symbol="x_0"/>
                                           <variable symbol="y_0"/>
                                          </function>
                                          <variable symbol="x_0"/>
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                                         <function symbol="∪">
                                          <variable symbol="x_0"/>
                                          <variable symbol="y_0"/>
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                                        </constantatomformula>
                                        <constantatomformula symbol="≤" definition="≤">
                                         <function symbol="∩">
                                          <function symbol="∪">
                                           <variable symbol="x_0"/>
                                           <variable symbol="y_0"/>
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                                          <variable symbol="x_0"/>
                                         </function>
                                         <variable symbol="x_0"/>
                                        </constantatomformula>
                                       </conjunctiveformula>
                                       <quantifiedformula type="all">
                                        <variable symbol="z"/>
                                        <conjunctiveformula type="impl">
                                         <conjunctiveformula type="and">
                                          <constantatomformula symbol="≤" definition="≤">
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                                           <function symbol="∪">
                                            <variable symbol="x_0"/>
                                            <variable symbol="y_0"/>
                                           </function>
                                          </constantatomformula>
                                          <constantatomformula symbol="≤" definition="≤">
                                           <variable symbol="z"/>
                                           <variable symbol="x_0"/>
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                                         </conjunctiveformula>
                                         <constantatomformula symbol="≤" definition="≤">
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                                           <function symbol="∪">
                                            <variable symbol="x_0"/>
                                            <variable symbol="y_0"/>
                                           </function>
                                           <variable symbol="x_0"/>
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                                         </constantatomformula>
                                        </conjunctiveformula>
                                       </quantifiedformula>
                                       <constantatomformula symbol="LUB" definition="LUB"/>
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                                      <formulalist>
                                       <conjunctiveformula type="and">
                                        <constantatomformula symbol="≤" definition="≤">
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                                          <function symbol="∪">
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                                           <variable symbol="y_0"/>
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                                          <variable symbol="x_0"/>
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                                         <variable symbol="x_0"/>
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                                        <constantatomformula symbol="≤" definition="≤">
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                                         <function symbol="∩">
                                          <function symbol="∪">
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                                           <variable symbol="y_0"/>
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                                          <variable symbol="x_0"/>
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                                        </constantatomformula>
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                                     <rule type="andr" symbol="\and:r">
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                                       <formulalist>
                                        <constantatomformula symbol="≤" definition="≤">
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                                          <function symbol="∪">
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                                         <variable symbol="x_0"/>
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                                        <quantifiedformula type="all">
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                                             <variable symbol="y_0"/>
                                            </function>
                                           </constantatomformula>
                                           <constantatomformula symbol="≤" definition="≤">
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                                            <variable symbol="x_0"/>
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                                          <constantatomformula symbol="≤" definition="≤">
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                                             <variable symbol="y_0"/>
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                                            <variable symbol="x_0"/>
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                                        <constantatomformula symbol="LUB" definition="LUB"/>
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                                       <formulalist>
                                        <conjunctiveformula type="and">
                                         <constantatomformula symbol="≤" definition="≤">
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                                            <variable symbol="x_0"/>
                                            <variable symbol="y_0"/>
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                                          <variable symbol="x_0"/>
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                                         <constantatomformula symbol="≤" definition="≤">
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                                            <variable symbol="y_0"/>
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                                           <variable symbol="x_0"/>
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                                         </constantatomformula>
                                        </conjunctiveformula>
                                       </formulalist>
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                                      <rule type="defl" symbol="d:l" param="();[(1)(1 1)(2)][(2)(1 2)];()">
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                                        <formulalist>
                                         <constantatomformula symbol="≤" definition="≤">
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                                            <variable symbol="y_0"/>
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                                          <variable symbol="x_0"/>
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                                        <formulalist>
                                         <constantatomformula symbol="≤" definition="≤">
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                                            <variable symbol="y_0"/>
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                                        <rule type="axiom" symbol="axiom">
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                                              <variable symbol="y_0"/>
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                                        </rule>
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                                      </rule>
                                      <rule type="foralll" symbol="\all:l">
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                                              <variable symbol="x_0"/>
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                                              <variable symbol="y_0"/>
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                                             <variable symbol="x_0"/>
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                                        <formulalist>
                                         <constantatomformula symbol="≤" definition="≤">
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                                            <variable symbol="y_0"/>
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                                        </formulalist>
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                                       <rule type="impll" symbol="\impl:l">
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                                              <variable symbol="y_0"/>
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                                         <formulalist>
                                          <constantatomformula symbol="≤" definition="≤">
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                                            <function symbol="∪">
                                             <variable symbol="x_0"/>
                                             <variable symbol="y_0"/>
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                                          <formulalist>
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                                              <variable symbol="y_0"/>
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                                            <constantatomformula symbol="≤" definition="≤">
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                                             <variable symbol="x_0"/>
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                                          </formulalist>
                                         </sequent>
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                                           <formulalist>
                                            <constantatomformula symbol="LUB" definition="LUB"/>
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                                           <formulalist>
                                            <constantatomformula symbol="≤" definition="≤">
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                                              <variable symbol="x_0"/>
                                              <variable symbol="y_0"/>
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                                           </formulalist>
                                          </sequent>
                                          <rule type="foralll" symbol="\all:l">
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                                                 <constantatomformula symbol="≤" definition="≤">
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                                                   <variable symbol="x"/>
                                                   <variable symbol="y"/>
                                                  </function>
                                                 </constantatomformula>
                                                 <constantatomformula symbol="≤" definition="≤">
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                                                   <variable symbol="x"/>
                                                   <variable symbol="y"/>
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                                                 </constantatomformula>
                                                </conjunctiveformula>
                                                <quantifiedformula type="all">
                                                 <variable symbol="z"/>
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                                                  <conjunctiveformula type="and">
                                                   <constantatomformula symbol="≤" definition="≤">
                                                    <variable symbol="x"/>
                                                    <variable symbol="z"/>
                                                   </constantatomformula>
                                                   <constantatomformula symbol="≤" definition="≤">
                                                    <variable symbol="y"/>
                                                    <variable symbol="z"/>
                                                   </constantatomformula>
                                                  </conjunctiveformula>
                                                  <constantatomformula symbol="≤" definition="≤">
                                                   <function symbol="∪">
                                                    <variable symbol="x"/>
                                                    <variable symbol="y"/>
                                                   </function>
                                                   <variable symbol="z"/>
                                                  </constantatomformula>
                                                 </conjunctiveformula>
                                                </quantifiedformula>
                                               </conjunctiveformula>
                                              </quantifiedformula>
                                             </quantifiedformula>
                                            </formulalist>
                                            <formulalist>
                                             <constantatomformula symbol="≤" definition="≤">
                                              <variable symbol="x_0"/>
                                              <function symbol="∪">
                                               <variable symbol="x_0"/>
                                               <variable symbol="y_0"/>
                                              </function>
                                             </constantatomformula>
                                            </formulalist>
                                           </sequent>
                                           <rule type="foralll" symbol="\all:l">
                                            <sequent>
                                             <formulalist>
                                              <quantifiedformula type="all">
                                               <variable symbol="y"/>
                                               <conjunctiveformula type="and">
                                                <conjunctiveformula type="and">
                                                 <constantatomformula symbol="≤" definition="≤">
                                                  <variable symbol="x_0"/>
                                                  <function symbol="∪">
                                                   <variable symbol="x_0"/>
                                                   <variable symbol="y"/>
                                                  </function>
                                                 </constantatomformula>
                                                 <constantatomformula symbol="≤" definition="≤">
                                                  <variable symbol="y"/>
                                                  <function symbol="∪">
                                                   <variable symbol="x_0"/>
                                                   <variable symbol="y"/>
                                                  </function>
                                                 </constantatomformula>
                                                </conjunctiveformula>
                                                <quantifiedformula type="all">
                                                 <variable symbol="z"/>
                                                 <conjunctiveformula type="impl">
                                                  <conjunctiveformula type="and">
                                                   <constantatomformula symbol="≤" definition="≤">
                                                    <variable symbol="x_0"/>
                                                    <variable symbol="z"/>
                                                   </constantatomformula>
                                                   <constantatomformula symbol="≤" definition="≤">
                                                    <variable symbol="y"/>
                                                    <variable symbol="z"/>
                                                   </constantatomformula>
                                                  </conjunctiveformula>
                                                  <constantatomformula symbol="≤" definition="≤">
                                                   <function symbol="∪">
                                                    <variable symbol="x_0"/>
                                                    <variable symbol="y"/>
                                                   </function>
                                                   <variable symbol="z"/>
                                                  </constantatomformula>
                                                 </conjunctiveformula>
                                                </quantifiedformula>
                                               </conjunctiveformula>
                                              </quantifiedformula>
                                             </formulalist>
                                             <formulalist>
                                              <constantatomformula symbol="≤" definition="≤">
                                               <variable symbol="x_0"/>
                                               <function symbol="∪">
                                                <variable symbol="x_0"/>
                                                <variable symbol="y_0"/>
                                               </function>
                                              </constantatomformula>
                                             </formulalist>
                                            </sequent>
                                            <rule type="andl1" symbol="\and:l1">
                                             <sequent>
                                              <formulalist>
                                               <conjunctiveformula type="and">
                                                <conjunctiveformula type="and">
                                                 <constantatomformula symbol="≤" definition="≤">
                                                  <variable symbol="x_0"/>
                                                  <function symbol="∪">
                                                   <variable symbol="x_0"/>
                                                   <variable symbol="y_0"/>
                                                  </function>
                                                 </constantatomformula>
                                                 <constantatomformula symbol="≤" definition="≤">
                                                  <variable symbol="y_0"/>
                                                  <function symbol="∪">
                                                   <variable symbol="x_0"/>
                                                   <variable symbol="y_0"/>
                                                  </function>
                                                 </constantatomformula>
                                                </conjunctiveformula>
                                                <quantifiedformula type="all">
                                                 <variable symbol="z"/>
                                                 <conjunctiveformula type="impl">
                                                  <conjunctiveformula type="and">
                                                   <constantatomformula symbol="≤" definition="≤">
                                                    <variable symbol="x_0"/>
                                                    <variable symbol="z"/>
                                                   </constantatomformula>
                                                   <constantatomformula symbol="≤" definition="≤">
                                                    <variable symbol="y_0"/>
                                                    <variable symbol="z"/>
                                                   </constantatomformula>
                                                  </conjunctiveformula>
                                                  <constantatomformula symbol="≤" definition="≤">
                                                   <function symbol="∪">
                                                    <variable symbol="x_0"/>
                                                    <variable symbol="y_0"/>
                                                   </function>
                                                   <variable symbol="z"/>
                                                  </constantatomformula>
                                                 </conjunctiveformula>
                                                </quantifiedformula>
                                               </conjunctiveformula>
                                              </formulalist>
                                              <formulalist>
                                               <constantatomformula symbol="≤" definition="≤">
                                                <variable symbol="x_0"/>
                                                <function symbol="∪">
                                                 <variable symbol="x_0"/>
                                                 <variable symbol="y_0"/>
                                                </function>
                                               </constantatomformula>
                                              </formulalist>
                                             </sequent>
                                             <rule type="andl1" symbol="\and:l1">
                                              <sequent>
                                               <formulalist>
                                                <conjunctiveformula type="and">
                                                 <constantatomformula symbol="≤" definition="≤">
                                                  <variable symbol="x_0"/>
                                                  <function symbol="∪">
                                                   <variable symbol="x_0"/>
                                                   <variable symbol="y_0"/>
                                                  </function>
                                                 </constantatomformula>
                                                 <constantatomformula symbol="≤" definition="≤">
                                                  <variable symbol="y_0"/>
                                                  <function symbol="∪">
                                                   <variable symbol="x_0"/>
                                                   <variable symbol="y_0"/>
                                                  </function>
                                                 </constantatomformula>
                                                </conjunctiveformula>
                                               </formulalist>
                                               <formulalist>
                                                <constantatomformula symbol="≤" definition="≤">
                                                 <variable symbol="x_0"/>
                                                 <function symbol="∪">
                                                  <variable symbol="x_0"/>
                                                  <variable symbol="y_0"/>
                                                 </function>
                                                </constantatomformula>
                                               </formulalist>
                                              </sequent>
                                              <rule type="defl" symbol="d:l" param="();[(1)(1 1)(2)][(2)(1 2)];()">
                                               <sequent>
                                                <formulalist>
                                                 <constantatomformula symbol="≤" definition="≤">
                                                  <variable symbol="x_0"/>
                                                  <function symbol="∪">
                                                   <variable symbol="x_0"/>
                                                   <variable symbol="y_0"/>
                                                  </function>
                                                 </constantatomformula>
                                                </formulalist>
                                                <formulalist>
                                                 <constantatomformula symbol="≤" definition="≤">
                                                  <variable symbol="x_0"/>
                                                  <function symbol="∪">
                                                   <variable symbol="x_0"/>
                                                   <variable symbol="y_0"/>
                                                  </function>
                                                 </constantatomformula>
                                                </formulalist>
                                               </sequent>
                                               <rule type="defr" symbol="d:r" param="();[(1)(1 1)(2)][(2)(1 2)];()">
                                                <sequent>
                                                 <formulalist>
                                                  <constantatomformula symbol="=">
                                                   <function symbol="∩">
                                                    <variable symbol="x_0"/>
                                                    <function symbol="∪">
                                                     <variable symbol="x_0"/>
                                                     <variable symbol="y_0"/>
                                                    </function>
                                                   </function>
                                                   <variable symbol="x_0"/>
                                                  </constantatomformula>
                                                 </formulalist>
                                                 <formulalist>
                                                  <constantatomformula symbol="≤" definition="≤">
                                                   <variable symbol="x_0"/>
                                                   <function symbol="∪">
                                                    <variable symbol="x_0"/>
                                                    <variable symbol="y_0"/>
                                                   </function>
                                                  </constantatomformula>
                                                 </formulalist>
                                                </sequent>
                                                <rule type="axiom" symbol="axiom">
                                                 <sequent>
                                                  <formulalist>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="∩">
                                                     <variable symbol="x_0"/>
                                                     <function symbol="∪">
                                                      <variable symbol="x_0"/>
                                                      <variable symbol="y_0"/>
                                                     </function>
                                                    </function>
                                                    <variable symbol="x_0"/>
                                                   </constantatomformula>
                                                  </formulalist>
                                                  <formulalist>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="∩">
                                                     <variable symbol="x_0"/>
                                                     <function symbol="∪">
                                                      <variable symbol="x_0"/>
                                                      <variable symbol="y_0"/>
                                                     </function>
                                                    </function>
                                                    <variable symbol="x_0"/>
                                                   </constantatomformula>
                                                  </formulalist>
                                                 </sequent>
                                                </rule>
                                               </rule>
                                              </rule>
                                             </rule>
                                            </rule>
                                           </rule>
                                          </rule>
                                         </rule>
                                         <rule type="defl" symbol="d:l" param="();[()];()">
                                          <sequent>
                                           <formulalist>
                                            <constantatomformula symbol="R" definition="R"/>
                                           </formulalist>
                                           <formulalist>
                                            <constantatomformula symbol="≤" definition="≤">
                                             <variable symbol="x_0"/>
                                             <variable symbol="x_0"/>
                                            </constantatomformula>
                                           </formulalist>
                                          </sequent>
                                          <rule type="foralll" symbol="\all:l">
                                           <sequent>
                                            <formulalist>
                                             <quantifiedformula type="all">
                                              <variable symbol="x"/>
                                              <constantatomformula symbol="≤" definition="≤">
                                               <variable symbol="x"/>
                                               <variable symbol="x"/>
                                              </constantatomformula>
                                             </quantifiedformula>
                                            </formulalist>
                                            <formulalist>
                                             <constantatomformula symbol="≤" definition="≤">
                                              <variable symbol="x_0"/>
                                              <variable symbol="x_0"/>
                                             </constantatomformula>
                                            </formulalist>
                                           </sequent>
                                           <rule type="defl" symbol="d:l" param="();[(1)(1 1)(2)][(2)(1 2)];()">
                                            <sequent>
                                             <formulalist>
                                              <constantatomformula symbol="≤" definition="≤">
                                               <variable symbol="x_0"/>
                                               <variable symbol="x_0"/>
                                              </constantatomformula>
                                             </formulalist>
                                             <formulalist>
                                              <constantatomformula symbol="≤" definition="≤">
                                               <variable symbol="x_0"/>
                                               <variable symbol="x_0"/>
                                              </constantatomformula>
                                             </formulalist>
                                            </sequent>
                                            <rule type="defr" symbol="d:r" param="();[(1)(1 1)(2)][(2)(1 2)];()">
                                             <sequent>
                                              <formulalist>
                                               <constantatomformula symbol="=">
                                                <function symbol="∩">
                                                 <variable symbol="x_0"/>
                                                 <variable symbol="x_0"/>
                                                </function>
                                                <variable symbol="x_0"/>
                                               </constantatomformula>
                                              </formulalist>
                                              <formulalist>
                                               <constantatomformula symbol="≤" definition="≤">
                                                <variable symbol="x_0"/>
                                                <variable symbol="x_0"/>
                                               </constantatomformula>
                                              </formulalist>
                                             </sequent>
                                             <rule type="axiom" symbol="axiom">
                                              <sequent>
                                               <formulalist>
                                                <constantatomformula symbol="=">
                                                 <function symbol="∩">
                                                  <variable symbol="x_0"/>
                                                  <variable symbol="x_0"/>
                                                 </function>
                                                 <variable symbol="x_0"/>
                                                </constantatomformula>
                                               </formulalist>
                                               <formulalist>
                                                <constantatomformula symbol="=">
                                                 <function symbol="∩">
                                                  <variable symbol="x_0"/>
                                                  <variable symbol="x_0"/>
                                                 </function>
                                                 <variable symbol="x_0"/>
                                                </constantatomformula>
                                               </formulalist>
                                              </sequent>
                                             </rule>
                                            </rule>
                                           </rule>
                                          </rule>
                                         </rule>
                                        </rule>
                                        <rule type="defl" symbol="d:l" param="();[(1)(1 1)(2)][(2)(1 2)];()">
                                         <sequent>
                                          <formulalist>
                                           <constantatomformula symbol="≤" definition="≤">
                                            <variable symbol="x_0"/>
                                            <function symbol="∩">
                                             <function symbol="∪">
                                              <variable symbol="x_0"/>
                                              <variable symbol="y_0"/>
                                             </function>
                                             <variable symbol="x_0"/>
                                            </function>
                                           </constantatomformula>
                                          </formulalist>
                                          <formulalist>
                                           <constantatomformula symbol="≤" definition="≤">
                                            <variable symbol="x_0"/>
                                            <function symbol="∩">
                                             <function symbol="∪">
                                              <variable symbol="x_0"/>
                                              <variable symbol="y_0"/>
                                             </function>
                                             <variable symbol="x_0"/>
                                            </function>
                                           </constantatomformula>
                                          </formulalist>
                                         </sequent>
                                         <rule type="defr" symbol="d:r" param="();[(1)(1 1)(2)][(2)(1 2)];()">
                                          <sequent>
                                           <formulalist>
                                            <constantatomformula symbol="=">
                                             <function symbol="∩">
                                              <variable symbol="x_0"/>
                                              <function symbol="∩">
                                               <function symbol="∪">
                                                <variable symbol="x_0"/>
                                                <variable symbol="y_0"/>
                                               </function>
                                               <variable symbol="x_0"/>
                                              </function>
                                             </function>
                                             <variable symbol="x_0"/>
                                            </constantatomformula>
                                           </formulalist>
                                           <formulalist>
                                            <constantatomformula symbol="≤" definition="≤">
                                             <variable symbol="x_0"/>
                                             <function symbol="∩">
                                              <function symbol="∪">
                                               <variable symbol="x_0"/>
                                               <variable symbol="y_0"/>
                                              </function>
                                              <variable symbol="x_0"/>
                                             </function>
                                            </constantatomformula>
                                           </formulalist>
                                          </sequent>
                                          <rule type="axiom" symbol="axiom">
                                           <sequent>
                                            <formulalist>
                                             <constantatomformula symbol="=">
                                              <function symbol="∩">
                                               <variable symbol="x_0"/>
                                               <function symbol="∩">
                                                <function symbol="∪">
                                                 <variable symbol="x_0"/>
                                                 <variable symbol="y_0"/>
                                                </function>
                                                <variable symbol="x_0"/>
                                               </function>
                                              </function>
                                              <variable symbol="x_0"/>
                                             </constantatomformula>
                                            </formulalist>
                                            <formulalist>
                                             <constantatomformula symbol="=">
                                              <function symbol="∩">
                                               <variable symbol="x_0"/>
                                               <function symbol="∩">
                                                <function symbol="∪">
                                                 <variable symbol="x_0"/>
                                                 <variable symbol="y_0"/>
                                                </function>
                                                <variable symbol="x_0"/>
                                               </function>
                                              </function>
                                              <variable symbol="x_0"/>
                                             </constantatomformula>
                                            </formulalist>
                                           </sequent>
                                          </rule>
                                         </rule>
                                        </rule>
                                       </rule>
                                      </rule>
                                     </rule>
                                    </rule>
                                   </rule>
                                  </rule>
                                 </rule>
                                </rule>
                               </rule>
                              </rule>
                             </rule>
                             <rule type="axiom" symbol="axiom">
                              <sequent>
                               <formulalist>
                                <constantatomformula symbol="=">
                                 <function symbol="∩">
                                  <function symbol="∪">
                                   <variable symbol="x_0"/>
                                   <variable symbol="y_0"/>
                                  </function>
                                  <variable symbol="x_0"/>
                                 </function>
                                 <variable symbol="x_0"/>
                                </constantatomformula>
                               </formulalist>
                               <formulalist>
                                <constantatomformula symbol="=">
                                 <function symbol="∩">
                                  <function symbol="∪">
                                   <variable symbol="x_0"/>
                                   <variable symbol="y_0"/>
                                  </function>
                                  <variable symbol="x_0"/>
                                 </function>
                                 <variable symbol="x_0"/>
                                </constantatomformula>
                               </formulalist>
                              </sequent>
                             </rule>
                            </rule>
                           </rule>
                          </rule>
                         </rule>
                        </rule>
                       </rule>
                      </rule>
                     </rule>
                    </rule>
                   </rule>
                  </rule>
                 </rule>
                </rule>
               </rule>
              </rule>
             </rule>
            </rule>
           </rule>
          </rule>
         </rule>
        </rule>
       </rule>
      </rule>
     </rule>
    </rule>
   </rule>
  </rule>
 </proof>
 <proof symbol="p_{sk}" calculus="LK">
  <rule type="cut" symbol="cut">
   <sequent>
    <formulalist>
     <constantatomformula symbol="L1" definition="L1"/>
    </formulalist>
    <formulalist>
     <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
    </formulalist>
   </sequent>
   <rule type="defr" symbol="d:r" param="();[()];()">
    <sequent>
     <formulalist>
      <constantatomformula symbol="L1" definition="L1"/>
     </formulalist>
     <formulalist>
      <constantatomformula symbol="L3" definition="L3"/>
     </formulalist>
    </sequent>
    <rule type="andr" symbol="\and:r">
     <sequent>
      <formulalist>
       <constantatomformula symbol="L1" definition="L1"/>
      </formulalist>
      <formulalist>
       <conjunctiveformula type="and">
        <constantatomformula symbol="POSET" definition="POSET"/>
        <conjunctiveformula type="and">
         <constantatomformula symbol="GLB" definition="GLB"/>
         <constantatomformula symbol="LUB" definition="LUB"/>
        </conjunctiveformula>
       </conjunctiveformula>
      </formulalist>
     </sequent>
     <rule type="defr" symbol="d:r" param="();[()];()">
      <sequent>
       <formulalist/>
       <formulalist>
        <constantatomformula symbol="POSET" definition="POSET"/>
       </formulalist>
      </sequent>
      <rule type="andr" symbol="\and:r">
       <sequent>
        <formulalist/>
        <formulalist>
         <conjunctiveformula type="and">
          <constantatomformula symbol="R" definition="R"/>
          <conjunctiveformula type="and">
           <constantatomformula symbol="AS" definition="AS"/>
           <constantatomformula symbol="T" definition="T"/>
          </conjunctiveformula>
         </conjunctiveformula>
        </formulalist>
       </sequent>
       <rule type="defr" symbol="d:r" param="();[()];()">
        <sequent>
         <formulalist/>
         <formulalist>
          <constantatomformula symbol="R" definition="R"/>
         </formulalist>
        </sequent>
        <rule type="forallr" symbol="\all:r">
         <sequent>
          <formulalist/>
          <formulalist>
           <quantifiedformula type="all">
            <variable symbol="x"/>
            <constantatomformula symbol="≤" definition="≤">
             <variable symbol="x"/>
             <variable symbol="x"/>
            </constantatomformula>
           </quantifiedformula>
          </formulalist>
         </sequent>
         <rule type="defr" symbol="d:r" param="();[(1)(1 1)(2)][(2)(1 2)];()">
          <sequent>
           <formulalist/>
           <formulalist>
            <constantatomformula symbol="≤" definition="≤">
             <variable symbol="x_0"/>
             <variable symbol="x_0"/>
            </constantatomformula>
           </formulalist>
          </sequent>
          <rule type="axiom" symbol="axiom">
           <sequent>
            <formulalist/>
            <formulalist>
             <constantatomformula symbol="=">
              <function symbol="∩">
               <variable symbol="x_0"/>
               <variable symbol="x_0"/>
              </function>
              <variable symbol="x_0"/>
             </constantatomformula>
            </formulalist>
           </sequent>
          </rule>
         </rule>
        </rule>
       </rule>
       <rule type="andr" symbol="\and:r">
        <sequent>
         <formulalist/>
         <formulalist>
          <conjunctiveformula type="and">
           <constantatomformula symbol="AS" definition="AS"/>
           <constantatomformula symbol="T" definition="T"/>
          </conjunctiveformula>
         </formulalist>
        </sequent>
        <rule type="defr" symbol="d:r" param="();[()];()">
         <sequent>
          <formulalist/>
          <formulalist>
           <constantatomformula symbol="AS" definition="AS"/>
          </formulalist>
         </sequent>
         <rule type="forallr" symbol="\all:r">
          <sequent>
           <formulalist/>
           <formulalist>
            <quantifiedformula type="all">
             <variable symbol="x"/>
             <quantifiedformula type="all">
              <variable symbol="y"/>
              <conjunctiveformula type="impl">
               <conjunctiveformula type="and">
                <constantatomformula symbol="≤" definition="≤">
                 <variable symbol="x"/>
                 <variable symbol="y"/>
                </constantatomformula>
                <constantatomformula symbol="≤" definition="≤">
                 <variable symbol="y"/>
                 <variable symbol="x"/>
                </constantatomformula>
               </conjunctiveformula>
               <constantatomformula symbol="=">
                <variable symbol="x"/>
                <variable symbol="y"/>
               </constantatomformula>
              </conjunctiveformula>
             </quantifiedformula>
            </quantifiedformula>
           </formulalist>
          </sequent>
          <rule type="forallr" symbol="\all:r">
           <sequent>
            <formulalist/>
            <formulalist>
             <quantifiedformula type="all">
              <variable symbol="y"/>
              <conjunctiveformula type="impl">
               <conjunctiveformula type="and">
                <constantatomformula symbol="≤" definition="≤">
                 <variable symbol="x_0"/>
                 <variable symbol="y"/>
                </constantatomformula>
                <constantatomformula symbol="≤" definition="≤">
                 <variable symbol="y"/>
                 <variable symbol="x_0"/>
                </constantatomformula>
               </conjunctiveformula>
               <constantatomformula symbol="=">
                <variable symbol="x_0"/>
                <variable symbol="y"/>
               </constantatomformula>
              </conjunctiveformula>
             </quantifiedformula>
            </formulalist>
           </sequent>
           <rule type="implr" symbol="\impl:r">
            <sequent>
             <formulalist/>
             <formulalist>
              <conjunctiveformula type="impl">
               <conjunctiveformula type="and">
                <constantatomformula symbol="≤" definition="≤">
                 <variable symbol="x_0"/>
                 <variable symbol="y_0"/>
                </constantatomformula>
                <constantatomformula symbol="≤" definition="≤">
                 <variable symbol="y_0"/>
                 <variable symbol="x_0"/>
                </constantatomformula>
               </conjunctiveformula>
               <constantatomformula symbol="=">
                <variable symbol="x_0"/>
                <variable symbol="y_0"/>
               </constantatomformula>
              </conjunctiveformula>
             </formulalist>
            </sequent>
            <rule type="contrl" symbol="c:l" param="2">
             <sequent>
              <formulalist>
               <conjunctiveformula type="and">
                <constantatomformula symbol="≤" definition="≤">
                 <variable symbol="x_0"/>
                 <variable symbol="y_0"/>
                </constantatomformula>
                <constantatomformula symbol="≤" definition="≤">
                 <variable symbol="y_0"/>
                 <variable symbol="x_0"/>
                </constantatomformula>
               </conjunctiveformula>
              </formulalist>
              <formulalist>
               <constantatomformula symbol="=">
                <variable symbol="x_0"/>
                <variable symbol="y_0"/>
               </constantatomformula>
              </formulalist>
             </sequent>
             <rule type="andl1" symbol="\and:l1">
              <sequent>
               <formulalist>
                <conjunctiveformula type="and">
                 <constantatomformula symbol="≤" definition="≤">
                  <variable symbol="x_0"/>
                  <variable symbol="y_0"/>
                 </constantatomformula>
                 <constantatomformula symbol="≤" definition="≤">
                  <variable symbol="y_0"/>
                  <variable symbol="x_0"/>
                 </constantatomformula>
                </conjunctiveformula>
                <conjunctiveformula type="and">
                 <constantatomformula symbol="≤" definition="≤">
                  <variable symbol="x_0"/>
                  <variable symbol="y_0"/>
                 </constantatomformula>
                 <constantatomformula symbol="≤" definition="≤">
                  <variable symbol="y_0"/>
                  <variable symbol="x_0"/>
                 </constantatomformula>
                </conjunctiveformula>
               </formulalist>
               <formulalist>
                <constantatomformula symbol="=">
                 <variable symbol="x_0"/>
                 <variable symbol="y_0"/>
                </constantatomformula>
               </formulalist>
              </sequent>
              <rule type="perml" symbol="\pi:l" param="(1 2)">
               <sequent>
                <formulalist>
                 <constantatomformula symbol="≤" definition="≤">
                  <variable symbol="x_0"/>
                  <variable symbol="y_0"/>
                 </constantatomformula>
                 <conjunctiveformula type="and">
                  <constantatomformula symbol="≤" definition="≤">
                   <variable symbol="x_0"/>
                   <variable symbol="y_0"/>
                  </constantatomformula>
                  <constantatomformula symbol="≤" definition="≤">
                   <variable symbol="y_0"/>
                   <variable symbol="x_0"/>
                  </constantatomformula>
                 </conjunctiveformula>
                </formulalist>
                <formulalist>
                 <constantatomformula symbol="=">
                  <variable symbol="x_0"/>
                  <variable symbol="y_0"/>
                 </constantatomformula>
                </formulalist>
               </sequent>
               <rule type="andl2" symbol="\and:l2">
                <sequent>
                 <formulalist>
                  <conjunctiveformula type="and">
                   <constantatomformula symbol="≤" definition="≤">
                    <variable symbol="x_0"/>
                    <variable symbol="y_0"/>
                   </constantatomformula>
                   <constantatomformula symbol="≤" definition="≤">
                    <variable symbol="y_0"/>
                    <variable symbol="x_0"/>
                   </constantatomformula>
                  </conjunctiveformula>
                  <constantatomformula symbol="≤" definition="≤">
                   <variable symbol="x_0"/>
                   <variable symbol="y_0"/>
                  </constantatomformula>
                 </formulalist>
                 <formulalist>
                  <constantatomformula symbol="=">
                   <variable symbol="x_0"/>
                   <variable symbol="y_0"/>
                  </constantatomformula>
                 </formulalist>
                </sequent>
                <rule type="perml" symbol="\pi:l" param="(1 2)">
                 <sequent>
                  <formulalist>
                   <constantatomformula symbol="≤" definition="≤">
                    <variable symbol="y_0"/>
                    <variable symbol="x_0"/>
                   </constantatomformula>
                   <constantatomformula symbol="≤" definition="≤">
                    <variable symbol="x_0"/>
                    <variable symbol="y_0"/>
                   </constantatomformula>
                  </formulalist>
                  <formulalist>
                   <constantatomformula symbol="=">
                    <variable symbol="x_0"/>
                    <variable symbol="y_0"/>
                   </constantatomformula>
                  </formulalist>
                 </sequent>
                 <rule type="defl" symbol="d:l" param="();[(1)(1 1)(2)][(2)(1 2)];()">
                  <sequent>
                   <formulalist>
                    <constantatomformula symbol="≤" definition="≤">
                     <variable symbol="x_0"/>
                     <variable symbol="y_0"/>
                    </constantatomformula>
                    <constantatomformula symbol="≤" definition="≤">
                     <variable symbol="y_0"/>
                     <variable symbol="x_0"/>
                    </constantatomformula>
                   </formulalist>
                   <formulalist>
                    <constantatomformula symbol="=">
                     <variable symbol="x_0"/>
                     <variable symbol="y_0"/>
                    </constantatomformula>
                   </formulalist>
                  </sequent>
                  <rule type="perml" symbol="\pi:l" param="(1 2)">
                   <sequent>
                    <formulalist>
                     <constantatomformula symbol="=">
                      <function symbol="∩">
                       <variable symbol="x_0"/>
                       <variable symbol="y_0"/>
                      </function>
                      <variable symbol="x_0"/>
                     </constantatomformula>
                     <constantatomformula symbol="≤" definition="≤">
                      <variable symbol="y_0"/>
                      <variable symbol="x_0"/>
                     </constantatomformula>
                    </formulalist>
                    <formulalist>
                     <constantatomformula symbol="=">
                      <variable symbol="x_0"/>
                      <variable symbol="y_0"/>
                     </constantatomformula>
                    </formulalist>
                   </sequent>
                   <rule type="defl" symbol="d:l" param="();[(1)(1 1)(2)][(2)(1 2)];()">
                    <sequent>
                     <formulalist>
                      <constantatomformula symbol="≤" definition="≤">
                       <variable symbol="y_0"/>
                       <variable symbol="x_0"/>
                      </constantatomformula>
                      <constantatomformula symbol="=">
                       <function symbol="∩">
                        <variable symbol="x_0"/>
                        <variable symbol="y_0"/>
                       </function>
                       <variable symbol="x_0"/>
                      </constantatomformula>
                     </formulalist>
                     <formulalist>
                      <constantatomformula symbol="=">
                       <variable symbol="x_0"/>
                       <variable symbol="y_0"/>
                      </constantatomformula>
                     </formulalist>
                    </sequent>
                    <rule type="eql1" symbol="=:l1" param="(1)">
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                      <formulalist>
                       <constantatomformula symbol="=">
                        <function symbol="∩">
                         <variable symbol="y_0"/>
                         <variable symbol="x_0"/>
                        </function>
                        <variable symbol="y_0"/>
                       </constantatomformula>
                       <constantatomformula symbol="=">
                        <function symbol="∩">
                         <variable symbol="x_0"/>
                         <variable symbol="y_0"/>
                        </function>
                        <variable symbol="x_0"/>
                       </constantatomformula>
                      </formulalist>
                      <formulalist>
                       <constantatomformula symbol="=">
                        <variable symbol="x_0"/>
                        <variable symbol="y_0"/>
                       </constantatomformula>
                      </formulalist>
                     </sequent>
                     <rule type="axiom" symbol="axiom">
                      <sequent>
                       <formulalist/>
                       <formulalist>
                        <constantatomformula symbol="=">
                         <function symbol="∩">
                          <variable symbol="x_0"/>
                          <variable symbol="y_0"/>
                         </function>
                         <function symbol="∩">
                          <variable symbol="y_0"/>
                          <variable symbol="x_0"/>
                         </function>
                        </constantatomformula>
                       </formulalist>
                      </sequent>
                     </rule>
                     <rule type="eql2" symbol="=:l2" param="(1)">
                      <sequent>
                       <formulalist>
                        <constantatomformula symbol="=">
                         <function symbol="∩">
                          <variable symbol="x_0"/>
                          <variable symbol="y_0"/>
                         </function>
                         <variable symbol="y_0"/>
                        </constantatomformula>
                        <constantatomformula symbol="=">
                         <function symbol="∩">
                          <variable symbol="x_0"/>
                          <variable symbol="y_0"/>
                         </function>
                         <variable symbol="x_0"/>
                        </constantatomformula>
                       </formulalist>
                       <formulalist>
                        <constantatomformula symbol="=">
                         <variable symbol="x_0"/>
                         <variable symbol="y_0"/>
                        </constantatomformula>
                       </formulalist>
                      </sequent>
                      <rule type="axiom" symbol="axiom">
                       <sequent>
                        <formulalist>
                         <constantatomformula symbol="=">
                          <function symbol="∩">
                           <variable symbol="x_0"/>
                           <variable symbol="y_0"/>
                          </function>
                          <variable symbol="x_0"/>
                         </constantatomformula>
                        </formulalist>
                        <formulalist>
                         <constantatomformula symbol="=">
                          <function symbol="∩">
                           <variable symbol="x_0"/>
                           <variable symbol="y_0"/>
                          </function>
                          <variable symbol="x_0"/>
                         </constantatomformula>
                        </formulalist>
                       </sequent>
                      </rule>
                      <rule type="axiom" symbol="axiom">
                       <sequent>
                        <formulalist>
                         <constantatomformula symbol="=">
                          <variable symbol="x_0"/>
                          <variable symbol="y_0"/>
                         </constantatomformula>
                        </formulalist>
                        <formulalist>
                         <constantatomformula symbol="=">
                          <variable symbol="x_0"/>
                          <variable symbol="y_0"/>
                         </constantatomformula>
                        </formulalist>
                       </sequent>
                      </rule>
                     </rule>
                    </rule>
                   </rule>
                  </rule>
                 </rule>
                </rule>
               </rule>
              </rule>
             </rule>
            </rule>
           </rule>
          </rule>
         </rule>
        </rule>
        <rule type="defr" symbol="d:r" param="();[()];()">
         <sequent>
          <formulalist/>
          <formulalist>
           <constantatomformula symbol="T" definition="T"/>
          </formulalist>
         </sequent>
         <rule type="forallr" symbol="\all:r">
          <sequent>
           <formulalist/>
           <formulalist>
            <quantifiedformula type="all">
             <variable symbol="x"/>
             <quantifiedformula type="all">
              <variable symbol="y"/>
              <quantifiedformula type="all">
               <variable symbol="z"/>
               <conjunctiveformula type="impl">
                <conjunctiveformula type="and">
                 <constantatomformula symbol="≤" definition="≤">
                  <variable symbol="x"/>
                  <variable symbol="y"/>
                 </constantatomformula>
                 <constantatomformula symbol="≤" definition="≤">
                  <variable symbol="y"/>
                  <variable symbol="z"/>
                 </constantatomformula>
                </conjunctiveformula>
                <constantatomformula symbol="≤" definition="≤">
                 <variable symbol="x"/>
                 <variable symbol="z"/>
                </constantatomformula>
               </conjunctiveformula>
              </quantifiedformula>
             </quantifiedformula>
            </quantifiedformula>
           </formulalist>
          </sequent>
          <rule type="forallr" symbol="\all:r">
           <sequent>
            <formulalist/>
            <formulalist>
             <quantifiedformula type="all">
              <variable symbol="y"/>
              <quantifiedformula type="all">
               <variable symbol="z"/>
               <conjunctiveformula type="impl">
                <conjunctiveformula type="and">
                 <constantatomformula symbol="≤" definition="≤">
                  <variable symbol="x_0"/>
                  <variable symbol="y"/>
                 </constantatomformula>
                 <constantatomformula symbol="≤" definition="≤">
                  <variable symbol="y"/>
                  <variable symbol="z"/>
                 </constantatomformula>
                </conjunctiveformula>
                <constantatomformula symbol="≤" definition="≤">
                 <variable symbol="x_0"/>
                 <variable symbol="z"/>
                </constantatomformula>
               </conjunctiveformula>
              </quantifiedformula>
             </quantifiedformula>
            </formulalist>
           </sequent>
           <rule type="forallr" symbol="\all:r">
            <sequent>
             <formulalist/>
             <formulalist>
              <quantifiedformula type="all">
               <variable symbol="z"/>
               <conjunctiveformula type="impl">
                <conjunctiveformula type="and">
                 <constantatomformula symbol="≤" definition="≤">
                  <variable symbol="x_0"/>
                  <variable symbol="y_0"/>
                 </constantatomformula>
                 <constantatomformula symbol="≤" definition="≤">
                  <variable symbol="y_0"/>
                  <variable symbol="z"/>
                 </constantatomformula>
                </conjunctiveformula>
                <constantatomformula symbol="≤" definition="≤">
                 <variable symbol="x_0"/>
                 <variable symbol="z"/>
                </constantatomformula>
               </conjunctiveformula>
              </quantifiedformula>
             </formulalist>
            </sequent>
            <rule type="implr" symbol="\impl:r">
             <sequent>
              <formulalist/>
              <formulalist>
               <conjunctiveformula type="impl">
                <conjunctiveformula type="and">
                 <constantatomformula symbol="≤" definition="≤">
                  <variable symbol="x_0"/>
                  <variable symbol="y_0"/>
                 </constantatomformula>
                 <constantatomformula symbol="≤" definition="≤">
                  <variable symbol="y_0"/>
                  <variable symbol="z_0"/>
                 </constantatomformula>
                </conjunctiveformula>
                <constantatomformula symbol="≤" definition="≤">
                 <variable symbol="x_0"/>
                 <variable symbol="z_0"/>
                </constantatomformula>
               </conjunctiveformula>
              </formulalist>
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             <rule type="contrl" symbol="c:l" param="2">
              <sequent>
               <formulalist>
                <conjunctiveformula type="and">
                 <constantatomformula symbol="≤" definition="≤">
                  <variable symbol="x_0"/>
                  <variable symbol="y_0"/>
                 </constantatomformula>
                 <constantatomformula symbol="≤" definition="≤">
                  <variable symbol="y_0"/>
                  <variable symbol="z_0"/>
                 </constantatomformula>
                </conjunctiveformula>
               </formulalist>
               <formulalist>
                <constantatomformula symbol="≤" definition="≤">
                 <variable symbol="x_0"/>
                 <variable symbol="z_0"/>
                </constantatomformula>
               </formulalist>
              </sequent>
              <rule type="andl1" symbol="\and:l1">
               <sequent>
                <formulalist>
                 <conjunctiveformula type="and">
                  <constantatomformula symbol="≤" definition="≤">
                   <variable symbol="x_0"/>
                   <variable symbol="y_0"/>
                  </constantatomformula>
                  <constantatomformula symbol="≤" definition="≤">
                   <variable symbol="y_0"/>
                   <variable symbol="z_0"/>
                  </constantatomformula>
                 </conjunctiveformula>
                 <conjunctiveformula type="and">
                  <constantatomformula symbol="≤" definition="≤">
                   <variable symbol="x_0"/>
                   <variable symbol="y_0"/>
                  </constantatomformula>
                  <constantatomformula symbol="≤" definition="≤">
                   <variable symbol="y_0"/>
                   <variable symbol="z_0"/>
                  </constantatomformula>
                 </conjunctiveformula>
                </formulalist>
                <formulalist>
                 <constantatomformula symbol="≤" definition="≤">
                  <variable symbol="x_0"/>
                  <variable symbol="z_0"/>
                 </constantatomformula>
                </formulalist>
               </sequent>
               <rule type="perml" symbol="\pi:l" param="(1 2)">
                <sequent>
                 <formulalist>
                  <constantatomformula symbol="≤" definition="≤">
                   <variable symbol="x_0"/>
                   <variable symbol="y_0"/>
                  </constantatomformula>
                  <conjunctiveformula type="and">
                   <constantatomformula symbol="≤" definition="≤">
                    <variable symbol="x_0"/>
                    <variable symbol="y_0"/>
                   </constantatomformula>
                   <constantatomformula symbol="≤" definition="≤">
                    <variable symbol="y_0"/>
                    <variable symbol="z_0"/>
                   </constantatomformula>
                  </conjunctiveformula>
                 </formulalist>
                 <formulalist>
                  <constantatomformula symbol="≤" definition="≤">
                   <variable symbol="x_0"/>
                   <variable symbol="z_0"/>
                  </constantatomformula>
                 </formulalist>
                </sequent>
                <rule type="andl2" symbol="\and:l2">
                 <sequent>
                  <formulalist>
                   <conjunctiveformula type="and">
                    <constantatomformula symbol="≤" definition="≤">
                     <variable symbol="x_0"/>
                     <variable symbol="y_0"/>
                    </constantatomformula>
                    <constantatomformula symbol="≤" definition="≤">
                     <variable symbol="y_0"/>
                     <variable symbol="z_0"/>
                    </constantatomformula>
                   </conjunctiveformula>
                   <constantatomformula symbol="≤" definition="≤">
                    <variable symbol="x_0"/>
                    <variable symbol="y_0"/>
                   </constantatomformula>
                  </formulalist>
                  <formulalist>
                   <constantatomformula symbol="≤" definition="≤">
                    <variable symbol="x_0"/>
                    <variable symbol="z_0"/>
                   </constantatomformula>
                  </formulalist>
                 </sequent>
                 <rule type="perml" symbol="\pi:l" param="(1 2)">
                  <sequent>
                   <formulalist>
                    <constantatomformula symbol="≤" definition="≤">
                     <variable symbol="y_0"/>
                     <variable symbol="z_0"/>
                    </constantatomformula>
                    <constantatomformula symbol="≤" definition="≤">
                     <variable symbol="x_0"/>
                     <variable symbol="y_0"/>
                    </constantatomformula>
                   </formulalist>
                   <formulalist>
                    <constantatomformula symbol="≤" definition="≤">
                     <variable symbol="x_0"/>
                     <variable symbol="z_0"/>
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                   <sequent>
                    <formulalist>
                     <constantatomformula symbol="≤" definition="≤">
                      <variable symbol="x_0"/>
                      <variable symbol="y_0"/>
                     </constantatomformula>
                     <constantatomformula symbol="≤" definition="≤">
                      <variable symbol="y_0"/>
                      <variable symbol="z_0"/>
                     </constantatomformula>
                    </formulalist>
                    <formulalist>
                     <constantatomformula symbol="≤" definition="≤">
                      <variable symbol="x_0"/>
                      <variable symbol="z_0"/>
                     </constantatomformula>
                    </formulalist>
                   </sequent>
                   <rule type="perml" symbol="\pi:l" param="(1 2)">
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                     <formulalist>
                      <constantatomformula symbol="=">
                       <function symbol="∩">
                        <variable symbol="x_0"/>
                        <variable symbol="y_0"/>
                       </function>
                       <variable symbol="x_0"/>
                      </constantatomformula>
                      <constantatomformula symbol="≤" definition="≤">
                       <variable symbol="y_0"/>
                       <variable symbol="z_0"/>
                      </constantatomformula>
                     </formulalist>
                     <formulalist>
                      <constantatomformula symbol="≤" definition="≤">
                       <variable symbol="x_0"/>
                       <variable symbol="z_0"/>
                      </constantatomformula>
                     </formulalist>
                    </sequent>
                    <rule type="defl" symbol="d:l" param="();[(1)(1 1)(2)][(2)(1 2)];()">
                     <sequent>
                      <formulalist>
                       <constantatomformula symbol="≤" definition="≤">
                        <variable symbol="y_0"/>
                        <variable symbol="z_0"/>
                       </constantatomformula>
                       <constantatomformula symbol="=">
                        <function symbol="∩">
                         <variable symbol="x_0"/>
                         <variable symbol="y_0"/>
                        </function>
                        <variable symbol="x_0"/>
                       </constantatomformula>
                      </formulalist>
                      <formulalist>
                       <constantatomformula symbol="≤" definition="≤">
                        <variable symbol="x_0"/>
                        <variable symbol="z_0"/>
                       </constantatomformula>
                      </formulalist>
                     </sequent>
                     <rule type="perml" symbol="\pi:l" param="(1 2)">
                      <sequent>
                       <formulalist>
                        <constantatomformula symbol="=">
                         <function symbol="∩">
                          <variable symbol="y_0"/>
                          <variable symbol="z_0"/>
                         </function>
                         <variable symbol="y_0"/>
                        </constantatomformula>
                        <constantatomformula symbol="=">
                         <function symbol="∩">
                          <variable symbol="x_0"/>
                          <variable symbol="y_0"/>
                         </function>
                         <variable symbol="x_0"/>
                        </constantatomformula>
                       </formulalist>
                       <formulalist>
                        <constantatomformula symbol="≤" definition="≤">
                         <variable symbol="x_0"/>
                         <variable symbol="z_0"/>
                        </constantatomformula>
                       </formulalist>
                      </sequent>
                      <rule type="defr" symbol="d:r" param="();[(1)(1 1)(2)][(2)(1 2)];()">
                       <sequent>
                        <formulalist>
                         <constantatomformula symbol="=">
                          <function symbol="∩">
                           <variable symbol="x_0"/>
                           <variable symbol="y_0"/>
                          </function>
                          <variable symbol="x_0"/>
                         </constantatomformula>
                         <constantatomformula symbol="=">
                          <function symbol="∩">
                           <variable symbol="y_0"/>
                           <variable symbol="z_0"/>
                          </function>
                          <variable symbol="y_0"/>
                         </constantatomformula>
                        </formulalist>
                        <formulalist>
                         <constantatomformula symbol="≤" definition="≤">
                          <variable symbol="x_0"/>
                          <variable symbol="z_0"/>
                         </constantatomformula>
                        </formulalist>
                       </sequent>
                       <rule type="eqr1" symbol="=:r1" param="(2)(1 1)">
                        <sequent>
                         <formulalist>
                          <constantatomformula symbol="=">
                           <function symbol="∩">
                            <variable symbol="x_0"/>
                            <variable symbol="y_0"/>
                           </function>
                           <variable symbol="x_0"/>
                          </constantatomformula>
                          <constantatomformula symbol="=">
                           <function symbol="∩">
                            <variable symbol="y_0"/>
                            <variable symbol="z_0"/>
                           </function>
                           <variable symbol="y_0"/>
                          </constantatomformula>
                         </formulalist>
                         <formulalist>
                          <constantatomformula symbol="=">
                           <function symbol="∩">
                            <variable symbol="x_0"/>
                            <variable symbol="z_0"/>
                           </function>
                           <variable symbol="x_0"/>
                          </constantatomformula>
                         </formulalist>
                        </sequent>
                        <rule type="axiom" symbol="axiom">
                         <sequent>
                          <formulalist>
                           <constantatomformula symbol="=">
                            <function symbol="∩">
                             <variable symbol="x_0"/>
                             <variable symbol="y_0"/>
                            </function>
                            <variable symbol="x_0"/>
                           </constantatomformula>
                          </formulalist>
                          <formulalist>
                           <constantatomformula symbol="=">
                            <function symbol="∩">
                             <variable symbol="x_0"/>
                             <variable symbol="y_0"/>
                            </function>
                            <variable symbol="x_0"/>
                           </constantatomformula>
                          </formulalist>
                         </sequent>
                        </rule>
                        <rule type="eqr1" symbol="=:r1" param="(2 2)">
                         <sequent>
                          <formulalist>
                           <constantatomformula symbol="=">
                            <function symbol="∩">
                             <variable symbol="y_0"/>
                             <variable symbol="z_0"/>
                            </function>
                            <variable symbol="y_0"/>
                           </constantatomformula>
                          </formulalist>
                          <formulalist>
                           <constantatomformula symbol="=">
                            <function symbol="∩">
                             <function symbol="∩">
                              <variable symbol="x_0"/>
                              <variable symbol="y_0"/>
                             </function>
                             <variable symbol="z_0"/>
                            </function>
                            <function symbol="∩">
                             <variable symbol="x_0"/>
                             <variable symbol="y_0"/>
                            </function>
                           </constantatomformula>
                          </formulalist>
                         </sequent>
                         <rule type="axiom" symbol="axiom">
                          <sequent>
                           <formulalist>
                            <constantatomformula symbol="=">
                             <function symbol="∩">
                              <variable symbol="y_0"/>
                              <variable symbol="z_0"/>
                             </function>
                             <variable symbol="y_0"/>
                            </constantatomformula>
                           </formulalist>
                           <formulalist>
                            <constantatomformula symbol="=">
                             <function symbol="∩">
                              <variable symbol="y_0"/>
                              <variable symbol="z_0"/>
                             </function>
                             <variable symbol="y_0"/>
                            </constantatomformula>
                           </formulalist>
                          </sequent>
                         </rule>
                         <rule type="axiom" symbol="axiom">
                          <sequent>
                           <formulalist/>
                           <formulalist>
                            <constantatomformula symbol="=">
                             <function symbol="∩">
                              <function symbol="∩">
                               <variable symbol="x_0"/>
                               <variable symbol="y_0"/>
                              </function>
                              <variable symbol="z_0"/>
                             </function>
                             <function symbol="∩">
                              <variable symbol="x_0"/>
                              <function symbol="∩">
                               <variable symbol="y_0"/>
                               <variable symbol="z_0"/>
                              </function>
                             </function>
                            </constantatomformula>
                           </formulalist>
                          </sequent>
                         </rule>
                        </rule>
                       </rule>
                      </rule>
                     </rule>
                    </rule>
                   </rule>
                  </rule>
                 </rule>
                </rule>
               </rule>
              </rule>
             </rule>
            </rule>
           </rule>
          </rule>
         </rule>
        </rule>
       </rule>
      </rule>
     </rule>
     <rule type="andr" symbol="\and:r">
      <sequent>
       <formulalist>
        <constantatomformula symbol="L1" definition="L1"/>
       </formulalist>
       <formulalist>
        <conjunctiveformula type="and">
         <constantatomformula symbol="GLB" definition="GLB"/>
         <constantatomformula symbol="LUB" definition="LUB"/>
        </conjunctiveformula>
       </formulalist>
      </sequent>
      <rule type="defr" symbol="d:r" param="();[()];()">
       <sequent>
        <formulalist/>
        <formulalist>
         <constantatomformula symbol="GLB" definition="GLB"/>
        </formulalist>
       </sequent>
       <rule type="forallr" symbol="\all:r">
        <sequent>
         <formulalist/>
         <formulalist>
          <quantifiedformula type="all">
           <variable symbol="x"/>
           <quantifiedformula type="all">
            <variable symbol="y"/>
            <conjunctiveformula type="and">
             <conjunctiveformula type="and">
              <constantatomformula symbol="≤" definition="≤">
               <function symbol="∩">
                <variable symbol="x"/>
                <variable symbol="y"/>
               </function>
               <variable symbol="x"/>
              </constantatomformula>
              <constantatomformula symbol="≤" definition="≤">
               <function symbol="∩">
                <variable symbol="x"/>
                <variable symbol="y"/>
               </function>
               <variable symbol="y"/>
              </constantatomformula>
             </conjunctiveformula>
             <quantifiedformula type="all">
              <variable symbol="z"/>
              <conjunctiveformula type="impl">
               <conjunctiveformula type="and">
                <constantatomformula symbol="≤" definition="≤">
                 <variable symbol="z"/>
                 <variable symbol="x"/>
                </constantatomformula>
                <constantatomformula symbol="≤" definition="≤">
                 <variable symbol="z"/>
                 <variable symbol="y"/>
                </constantatomformula>
               </conjunctiveformula>
               <constantatomformula symbol="≤" definition="≤">
                <variable symbol="z"/>
                <function symbol="∩">
                 <variable symbol="x"/>
                 <variable symbol="y"/>
                </function>
               </constantatomformula>
              </conjunctiveformula>
             </quantifiedformula>
            </conjunctiveformula>
           </quantifiedformula>
          </quantifiedformula>
         </formulalist>
        </sequent>
        <rule type="forallr" symbol="\all:r">
         <sequent>
          <formulalist/>
          <formulalist>
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                  <variable symbol="y_0"/>
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                 <variable symbol="y_0"/>
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                  <function symbol="∩">
                   <variable symbol="x_0"/>
                   <variable symbol="y_0"/>
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                  <variable symbol="x_0"/>
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                 <function symbol="∩">
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                  <function symbol="∩">
                   <variable symbol="x_0"/>
                   <variable symbol="y_0"/>
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               <formulalist/>
               <formulalist>
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                  <variable symbol="x_0"/>
                  <function symbol="∩">
                   <variable symbol="x_0"/>
                   <variable symbol="y_0"/>
                  </function>
                 </function>
                 <function symbol="∩">
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                  <variable symbol="y_0"/>
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                   <function symbol="∩">
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                    <variable symbol="x_0"/>
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                   <variable symbol="y_0"/>
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                  <function symbol="∩">
                   <variable symbol="x_0"/>
                   <function symbol="∩">
                    <variable symbol="x_0"/>
                    <variable symbol="y_0"/>
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                <formulalist/>
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                 <constantatomformula symbol="=">
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                   <function symbol="∩">
                    <variable symbol="x_0"/>
                    <variable symbol="x_0"/>
                   </function>
                   <variable symbol="y_0"/>
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                   <variable symbol="x_0"/>
                   <variable symbol="y_0"/>
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                 <formulalist/>
                 <formulalist>
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                    <variable symbol="x_0"/>
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                   <variable symbol="x_0"/>
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                 <formulalist>
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                    <variable symbol="y_0"/>
                   </function>
                   <function symbol="∩">
                    <variable symbol="x_0"/>
                    <variable symbol="y_0"/>
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             <formulalist>
              <constantatomformula symbol="≤" definition="≤">
               <function symbol="∩">
                <variable symbol="x_0"/>
                <variable symbol="y_0"/>
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                 <function symbol="∩">
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                  <variable symbol="y_0"/>
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                <function symbol="∩">
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                 <variable symbol="y_0"/>
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               <formulalist/>
               <formulalist>
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                  <function symbol="∩">
                   <variable symbol="x_0"/>
                   <variable symbol="y_0"/>
                  </function>
                  <variable symbol="y_0"/>
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                 <function symbol="∩">
                  <variable symbol="x_0"/>
                  <function symbol="∩">
                   <variable symbol="y_0"/>
                   <variable symbol="y_0"/>
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               <formulalist/>
               <formulalist>
                <constantatomformula symbol="=">
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                  <variable symbol="x_0"/>
                  <function symbol="∩">
                   <variable symbol="y_0"/>
                   <variable symbol="y_0"/>
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                 <function symbol="∩">
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                  <variable symbol="y_0"/>
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                <formulalist/>
                <formulalist>
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                   <variable symbol="y_0"/>
                   <variable symbol="y_0"/>
                  </function>
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                <formulalist/>
                <formulalist>
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                   <variable symbol="x_0"/>
                   <variable symbol="y_0"/>
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                   <variable symbol="y_0"/>
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                        <formulalist>
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                            <variable symbol="x_0"/>
                            <variable symbol="y_0"/>
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                             <variable symbol="x_0"/>
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                           <function symbol="∩">
                            <variable symbol="z_0"/>
                            <function symbol="∩">
                             <variable symbol="x_0"/>
                             <variable symbol="y_0"/>
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                            <variable symbol="x_0"/>
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                           <variable symbol="z_0"/>
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                          <constantatomformula symbol="=">
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                            <variable symbol="z_0"/>
                            <variable symbol="y_0"/>
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                           <variable symbol="z_0"/>
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                         <formulalist>
                          <constantatomformula symbol="=">
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                            <function symbol="∩">
                             <variable symbol="z_0"/>
                             <variable symbol="x_0"/>
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                            <variable symbol="y_0"/>
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                           <variable symbol="z_0"/>
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                          <formulalist>
                           <constantatomformula symbol="=">
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                             <variable symbol="z_0"/>
                             <variable symbol="x_0"/>
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                            <variable symbol="z_0"/>
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                          </formulalist>
                          <formulalist>
                           <constantatomformula symbol="=">
                            <function symbol="∩">
                             <variable symbol="z_0"/>
                             <variable symbol="x_0"/>
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                            <variable symbol="z_0"/>
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                         <sequent>
                          <formulalist>
                           <constantatomformula symbol="=">
                            <function symbol="∩">
                             <variable symbol="z_0"/>
                             <variable symbol="y_0"/>
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                            <variable symbol="z_0"/>
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                          </formulalist>
                          <formulalist>
                           <constantatomformula symbol="=">
                            <function symbol="∩">
                             <variable symbol="z_0"/>
                             <variable symbol="y_0"/>
                            </function>
                            <variable symbol="z_0"/>
                           </constantatomformula>
                          </formulalist>
                         </sequent>
                        </rule>
                       </rule>
                      </rule>
                     </rule>
                    </rule>
                   </rule>
                  </rule>
                 </rule>
                </rule>
               </rule>
              </rule>
             </rule>
            </rule>
           </rule>
          </rule>
         </rule>
        </rule>
       </rule>
      </rule>
      <rule type="defr" symbol="d:r" param="();[()];()">
       <sequent>
        <formulalist>
         <constantatomformula symbol="L1" definition="L1"/>
        </formulalist>
        <formulalist>
         <constantatomformula symbol="LUB" definition="LUB"/>
        </formulalist>
       </sequent>
       <rule type="forallr" symbol="\all:r">
        <sequent>
         <formulalist>
          <constantatomformula symbol="L1" definition="L1"/>
         </formulalist>
         <formulalist>
          <quantifiedformula type="all">
           <variable symbol="x"/>
           <quantifiedformula type="all">
            <variable symbol="y"/>
            <conjunctiveformula type="and">
             <conjunctiveformula type="and">
              <constantatomformula symbol="≤" definition="≤">
               <variable symbol="x"/>
               <function symbol="∪">
                <variable symbol="x"/>
                <variable symbol="y"/>
               </function>
              </constantatomformula>
              <constantatomformula symbol="≤" definition="≤">
               <variable symbol="y"/>
               <function symbol="∪">
                <variable symbol="x"/>
                <variable symbol="y"/>
               </function>
              </constantatomformula>
             </conjunctiveformula>
             <quantifiedformula type="all">
              <variable symbol="z"/>
              <conjunctiveformula type="impl">
               <conjunctiveformula type="and">
                <constantatomformula symbol="≤" definition="≤">
                 <variable symbol="x"/>
                 <variable symbol="z"/>
                </constantatomformula>
                <constantatomformula symbol="≤" definition="≤">
                 <variable symbol="y"/>
                 <variable symbol="z"/>
                </constantatomformula>
               </conjunctiveformula>
               <constantatomformula symbol="≤" definition="≤">
                <function symbol="∪">
                 <variable symbol="x"/>
                 <variable symbol="y"/>
                </function>
                <variable symbol="z"/>
               </constantatomformula>
              </conjunctiveformula>
             </quantifiedformula>
            </conjunctiveformula>
           </quantifiedformula>
          </quantifiedformula>
         </formulalist>
        </sequent>
        <rule type="forallr" symbol="\all:r">
         <sequent>
          <formulalist>
           <constantatomformula symbol="L1" definition="L1"/>
          </formulalist>
          <formulalist>
           <quantifiedformula type="all">
            <variable symbol="y"/>
            <conjunctiveformula type="and">
             <conjunctiveformula type="and">
              <constantatomformula symbol="≤" definition="≤">
               <variable symbol="x_0"/>
               <function symbol="∪">
                <variable symbol="x_0"/>
                <variable symbol="y"/>
               </function>
              </constantatomformula>
              <constantatomformula symbol="≤" definition="≤">
               <variable symbol="y"/>
               <function symbol="∪">
                <variable symbol="x_0"/>
                <variable symbol="y"/>
               </function>
              </constantatomformula>
             </conjunctiveformula>
             <quantifiedformula type="all">
              <variable symbol="z"/>
              <conjunctiveformula type="impl">
               <conjunctiveformula type="and">
                <constantatomformula symbol="≤" definition="≤">
                 <variable symbol="x_0"/>
                 <variable symbol="z"/>
                </constantatomformula>
                <constantatomformula symbol="≤" definition="≤">
                 <variable symbol="y"/>
                 <variable symbol="z"/>
                </constantatomformula>
               </conjunctiveformula>
               <constantatomformula symbol="≤" definition="≤">
                <function symbol="∪">
                 <variable symbol="x_0"/>
                 <variable symbol="y"/>
                </function>
                <variable symbol="z"/>
               </constantatomformula>
              </conjunctiveformula>
             </quantifiedformula>
            </conjunctiveformula>
           </quantifiedformula>
          </formulalist>
         </sequent>
         <rule type="contrl" symbol="c:l" param="2">
          <sequent>
           <formulalist>
            <constantatomformula symbol="L1" definition="L1"/>
           </formulalist>
           <formulalist>
            <conjunctiveformula type="and">
             <conjunctiveformula type="and">
              <constantatomformula symbol="≤" definition="≤">
               <variable symbol="x_0"/>
               <function symbol="∪">
                <variable symbol="x_0"/>
                <variable symbol="y_0"/>
               </function>
              </constantatomformula>
              <constantatomformula symbol="≤" definition="≤">
               <variable symbol="y_0"/>
               <function symbol="∪">
                <variable symbol="x_0"/>
                <variable symbol="y_0"/>
               </function>
              </constantatomformula>
             </conjunctiveformula>
             <quantifiedformula type="all">
              <variable symbol="z"/>
              <conjunctiveformula type="impl">
               <conjunctiveformula type="and">
                <constantatomformula symbol="≤" definition="≤">
                 <variable symbol="x_0"/>
                 <variable symbol="z"/>
                </constantatomformula>
                <constantatomformula symbol="≤" definition="≤">
                 <variable symbol="y_0"/>
                 <variable symbol="z"/>
                </constantatomformula>
               </conjunctiveformula>
               <constantatomformula symbol="≤" definition="≤">
                <function symbol="∪">
                 <variable symbol="x_0"/>
                 <variable symbol="y_0"/>
                </function>
                <variable symbol="z"/>
               </constantatomformula>
              </conjunctiveformula>
             </quantifiedformula>
            </conjunctiveformula>
           </formulalist>
          </sequent>
          <rule type="andr" symbol="\and:r">
           <sequent>
            <formulalist>
             <constantatomformula symbol="L1" definition="L1"/>
             <constantatomformula symbol="L1" definition="L1"/>
            </formulalist>
            <formulalist>
             <conjunctiveformula type="and">
              <conjunctiveformula type="and">
               <constantatomformula symbol="≤" definition="≤">
                <variable symbol="x_0"/>
                <function symbol="∪">
                 <variable symbol="x_0"/>
                 <variable symbol="y_0"/>
                </function>
               </constantatomformula>
               <constantatomformula symbol="≤" definition="≤">
                <variable symbol="y_0"/>
                <function symbol="∪">
                 <variable symbol="x_0"/>
                 <variable symbol="y_0"/>
                </function>
               </constantatomformula>
              </conjunctiveformula>
              <quantifiedformula type="all">
               <variable symbol="z"/>
               <conjunctiveformula type="impl">
                <conjunctiveformula type="and">
                 <constantatomformula symbol="≤" definition="≤">
                  <variable symbol="x_0"/>
                  <variable symbol="z"/>
                 </constantatomformula>
                 <constantatomformula symbol="≤" definition="≤">
                  <variable symbol="y_0"/>
                  <variable symbol="z"/>
                 </constantatomformula>
                </conjunctiveformula>
                <constantatomformula symbol="≤" definition="≤">
                 <function symbol="∪">
                  <variable symbol="x_0"/>
                  <variable symbol="y_0"/>
                 </function>
                 <variable symbol="z"/>
                </constantatomformula>
               </conjunctiveformula>
              </quantifiedformula>
             </conjunctiveformula>
            </formulalist>
           </sequent>
           <rule type="contrl" symbol="c:l" param="2">
            <sequent>
             <formulalist>
              <constantatomformula symbol="L1" definition="L1"/>
             </formulalist>
             <formulalist>
              <conjunctiveformula type="and">
               <constantatomformula symbol="≤" definition="≤">
                <variable symbol="x_0"/>
                <function symbol="∪">
                 <variable symbol="x_0"/>
                 <variable symbol="y_0"/>
                </function>
               </constantatomformula>
               <constantatomformula symbol="≤" definition="≤">
                <variable symbol="y_0"/>
                <function symbol="∪">
                 <variable symbol="x_0"/>
                 <variable symbol="y_0"/>
                </function>
               </constantatomformula>
              </conjunctiveformula>
             </formulalist>
            </sequent>
            <rule type="andr" symbol="\and:r">
             <sequent>
              <formulalist>
               <constantatomformula symbol="L1" definition="L1"/>
               <constantatomformula symbol="L1" definition="L1"/>
              </formulalist>
              <formulalist>
               <conjunctiveformula type="and">
                <constantatomformula symbol="≤" definition="≤">
                 <variable symbol="x_0"/>
                 <function symbol="∪">
                  <variable symbol="x_0"/>
                  <variable symbol="y_0"/>
                 </function>
                </constantatomformula>
                <constantatomformula symbol="≤" definition="≤">
                 <variable symbol="y_0"/>
                 <function symbol="∪">
                  <variable symbol="x_0"/>
                  <variable symbol="y_0"/>
                 </function>
                </constantatomformula>
               </conjunctiveformula>
              </formulalist>
             </sequent>
             <rule type="defr" symbol="d:r" param="();[(1)(1 1)(2)][(2)(1 2)];()">
              <sequent>
               <formulalist>
                <constantatomformula symbol="L1" definition="L1"/>
               </formulalist>
               <formulalist>
                <constantatomformula symbol="≤" definition="≤">
                 <variable symbol="x_0"/>
                 <function symbol="∪">
                  <variable symbol="x_0"/>
                  <variable symbol="y_0"/>
                 </function>
                </constantatomformula>
               </formulalist>
              </sequent>
              <rule type="defl" symbol="d:l" param="();[()];()">
               <sequent>
                <formulalist>
                 <constantatomformula symbol="L1" definition="L1"/>
                </formulalist>
                <formulalist>
                 <constantatomformula symbol="=">
                  <function symbol="∩">
                   <variable symbol="x_0"/>
                   <function symbol="∪">
                    <variable symbol="x_0"/>
                    <variable symbol="y_0"/>
                   </function>
                  </function>
                  <variable symbol="x_0"/>
                 </constantatomformula>
                </formulalist>
               </sequent>
               <rule type="foralll" symbol="\all:l">
                <sequent>
                 <formulalist>
                  <quantifiedformula type="all">
                   <variable symbol="x"/>
                   <quantifiedformula type="all">
                    <variable symbol="y"/>
                    <conjunctiveformula type="and">
                     <conjunctiveformula type="impl">
                      <constantatomformula symbol="=">
                       <function symbol="∩">
                        <variable symbol="x"/>
                        <variable symbol="y"/>
                       </function>
                       <variable symbol="x"/>
                      </constantatomformula>
                      <constantatomformula symbol="=">
                       <function symbol="∪">
                        <variable symbol="x"/>
                        <variable symbol="y"/>
                       </function>
                       <variable symbol="y"/>
                      </constantatomformula>
                     </conjunctiveformula>
                     <conjunctiveformula type="impl">
                      <constantatomformula symbol="=">
                       <function symbol="∪">
                        <variable symbol="x"/>
                        <variable symbol="y"/>
                       </function>
                       <variable symbol="y"/>
                      </constantatomformula>
                      <constantatomformula symbol="=">
                       <function symbol="∩">
                        <variable symbol="x"/>
                        <variable symbol="y"/>
                       </function>
                       <variable symbol="x"/>
                      </constantatomformula>
                     </conjunctiveformula>
                    </conjunctiveformula>
                   </quantifiedformula>
                  </quantifiedformula>
                 </formulalist>
                 <formulalist>
                  <constantatomformula symbol="=">
                   <function symbol="∩">
                    <variable symbol="x_0"/>
                    <function symbol="∪">
                     <variable symbol="x_0"/>
                     <variable symbol="y_0"/>
                    </function>
                   </function>
                   <variable symbol="x_0"/>
                  </constantatomformula>
                 </formulalist>
                </sequent>
                <rule type="foralll" symbol="\all:l">
                 <sequent>
                  <formulalist>
                   <quantifiedformula type="all">
                    <variable symbol="y"/>
                    <conjunctiveformula type="and">
                     <conjunctiveformula type="impl">
                      <constantatomformula symbol="=">
                       <function symbol="∩">
                        <variable symbol="x_0"/>
                        <variable symbol="y"/>
                       </function>
                       <variable symbol="x_0"/>
                      </constantatomformula>
                      <constantatomformula symbol="=">
                       <function symbol="∪">
                        <variable symbol="x_0"/>
                        <variable symbol="y"/>
                       </function>
                       <variable symbol="y"/>
                      </constantatomformula>
                     </conjunctiveformula>
                     <conjunctiveformula type="impl">
                      <constantatomformula symbol="=">
                       <function symbol="∪">
                        <variable symbol="x_0"/>
                        <variable symbol="y"/>
                       </function>
                       <variable symbol="y"/>
                      </constantatomformula>
                      <constantatomformula symbol="=">
                       <function symbol="∩">
                        <variable symbol="x_0"/>
                        <variable symbol="y"/>
                       </function>
                       <variable symbol="x_0"/>
                      </constantatomformula>
                     </conjunctiveformula>
                    </conjunctiveformula>
                   </quantifiedformula>
                  </formulalist>
                  <formulalist>
                   <constantatomformula symbol="=">
                    <function symbol="∩">
                     <variable symbol="x_0"/>
                     <function symbol="∪">
                      <variable symbol="x_0"/>
                      <variable symbol="y_0"/>
                     </function>
                    </function>
                    <variable symbol="x_0"/>
                   </constantatomformula>
                  </formulalist>
                 </sequent>
                 <rule type="andl2" symbol="\and:l2">
                  <sequent>
                   <formulalist>
                    <conjunctiveformula type="and">
                     <conjunctiveformula type="impl">
                      <constantatomformula symbol="=">
                       <function symbol="∩">
                        <variable symbol="x_0"/>
                        <function symbol="∪">
                         <variable symbol="x_0"/>
                         <variable symbol="y_0"/>
                        </function>
                       </function>
                       <variable symbol="x_0"/>
                      </constantatomformula>
                      <constantatomformula symbol="=">
                       <function symbol="∪">
                        <variable symbol="x_0"/>
                        <function symbol="∪">
                         <variable symbol="x_0"/>
                         <variable symbol="y_0"/>
                        </function>
                       </function>
                       <function symbol="∪">
                        <variable symbol="x_0"/>
                        <variable symbol="y_0"/>
                       </function>
                      </constantatomformula>
                     </conjunctiveformula>
                     <conjunctiveformula type="impl">
                      <constantatomformula symbol="=">
                       <function symbol="∪">
                        <variable symbol="x_0"/>
                        <function symbol="∪">
                         <variable symbol="x_0"/>
                         <variable symbol="y_0"/>
                        </function>
                       </function>
                       <function symbol="∪">
                        <variable symbol="x_0"/>
                        <variable symbol="y_0"/>
                       </function>
                      </constantatomformula>
                      <constantatomformula symbol="=">
                       <function symbol="∩">
                        <variable symbol="x_0"/>
                        <function symbol="∪">
                         <variable symbol="x_0"/>
                         <variable symbol="y_0"/>
                        </function>
                       </function>
                       <variable symbol="x_0"/>
                      </constantatomformula>
                     </conjunctiveformula>
                    </conjunctiveformula>
                   </formulalist>
                   <formulalist>
                    <constantatomformula symbol="=">
                     <function symbol="∩">
                      <variable symbol="x_0"/>
                      <function symbol="∪">
                       <variable symbol="x_0"/>
                       <variable symbol="y_0"/>
                      </function>
                     </function>
                     <variable symbol="x_0"/>
                    </constantatomformula>
                   </formulalist>
                  </sequent>
                  <rule type="impll" symbol="\impl:l">
                   <sequent>
                    <formulalist>
                     <conjunctiveformula type="impl">
                      <constantatomformula symbol="=">
                       <function symbol="∪">
                        <variable symbol="x_0"/>
                        <function symbol="∪">
                         <variable symbol="x_0"/>
                         <variable symbol="y_0"/>
                        </function>
                       </function>
                       <function symbol="∪">
                        <variable symbol="x_0"/>
                        <variable symbol="y_0"/>
                       </function>
                      </constantatomformula>
                      <constantatomformula symbol="=">
                       <function symbol="∩">
                        <variable symbol="x_0"/>
                        <function symbol="∪">
                         <variable symbol="x_0"/>
                         <variable symbol="y_0"/>
                        </function>
                       </function>
                       <variable symbol="x_0"/>
                      </constantatomformula>
                     </conjunctiveformula>
                    </formulalist>
                    <formulalist>
                     <constantatomformula symbol="=">
                      <function symbol="∩">
                       <variable symbol="x_0"/>
                       <function symbol="∪">
                        <variable symbol="x_0"/>
                        <variable symbol="y_0"/>
                       </function>
                      </function>
                      <variable symbol="x_0"/>
                     </constantatomformula>
                    </formulalist>
                   </sequent>
                   <rule type="eqr1" symbol="=:r1" param="(1)">
                    <sequent>
                     <formulalist/>
                     <formulalist>
                      <constantatomformula symbol="=">
                       <function symbol="∪">
                        <variable symbol="x_0"/>
                        <function symbol="∪">
                         <variable symbol="x_0"/>
                         <variable symbol="y_0"/>
                        </function>
                       </function>
                       <function symbol="∪">
                        <variable symbol="x_0"/>
                        <variable symbol="y_0"/>
                       </function>
                      </constantatomformula>
                     </formulalist>
                    </sequent>
                    <rule type="axiom" symbol="axiom">
                     <sequent>
                      <formulalist/>
                      <formulalist>
                       <constantatomformula symbol="=">
                        <function symbol="∪">
                         <function symbol="∪">
                          <variable symbol="x_0"/>
                          <variable symbol="x_0"/>
                         </function>
                         <variable symbol="y_0"/>
                        </function>
                        <function symbol="∪">
                         <variable symbol="x_0"/>
                         <function symbol="∪">
                          <variable symbol="x_0"/>
                          <variable symbol="y_0"/>
                         </function>
                        </function>
                       </constantatomformula>
                      </formulalist>
                     </sequent>
                    </rule>
                    <rule type="eqr2" symbol="=:r2" param="(1 1)">
                     <sequent>
                      <formulalist/>
                      <formulalist>
                       <constantatomformula symbol="=">
                        <function symbol="∪">
                         <function symbol="∪">
                          <variable symbol="x_0"/>
                          <variable symbol="x_0"/>
                         </function>
                         <variable symbol="y_0"/>
                        </function>
                        <function symbol="∪">
                         <variable symbol="x_0"/>
                         <variable symbol="y_0"/>
                        </function>
                       </constantatomformula>
                      </formulalist>
                     </sequent>
                     <rule type="axiom" symbol="axiom">
                      <sequent>
                       <formulalist/>
                       <formulalist>
                        <constantatomformula symbol="=">
                         <function symbol="∪">
                          <variable symbol="x_0"/>
                          <variable symbol="x_0"/>
                         </function>
                         <variable symbol="x_0"/>
                        </constantatomformula>
                       </formulalist>
                      </sequent>
                     </rule>
                     <rule type="axiom" symbol="axiom">
                      <sequent>
                       <formulalist/>
                       <formulalist>
                        <constantatomformula symbol="=">
                         <function symbol="∪">
                          <variable symbol="x_0"/>
                          <variable symbol="y_0"/>
                         </function>
                         <function symbol="∪">
                          <variable symbol="x_0"/>
                          <variable symbol="y_0"/>
                         </function>
                        </constantatomformula>
                       </formulalist>
                      </sequent>
                     </rule>
                    </rule>
                   </rule>
                   <rule type="axiom" symbol="axiom">
                    <sequent>
                     <formulalist>
                      <constantatomformula symbol="=">
                       <function symbol="∩">
                        <variable symbol="x_0"/>
                        <function symbol="∪">
                         <variable symbol="x_0"/>
                         <variable symbol="y_0"/>
                        </function>
                       </function>
                       <variable symbol="x_0"/>
                      </constantatomformula>
                     </formulalist>
                     <formulalist>
                      <constantatomformula symbol="=">
                       <function symbol="∩">
                        <variable symbol="x_0"/>
                        <function symbol="∪">
                         <variable symbol="x_0"/>
                         <variable symbol="y_0"/>
                        </function>
                       </function>
                       <variable symbol="x_0"/>
                      </constantatomformula>
                     </formulalist>
                    </sequent>
                   </rule>
                  </rule>
                 </rule>
                </rule>
               </rule>
              </rule>
             </rule>
             <rule type="eqr2" symbol="=:r2" param="(2)">
              <sequent>
               <formulalist>
                <constantatomformula symbol="L1" definition="L1"/>
               </formulalist>
               <formulalist>
                <constantatomformula symbol="≤" definition="≤">
                 <variable symbol="y_0"/>
                 <function symbol="∪">
                  <variable symbol="x_0"/>
                  <variable symbol="y_0"/>
                 </function>
                </constantatomformula>
               </formulalist>
              </sequent>
              <rule type="axiom" symbol="axiom">
               <sequent>
                <formulalist/>
                <formulalist>
                 <constantatomformula symbol="=">
                  <function symbol="∪">
                   <variable symbol="x_0"/>
                   <variable symbol="y_0"/>
                  </function>
                  <function symbol="∪">
                   <variable symbol="y_0"/>
                   <variable symbol="x_0"/>
                  </function>
                 </constantatomformula>
                </formulalist>
               </sequent>
              </rule>
              <rule type="defr" symbol="d:r" param="();[(1)(1 1)(2)][(2)(1 2)];()">
               <sequent>
                <formulalist>
                 <constantatomformula symbol="L1" definition="L1"/>
                </formulalist>
                <formulalist>
                 <constantatomformula symbol="≤" definition="≤">
                  <variable symbol="y_0"/>
                  <function symbol="∪">
                   <variable symbol="y_0"/>
                   <variable symbol="x_0"/>
                  </function>
                 </constantatomformula>
                </formulalist>
               </sequent>
               <rule type="defl" symbol="d:l" param="();[()];()">
                <sequent>
                 <formulalist>
                  <constantatomformula symbol="L1" definition="L1"/>
                 </formulalist>
                 <formulalist>
                  <constantatomformula symbol="=">
                   <function symbol="∩">
                    <variable symbol="y_0"/>
                    <function symbol="∪">
                     <variable symbol="y_0"/>
                     <variable symbol="x_0"/>
                    </function>
                   </function>
                   <variable symbol="y_0"/>
                  </constantatomformula>
                 </formulalist>
                </sequent>
                <rule type="foralll" symbol="\all:l">
                 <sequent>
                  <formulalist>
                   <quantifiedformula type="all">
                    <variable symbol="x"/>
                    <quantifiedformula type="all">
                     <variable symbol="y"/>
                     <conjunctiveformula type="and">
                      <conjunctiveformula type="impl">
                       <constantatomformula symbol="=">
                        <function symbol="∩">
                         <variable symbol="x"/>
                         <variable symbol="y"/>
                        </function>
                        <variable symbol="x"/>
                       </constantatomformula>
                       <constantatomformula symbol="=">
                        <function symbol="∪">
                         <variable symbol="x"/>
                         <variable symbol="y"/>
                        </function>
                        <variable symbol="y"/>
                       </constantatomformula>
                      </conjunctiveformula>
                      <conjunctiveformula type="impl">
                       <constantatomformula symbol="=">
                        <function symbol="∪">
                         <variable symbol="x"/>
                         <variable symbol="y"/>
                        </function>
                        <variable symbol="y"/>
                       </constantatomformula>
                       <constantatomformula symbol="=">
                        <function symbol="∩">
                         <variable symbol="x"/>
                         <variable symbol="y"/>
                        </function>
                        <variable symbol="x"/>
                       </constantatomformula>
                      </conjunctiveformula>
                     </conjunctiveformula>
                    </quantifiedformula>
                   </quantifiedformula>
                  </formulalist>
                  <formulalist>
                   <constantatomformula symbol="=">
                    <function symbol="∩">
                     <variable symbol="y_0"/>
                     <function symbol="∪">
                      <variable symbol="y_0"/>
                      <variable symbol="x_0"/>
                     </function>
                    </function>
                    <variable symbol="y_0"/>
                   </constantatomformula>
                  </formulalist>
                 </sequent>
                 <rule type="foralll" symbol="\all:l">
                  <sequent>
                   <formulalist>
                    <quantifiedformula type="all">
                     <variable symbol="y"/>
                     <conjunctiveformula type="and">
                      <conjunctiveformula type="impl">
                       <constantatomformula symbol="=">
                        <function symbol="∩">
                         <variable symbol="y_0"/>
                         <variable symbol="y"/>
                        </function>
                        <variable symbol="y_0"/>
                       </constantatomformula>
                       <constantatomformula symbol="=">
                        <function symbol="∪">
                         <variable symbol="y_0"/>
                         <variable symbol="y"/>
                        </function>
                        <variable symbol="y"/>
                       </constantatomformula>
                      </conjunctiveformula>
                      <conjunctiveformula type="impl">
                       <constantatomformula symbol="=">
                        <function symbol="∪">
                         <variable symbol="y_0"/>
                         <variable symbol="y"/>
                        </function>
                        <variable symbol="y"/>
                       </constantatomformula>
                       <constantatomformula symbol="=">
                        <function symbol="∩">
                         <variable symbol="y_0"/>
                         <variable symbol="y"/>
                        </function>
                        <variable symbol="y_0"/>
                       </constantatomformula>
                      </conjunctiveformula>
                     </conjunctiveformula>
                    </quantifiedformula>
                   </formulalist>
                   <formulalist>
                    <constantatomformula symbol="=">
                     <function symbol="∩">
                      <variable symbol="y_0"/>
                      <function symbol="∪">
                       <variable symbol="y_0"/>
                       <variable symbol="x_0"/>
                      </function>
                     </function>
                     <variable symbol="y_0"/>
                    </constantatomformula>
                   </formulalist>
                  </sequent>
                  <rule type="andl2" symbol="\and:l2">
                   <sequent>
                    <formulalist>
                     <conjunctiveformula type="and">
                      <conjunctiveformula type="impl">
                       <constantatomformula symbol="=">
                        <function symbol="∩">
                         <variable symbol="y_0"/>
                         <function symbol="∪">
                          <variable symbol="y_0"/>
                          <variable symbol="x_0"/>
                         </function>
                        </function>
                        <variable symbol="y_0"/>
                       </constantatomformula>
                       <constantatomformula symbol="=">
                        <function symbol="∪">
                         <variable symbol="y_0"/>
                         <function symbol="∪">
                          <variable symbol="y_0"/>
                          <variable symbol="x_0"/>
                         </function>
                        </function>
                        <function symbol="∪">
                         <variable symbol="y_0"/>
                         <variable symbol="x_0"/>
                        </function>
                       </constantatomformula>
                      </conjunctiveformula>
                      <conjunctiveformula type="impl">
                       <constantatomformula symbol="=">
                        <function symbol="∪">
                         <variable symbol="y_0"/>
                         <function symbol="∪">
                          <variable symbol="y_0"/>
                          <variable symbol="x_0"/>
                         </function>
                        </function>
                        <function symbol="∪">
                         <variable symbol="y_0"/>
                         <variable symbol="x_0"/>
                        </function>
                       </constantatomformula>
                       <constantatomformula symbol="=">
                        <function symbol="∩">
                         <variable symbol="y_0"/>
                         <function symbol="∪">
                          <variable symbol="y_0"/>
                          <variable symbol="x_0"/>
                         </function>
                        </function>
                        <variable symbol="y_0"/>
                       </constantatomformula>
                      </conjunctiveformula>
                     </conjunctiveformula>
                    </formulalist>
                    <formulalist>
                     <constantatomformula symbol="=">
                      <function symbol="∩">
                       <variable symbol="y_0"/>
                       <function symbol="∪">
                        <variable symbol="y_0"/>
                        <variable symbol="x_0"/>
                       </function>
                      </function>
                      <variable symbol="y_0"/>
                     </constantatomformula>
                    </formulalist>
                   </sequent>
                   <rule type="impll" symbol="\impl:l">
                    <sequent>
                     <formulalist>
                      <conjunctiveformula type="impl">
                       <constantatomformula symbol="=">
                        <function symbol="∪">
                         <variable symbol="y_0"/>
                         <function symbol="∪">
                          <variable symbol="y_0"/>
                          <variable symbol="x_0"/>
                         </function>
                        </function>
                        <function symbol="∪">
                         <variable symbol="y_0"/>
                         <variable symbol="x_0"/>
                        </function>
                       </constantatomformula>
                       <constantatomformula symbol="=">
                        <function symbol="∩">
                         <variable symbol="y_0"/>
                         <function symbol="∪">
                          <variable symbol="y_0"/>
                          <variable symbol="x_0"/>
                         </function>
                        </function>
                        <variable symbol="y_0"/>
                       </constantatomformula>
                      </conjunctiveformula>
                     </formulalist>
                     <formulalist>
                      <constantatomformula symbol="=">
                       <function symbol="∩">
                        <variable symbol="y_0"/>
                        <function symbol="∪">
                         <variable symbol="y_0"/>
                         <variable symbol="x_0"/>
                        </function>
                       </function>
                       <variable symbol="y_0"/>
                      </constantatomformula>
                     </formulalist>
                    </sequent>
                    <rule type="eqr1" symbol="=:r1" param="(1)">
                     <sequent>
                      <formulalist/>
                      <formulalist>
                       <constantatomformula symbol="=">
                        <function symbol="∪">
                         <variable symbol="y_0"/>
                         <function symbol="∪">
                          <variable symbol="y_0"/>
                          <variable symbol="x_0"/>
                         </function>
                        </function>
                        <function symbol="∪">
                         <variable symbol="y_0"/>
                         <variable symbol="x_0"/>
                        </function>
                       </constantatomformula>
                      </formulalist>
                     </sequent>
                     <rule type="axiom" symbol="axiom">
                      <sequent>
                       <formulalist/>
                       <formulalist>
                        <constantatomformula symbol="=">
                         <function symbol="∪">
                          <function symbol="∪">
                           <variable symbol="y_0"/>
                           <variable symbol="y_0"/>
                          </function>
                          <variable symbol="x_0"/>
                         </function>
                         <function symbol="∪">
                          <variable symbol="y_0"/>
                          <function symbol="∪">
                           <variable symbol="y_0"/>
                           <variable symbol="x_0"/>
                          </function>
                         </function>
                        </constantatomformula>
                       </formulalist>
                      </sequent>
                     </rule>
                     <rule type="eqr2" symbol="=:r2" param="(1 1)">
                      <sequent>
                       <formulalist/>
                       <formulalist>
                        <constantatomformula symbol="=">
                         <function symbol="∪">
                          <function symbol="∪">
                           <variable symbol="y_0"/>
                           <variable symbol="y_0"/>
                          </function>
                          <variable symbol="x_0"/>
                         </function>
                         <function symbol="∪">
                          <variable symbol="y_0"/>
                          <variable symbol="x_0"/>
                         </function>
                        </constantatomformula>
                       </formulalist>
                      </sequent>
                      <rule type="axiom" symbol="axiom">
                       <sequent>
                        <formulalist/>
                        <formulalist>
                         <constantatomformula symbol="=">
                          <function symbol="∪">
                           <variable symbol="y_0"/>
                           <variable symbol="y_0"/>
                          </function>
                          <variable symbol="y_0"/>
                         </constantatomformula>
                        </formulalist>
                       </sequent>
                      </rule>
                      <rule type="axiom" symbol="axiom">
                       <sequent>
                        <formulalist/>
                        <formulalist>
                         <constantatomformula symbol="=">
                          <function symbol="∪">
                           <variable symbol="y_0"/>
                           <variable symbol="x_0"/>
                          </function>
                          <function symbol="∪">
                           <variable symbol="y_0"/>
                           <variable symbol="x_0"/>
                          </function>
                         </constantatomformula>
                        </formulalist>
                       </sequent>
                      </rule>
                     </rule>
                    </rule>
                    <rule type="axiom" symbol="axiom">
                     <sequent>
                      <formulalist>
                       <constantatomformula symbol="=">
                        <function symbol="∩">
                         <variable symbol="y_0"/>
                         <function symbol="∪">
                          <variable symbol="y_0"/>
                          <variable symbol="x_0"/>
                         </function>
                        </function>
                        <variable symbol="y_0"/>
                       </constantatomformula>
                      </formulalist>
                      <formulalist>
                       <constantatomformula symbol="=">
                        <function symbol="∩">
                         <variable symbol="y_0"/>
                         <function symbol="∪">
                          <variable symbol="y_0"/>
                          <variable symbol="x_0"/>
                         </function>
                        </function>
                        <variable symbol="y_0"/>
                       </constantatomformula>
                      </formulalist>
                     </sequent>
                    </rule>
                   </rule>
                  </rule>
                 </rule>
                </rule>
               </rule>
              </rule>
             </rule>
            </rule>
           </rule>
           <rule type="forallr" symbol="\all:r">
            <sequent>
             <formulalist>
              <constantatomformula symbol="L1" definition="L1"/>
             </formulalist>
             <formulalist>
              <quantifiedformula type="all">
               <variable symbol="z"/>
               <conjunctiveformula type="impl">
                <conjunctiveformula type="and">
                 <constantatomformula symbol="≤" definition="≤">
                  <variable symbol="x_0"/>
                  <variable symbol="z"/>
                 </constantatomformula>
                 <constantatomformula symbol="≤" definition="≤">
                  <variable symbol="y_0"/>
                  <variable symbol="z"/>
                 </constantatomformula>
                </conjunctiveformula>
                <constantatomformula symbol="≤" definition="≤">
                 <function symbol="∪">
                  <variable symbol="x_0"/>
                  <variable symbol="y_0"/>
                 </function>
                 <variable symbol="z"/>
                </constantatomformula>
               </conjunctiveformula>
              </quantifiedformula>
             </formulalist>
            </sequent>
            <rule type="implr" symbol="\impl:r">
             <sequent>
              <formulalist>
               <constantatomformula symbol="L1" definition="L1"/>
              </formulalist>
              <formulalist>
               <conjunctiveformula type="impl">
                <conjunctiveformula type="and">
                 <constantatomformula symbol="≤" definition="≤">
                  <variable symbol="x_0"/>
                  <variable symbol="z_0"/>
                 </constantatomformula>
                 <constantatomformula symbol="≤" definition="≤">
                  <variable symbol="y_0"/>
                  <variable symbol="z_0"/>
                 </constantatomformula>
                </conjunctiveformula>
                <constantatomformula symbol="≤" definition="≤">
                 <function symbol="∪">
                  <variable symbol="x_0"/>
                  <variable symbol="y_0"/>
                 </function>
                 <variable symbol="z_0"/>
                </constantatomformula>
               </conjunctiveformula>
              </formulalist>
             </sequent>
             <rule type="contrl" symbol="c:l" param="2,1">
              <sequent>
               <formulalist>
                <conjunctiveformula type="and">
                 <constantatomformula symbol="≤" definition="≤">
                  <variable symbol="x_0"/>
                  <variable symbol="z_0"/>
                 </constantatomformula>
                 <constantatomformula symbol="≤" definition="≤">
                  <variable symbol="y_0"/>
                  <variable symbol="z_0"/>
                 </constantatomformula>
                </conjunctiveformula>
                <constantatomformula symbol="L1" definition="L1"/>
               </formulalist>
               <formulalist>
                <constantatomformula symbol="≤" definition="≤">
                 <function symbol="∪">
                  <variable symbol="x_0"/>
                  <variable symbol="y_0"/>
                 </function>
                 <variable symbol="z_0"/>
                </constantatomformula>
               </formulalist>
              </sequent>
              <rule type="perml" symbol="\pi:l" param="(2 3)">
               <sequent>
                <formulalist>
                 <conjunctiveformula type="and">
                  <constantatomformula symbol="≤" definition="≤">
                   <variable symbol="x_0"/>
                   <variable symbol="z_0"/>
                  </constantatomformula>
                  <constantatomformula symbol="≤" definition="≤">
                   <variable symbol="y_0"/>
                   <variable symbol="z_0"/>
                  </constantatomformula>
                 </conjunctiveformula>
                 <conjunctiveformula type="and">
                  <constantatomformula symbol="≤" definition="≤">
                   <variable symbol="x_0"/>
                   <variable symbol="z_0"/>
                  </constantatomformula>
                  <constantatomformula symbol="≤" definition="≤">
                   <variable symbol="y_0"/>
                   <variable symbol="z_0"/>
                  </constantatomformula>
                 </conjunctiveformula>
                 <constantatomformula symbol="L1" definition="L1"/>
                </formulalist>
                <formulalist>
                 <constantatomformula symbol="≤" definition="≤">
                  <function symbol="∪">
                   <variable symbol="x_0"/>
                   <variable symbol="y_0"/>
                  </function>
                  <variable symbol="z_0"/>
                 </constantatomformula>
                </formulalist>
               </sequent>
               <rule type="andl1" symbol="\and:l1">
                <sequent>
                 <formulalist>
                  <conjunctiveformula type="and">
                   <constantatomformula symbol="≤" definition="≤">
                    <variable symbol="x_0"/>
                    <variable symbol="z_0"/>
                   </constantatomformula>
                   <constantatomformula symbol="≤" definition="≤">
                    <variable symbol="y_0"/>
                    <variable symbol="z_0"/>
                   </constantatomformula>
                  </conjunctiveformula>
                  <constantatomformula symbol="L1" definition="L1"/>
                  <conjunctiveformula type="and">
                   <constantatomformula symbol="≤" definition="≤">
                    <variable symbol="x_0"/>
                    <variable symbol="z_0"/>
                   </constantatomformula>
                   <constantatomformula symbol="≤" definition="≤">
                    <variable symbol="y_0"/>
                    <variable symbol="z_0"/>
                   </constantatomformula>
                  </conjunctiveformula>
                 </formulalist>
                 <formulalist>
                  <constantatomformula symbol="≤" definition="≤">
                   <function symbol="∪">
                    <variable symbol="x_0"/>
                    <variable symbol="y_0"/>
                   </function>
                   <variable symbol="z_0"/>
                  </constantatomformula>
                 </formulalist>
                </sequent>
                <rule type="perml" symbol="\pi:l" param="(1 3)">
                 <sequent>
                  <formulalist>
                   <constantatomformula symbol="≤" definition="≤">
                    <variable symbol="x_0"/>
                    <variable symbol="z_0"/>
                   </constantatomformula>
                   <constantatomformula symbol="L1" definition="L1"/>
                   <conjunctiveformula type="and">
                    <constantatomformula symbol="≤" definition="≤">
                     <variable symbol="x_0"/>
                     <variable symbol="z_0"/>
                    </constantatomformula>
                    <constantatomformula symbol="≤" definition="≤">
                     <variable symbol="y_0"/>
                     <variable symbol="z_0"/>
                    </constantatomformula>
                   </conjunctiveformula>
                  </formulalist>
                  <formulalist>
                   <constantatomformula symbol="≤" definition="≤">
                    <function symbol="∪">
                     <variable symbol="x_0"/>
                     <variable symbol="y_0"/>
                    </function>
                    <variable symbol="z_0"/>
                   </constantatomformula>
                  </formulalist>
                 </sequent>
                 <rule type="andl2" symbol="\and:l2">
                  <sequent>
                   <formulalist>
                    <conjunctiveformula type="and">
                     <constantatomformula symbol="≤" definition="≤">
                      <variable symbol="x_0"/>
                      <variable symbol="z_0"/>
                     </constantatomformula>
                     <constantatomformula symbol="≤" definition="≤">
                      <variable symbol="y_0"/>
                      <variable symbol="z_0"/>
                     </constantatomformula>
                    </conjunctiveformula>
                    <constantatomformula symbol="L1" definition="L1"/>
                    <constantatomformula symbol="≤" definition="≤">
                     <variable symbol="x_0"/>
                     <variable symbol="z_0"/>
                    </constantatomformula>
                   </formulalist>
                   <formulalist>
                    <constantatomformula symbol="≤" definition="≤">
                     <function symbol="∪">
                      <variable symbol="x_0"/>
                      <variable symbol="y_0"/>
                     </function>
                     <variable symbol="z_0"/>
                    </constantatomformula>
                   </formulalist>
                  </sequent>
                  <rule type="perml" symbol="\pi:l" param="(1 3)">
                   <sequent>
                    <formulalist>
                     <constantatomformula symbol="≤" definition="≤">
                      <variable symbol="y_0"/>
                      <variable symbol="z_0"/>
                     </constantatomformula>
                     <constantatomformula symbol="L1" definition="L1"/>
                     <constantatomformula symbol="≤" definition="≤">
                      <variable symbol="x_0"/>
                      <variable symbol="z_0"/>
                     </constantatomformula>
                    </formulalist>
                    <formulalist>
                     <constantatomformula symbol="≤" definition="≤">
                      <function symbol="∪">
                       <variable symbol="x_0"/>
                       <variable symbol="y_0"/>
                      </function>
                      <variable symbol="z_0"/>
                     </constantatomformula>
                    </formulalist>
                   </sequent>
                   <rule type="defl" symbol="d:l" param="();[(1)(1 1)(2)][(2)(1 2)];()">
                    <sequent>
                     <formulalist>
                      <constantatomformula symbol="≤" definition="≤">
                       <variable symbol="x_0"/>
                       <variable symbol="z_0"/>
                      </constantatomformula>
                      <constantatomformula symbol="L1" definition="L1"/>
                      <constantatomformula symbol="≤" definition="≤">
                       <variable symbol="y_0"/>
                       <variable symbol="z_0"/>
                      </constantatomformula>
                     </formulalist>
                     <formulalist>
                      <constantatomformula symbol="≤" definition="≤">
                       <function symbol="∪">
                        <variable symbol="x_0"/>
                        <variable symbol="y_0"/>
                       </function>
                       <variable symbol="z_0"/>
                      </constantatomformula>
                     </formulalist>
                    </sequent>
                    <rule type="perml" symbol="\pi:l" param="(1 3)">
                     <sequent>
                      <formulalist>
                       <constantatomformula symbol="=">
                        <function symbol="∩">
                         <variable symbol="x_0"/>
                         <variable symbol="z_0"/>
                        </function>
                        <variable symbol="x_0"/>
                       </constantatomformula>
                       <constantatomformula symbol="L1" definition="L1"/>
                       <constantatomformula symbol="≤" definition="≤">
                        <variable symbol="y_0"/>
                        <variable symbol="z_0"/>
                       </constantatomformula>
                      </formulalist>
                      <formulalist>
                       <constantatomformula symbol="≤" definition="≤">
                        <function symbol="∪">
                         <variable symbol="x_0"/>
                         <variable symbol="y_0"/>
                        </function>
                        <variable symbol="z_0"/>
                       </constantatomformula>
                      </formulalist>
                     </sequent>
                     <rule type="defl" symbol="d:l" param="();[(1)(1 1)(2)][(2)(1 2)];()">
                      <sequent>
                       <formulalist>
                        <constantatomformula symbol="≤" definition="≤">
                         <variable symbol="y_0"/>
                         <variable symbol="z_0"/>
                        </constantatomformula>
                        <constantatomformula symbol="L1" definition="L1"/>
                        <constantatomformula symbol="=">
                         <function symbol="∩">
                          <variable symbol="x_0"/>
                          <variable symbol="z_0"/>
                         </function>
                         <variable symbol="x_0"/>
                        </constantatomformula>
                       </formulalist>
                       <formulalist>
                        <constantatomformula symbol="≤" definition="≤">
                         <function symbol="∪">
                          <variable symbol="x_0"/>
                          <variable symbol="y_0"/>
                         </function>
                         <variable symbol="z_0"/>
                        </constantatomformula>
                       </formulalist>
                      </sequent>
                      <rule type="perml" symbol="\pi:l" param="(1 2)">
                       <sequent>
                        <formulalist>
                         <constantatomformula symbol="=">
                          <function symbol="∩">
                           <variable symbol="y_0"/>
                           <variable symbol="z_0"/>
                          </function>
                          <variable symbol="y_0"/>
                         </constantatomformula>
                         <constantatomformula symbol="L1" definition="L1"/>
                         <constantatomformula symbol="=">
                          <function symbol="∩">
                           <variable symbol="x_0"/>
                           <variable symbol="z_0"/>
                          </function>
                          <variable symbol="x_0"/>
                         </constantatomformula>
                        </formulalist>
                        <formulalist>
                         <constantatomformula symbol="≤" definition="≤">
                          <function symbol="∪">
                           <variable symbol="x_0"/>
                           <variable symbol="y_0"/>
                          </function>
                          <variable symbol="z_0"/>
                         </constantatomformula>
                        </formulalist>
                       </sequent>
                       <rule type="defr" symbol="d:r" param="();[(1)(1 1)(2)][(2)(1 2)];()">
                        <sequent>
                         <formulalist>
                          <constantatomformula symbol="L1" definition="L1"/>
                          <constantatomformula symbol="=">
                           <function symbol="∩">
                            <variable symbol="y_0"/>
                            <variable symbol="z_0"/>
                           </function>
                           <variable symbol="y_0"/>
                          </constantatomformula>
                          <constantatomformula symbol="=">
                           <function symbol="∩">
                            <variable symbol="x_0"/>
                            <variable symbol="z_0"/>
                           </function>
                           <variable symbol="x_0"/>
                          </constantatomformula>
                         </formulalist>
                         <formulalist>
                          <constantatomformula symbol="≤" definition="≤">
                           <function symbol="∪">
                            <variable symbol="x_0"/>
                            <variable symbol="y_0"/>
                           </function>
                           <variable symbol="z_0"/>
                          </constantatomformula>
                         </formulalist>
                        </sequent>
                        <rule type="defl" symbol="d:l" param="();[()];()">
                         <sequent>
                          <formulalist>
                           <constantatomformula symbol="L1" definition="L1"/>
                           <constantatomformula symbol="=">
                            <function symbol="∩">
                             <variable symbol="y_0"/>
                             <variable symbol="z_0"/>
                            </function>
                            <variable symbol="y_0"/>
                           </constantatomformula>
                           <constantatomformula symbol="=">
                            <function symbol="∩">
                             <variable symbol="x_0"/>
                             <variable symbol="z_0"/>
                            </function>
                            <variable symbol="x_0"/>
                           </constantatomformula>
                          </formulalist>
                          <formulalist>
                           <constantatomformula symbol="=">
                            <function symbol="∩">
                             <function symbol="∪">
                              <variable symbol="x_0"/>
                              <variable symbol="y_0"/>
                             </function>
                             <variable symbol="z_0"/>
                            </function>
                            <function symbol="∪">
                             <variable symbol="x_0"/>
                             <variable symbol="y_0"/>
                            </function>
                           </constantatomformula>
                          </formulalist>
                         </sequent>
                         <rule type="contrl" symbol="c:l" param="2,1,1">
                          <sequent>
                           <formulalist>
                            <quantifiedformula type="all">
                             <variable symbol="x"/>
                             <quantifiedformula type="all">
                              <variable symbol="y"/>
                              <conjunctiveformula type="and">
                               <conjunctiveformula type="impl">
                                <constantatomformula symbol="=">
                                 <function symbol="∩">
                                  <variable symbol="x"/>
                                  <variable symbol="y"/>
                                 </function>
                                 <variable symbol="x"/>
                                </constantatomformula>
                                <constantatomformula symbol="=">
                                 <function symbol="∪">
                                  <variable symbol="x"/>
                                  <variable symbol="y"/>
                                 </function>
                                 <variable symbol="y"/>
                                </constantatomformula>
                               </conjunctiveformula>
                               <conjunctiveformula type="impl">
                                <constantatomformula symbol="=">
                                 <function symbol="∪">
                                  <variable symbol="x"/>
                                  <variable symbol="y"/>
                                 </function>
                                 <variable symbol="y"/>
                                </constantatomformula>
                                <constantatomformula symbol="=">
                                 <function symbol="∩">
                                  <variable symbol="x"/>
                                  <variable symbol="y"/>
                                 </function>
                                 <variable symbol="x"/>
                                </constantatomformula>
                               </conjunctiveformula>
                              </conjunctiveformula>
                             </quantifiedformula>
                            </quantifiedformula>
                            <constantatomformula symbol="=">
                             <function symbol="∩">
                              <variable symbol="y_0"/>
                              <variable symbol="z_0"/>
                             </function>
                             <variable symbol="y_0"/>
                            </constantatomformula>
                            <constantatomformula symbol="=">
                             <function symbol="∩">
                              <variable symbol="x_0"/>
                              <variable symbol="z_0"/>
                             </function>
                             <variable symbol="x_0"/>
                            </constantatomformula>
                           </formulalist>
                           <formulalist>
                            <constantatomformula symbol="=">
                             <function symbol="∩">
                              <function symbol="∪">
                               <variable symbol="x_0"/>
                               <variable symbol="y_0"/>
                              </function>
                              <variable symbol="z_0"/>
                             </function>
                             <function symbol="∪">
                              <variable symbol="x_0"/>
                              <variable symbol="y_0"/>
                             </function>
                            </constantatomformula>
                           </formulalist>
                          </sequent>
                          <rule type="perml" symbol="\pi:l" param="(2 4)">
                           <sequent>
                            <formulalist>
                             <quantifiedformula type="all">
                              <variable symbol="x"/>
                              <quantifiedformula type="all">
                               <variable symbol="y"/>
                               <conjunctiveformula type="and">
                                <conjunctiveformula type="impl">
                                 <constantatomformula symbol="=">
                                  <function symbol="∩">
                                   <variable symbol="x"/>
                                   <variable symbol="y"/>
                                  </function>
                                  <variable symbol="x"/>
                                 </constantatomformula>
                                 <constantatomformula symbol="=">
                                  <function symbol="∪">
                                   <variable symbol="x"/>
                                   <variable symbol="y"/>
                                  </function>
                                  <variable symbol="y"/>
                                 </constantatomformula>
                                </conjunctiveformula>
                                <conjunctiveformula type="impl">
                                 <constantatomformula symbol="=">
                                  <function symbol="∪">
                                   <variable symbol="x"/>
                                   <variable symbol="y"/>
                                  </function>
                                  <variable symbol="y"/>
                                 </constantatomformula>
                                 <constantatomformula symbol="=">
                                  <function symbol="∩">
                                   <variable symbol="x"/>
                                   <variable symbol="y"/>
                                  </function>
                                  <variable symbol="x"/>
                                 </constantatomformula>
                                </conjunctiveformula>
                               </conjunctiveformula>
                              </quantifiedformula>
                             </quantifiedformula>
                             <quantifiedformula type="all">
                              <variable symbol="x"/>
                              <quantifiedformula type="all">
                               <variable symbol="y"/>
                               <conjunctiveformula type="and">
                                <conjunctiveformula type="impl">
                                 <constantatomformula symbol="=">
                                  <function symbol="∩">
                                   <variable symbol="x"/>
                                   <variable symbol="y"/>
                                  </function>
                                  <variable symbol="x"/>
                                 </constantatomformula>
                                 <constantatomformula symbol="=">
                                  <function symbol="∪">
                                   <variable symbol="x"/>
                                   <variable symbol="y"/>
                                  </function>
                                  <variable symbol="y"/>
                                 </constantatomformula>
                                </conjunctiveformula>
                                <conjunctiveformula type="impl">
                                 <constantatomformula symbol="=">
                                  <function symbol="∪">
                                   <variable symbol="x"/>
                                   <variable symbol="y"/>
                                  </function>
                                  <variable symbol="y"/>
                                 </constantatomformula>
                                 <constantatomformula symbol="=">
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                                            </formulalist>
                                           </sequent>
                                           <rule type="eqr2" symbol="=:r2" param="(1)">
                                            <sequent>
                                             <formulalist>
                                              <constantatomformula symbol="=">
                                               <function symbol="∪">
                                                <variable symbol="y_0"/>
                                                <variable symbol="z_0"/>
                                               </function>
                                               <variable symbol="z_0"/>
                                              </constantatomformula>
                                              <constantatomformula symbol="=">
                                               <function symbol="∪">
                                                <variable symbol="x_0"/>
                                                <variable symbol="z_0"/>
                                               </function>
                                               <variable symbol="z_0"/>
                                              </constantatomformula>
                                             </formulalist>
                                             <formulalist>
                                              <constantatomformula symbol="=">
                                               <function symbol="∪">
                                                <function symbol="∪">
                                                 <variable symbol="x_0"/>
                                                 <variable symbol="y_0"/>
                                                </function>
                                                <variable symbol="z_0"/>
                                               </function>
                                               <variable symbol="z_0"/>
                                              </constantatomformula>
                                             </formulalist>
                                            </sequent>
                                            <rule type="axiom" symbol="axiom">
                                             <sequent>
                                              <formulalist/>
                                              <formulalist>
                                               <constantatomformula symbol="=">
                                                <function symbol="∪">
                                                 <function symbol="∪">
                                                  <variable symbol="x_0"/>
                                                  <variable symbol="y_0"/>
                                                 </function>
                                                 <variable symbol="z_0"/>
                                                </function>
                                                <function symbol="∪">
                                                 <variable symbol="x_0"/>
                                                 <function symbol="∪">
                                                  <variable symbol="y_0"/>
                                                  <variable symbol="z_0"/>
                                                 </function>
                                                </function>
                                               </constantatomformula>
                                              </formulalist>
                                             </sequent>
                                            </rule>
                                            <rule type="eqr2" symbol="=:r2" param="(1 2)">
                                             <sequent>
                                              <formulalist>
                                               <constantatomformula symbol="=">
                                                <function symbol="∪">
                                                 <variable symbol="y_0"/>
                                                 <variable symbol="z_0"/>
                                                </function>
                                                <variable symbol="z_0"/>
                                               </constantatomformula>
                                               <constantatomformula symbol="=">
                                                <function symbol="∪">
                                                 <variable symbol="x_0"/>
                                                 <variable symbol="z_0"/>
                                                </function>
                                                <variable symbol="z_0"/>
                                               </constantatomformula>
                                              </formulalist>
                                              <formulalist>
                                               <constantatomformula symbol="=">
                                                <function symbol="∪">
                                                 <variable symbol="x_0"/>
                                                 <function symbol="∪">
                                                  <variable symbol="y_0"/>
                                                  <variable symbol="z_0"/>
                                                 </function>
                                                </function>
                                                <variable symbol="z_0"/>
                                               </constantatomformula>
                                              </formulalist>
                                             </sequent>
                                             <rule type="axiom" symbol="axiom">
                                              <sequent>
                                               <formulalist>
                                                <constantatomformula symbol="=">
                                                 <function symbol="∪">
                                                  <variable symbol="y_0"/>
                                                  <variable symbol="z_0"/>
                                                 </function>
                                                 <variable symbol="z_0"/>
                                                </constantatomformula>
                                               </formulalist>
                                               <formulalist>
                                                <constantatomformula symbol="=">
                                                 <function symbol="∪">
                                                  <variable symbol="y_0"/>
                                                  <variable symbol="z_0"/>
                                                 </function>
                                                 <variable symbol="z_0"/>
                                                </constantatomformula>
                                               </formulalist>
                                              </sequent>
                                             </rule>
                                             <rule type="contrr" symbol="c:r" param="2">
                                              <sequent>
                                               <formulalist>
                                                <constantatomformula symbol="=">
                                                 <function symbol="∪">
                                                  <variable symbol="x_0"/>
                                                  <variable symbol="z_0"/>
                                                 </function>
                                                 <variable symbol="z_0"/>
                                                </constantatomformula>
                                               </formulalist>
                                               <formulalist>
                                                <constantatomformula symbol="=">
                                                 <function symbol="∪">
                                                  <variable symbol="x_0"/>
                                                  <variable symbol="z_0"/>
                                                 </function>
                                                 <variable symbol="z_0"/>
                                                </constantatomformula>
                                               </formulalist>
                                              </sequent>
                                              <rule type="contrl" symbol="c:l" param="2">
                                               <sequent>
                                                <formulalist>
                                                 <constantatomformula symbol="=">
                                                  <function symbol="∪">
                                                   <variable symbol="x_0"/>
                                                   <variable symbol="z_0"/>
                                                  </function>
                                                  <variable symbol="z_0"/>
                                                 </constantatomformula>
                                                </formulalist>
                                                <formulalist>
                                                 <constantatomformula symbol="=">
                                                  <function symbol="∪">
                                                   <variable symbol="x_0"/>
                                                   <variable symbol="z_0"/>
                                                  </function>
                                                  <variable symbol="z_0"/>
                                                 </constantatomformula>
                                                 <constantatomformula symbol="=">
                                                  <function symbol="∪">
                                                   <variable symbol="x_0"/>
                                                   <variable symbol="z_0"/>
                                                  </function>
                                                  <variable symbol="z_0"/>
                                                 </constantatomformula>
                                                </formulalist>
                                               </sequent>
                                               <rule type="eqr2" symbol="=:r2" param="(1)">
                                                <sequent>
                                                 <formulalist>
                                                  <constantatomformula symbol="=">
                                                   <function symbol="∪">
                                                    <variable symbol="x_0"/>
                                                    <variable symbol="z_0"/>
                                                   </function>
                                                   <variable symbol="z_0"/>
                                                  </constantatomformula>
                                                  <constantatomformula symbol="=">
                                                   <function symbol="∪">
                                                    <variable symbol="x_0"/>
                                                    <variable symbol="z_0"/>
                                                   </function>
                                                   <variable symbol="z_0"/>
                                                  </constantatomformula>
                                                 </formulalist>
                                                 <formulalist>
                                                  <constantatomformula symbol="=">
                                                   <function symbol="∪">
                                                    <variable symbol="x_0"/>
                                                    <variable symbol="z_0"/>
                                                   </function>
                                                   <variable symbol="z_0"/>
                                                  </constantatomformula>
                                                  <constantatomformula symbol="=">
                                                   <function symbol="∪">
                                                    <variable symbol="x_0"/>
                                                    <variable symbol="z_0"/>
                                                   </function>
                                                   <variable symbol="z_0"/>
                                                  </constantatomformula>
                                                 </formulalist>
                                                </sequent>
                                                <rule type="axiom" symbol="axiom">
                                                 <sequent>
                                                  <formulalist>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="∪">
                                                     <variable symbol="x_0"/>
                                                     <variable symbol="z_0"/>
                                                    </function>
                                                    <variable symbol="z_0"/>
                                                   </constantatomformula>
                                                  </formulalist>
                                                  <formulalist>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="∪">
                                                     <variable symbol="x_0"/>
                                                     <variable symbol="z_0"/>
                                                    </function>
                                                    <variable symbol="z_0"/>
                                                   </constantatomformula>
                                                  </formulalist>
                                                 </sequent>
                                                </rule>
                                                <rule type="weakr" symbol="w:r">
                                                 <sequent>
                                                  <formulalist>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="∪">
                                                     <variable symbol="x_0"/>
                                                     <variable symbol="z_0"/>
                                                    </function>
                                                    <variable symbol="z_0"/>
                                                   </constantatomformula>
                                                  </formulalist>
                                                  <formulalist>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="∪">
                                                     <variable symbol="x_0"/>
                                                     <variable symbol="z_0"/>
                                                    </function>
                                                    <variable symbol="z_0"/>
                                                   </constantatomformula>
                                                   <constantatomformula symbol="=">
                                                    <variable symbol="z_0"/>
                                                    <variable symbol="z_0"/>
                                                   </constantatomformula>
                                                  </formulalist>
                                                 </sequent>
                                                 <rule type="axiom" symbol="axiom">
                                                  <sequent>
                                                   <formulalist>
                                                    <constantatomformula symbol="=">
                                                     <function symbol="∪">
                                                      <variable symbol="x_0"/>
                                                      <variable symbol="z_0"/>
                                                     </function>
                                                     <variable symbol="z_0"/>
                                                    </constantatomformula>
                                                   </formulalist>
                                                   <formulalist>
                                                    <constantatomformula symbol="=">
                                                     <function symbol="∪">
                                                      <variable symbol="x_0"/>
                                                      <variable symbol="z_0"/>
                                                     </function>
                                                     <variable symbol="z_0"/>
                                                    </constantatomformula>
                                                   </formulalist>
                                                  </sequent>
                                                 </rule>
                                                </rule>
                                               </rule>
                                              </rule>
                                             </rule>
                                            </rule>
                                           </rule>
                                           <rule type="axiom" symbol="axiom">
                                            <sequent>
                                             <formulalist>
                                              <constantatomformula symbol="=">
                                               <function symbol="∩">
                                                <function symbol="∪">
                                                 <variable symbol="x_0"/>
                                                 <variable symbol="y_0"/>
                                                </function>
                                                <variable symbol="z_0"/>
                                               </function>
                                               <function symbol="∪">
                                                <variable symbol="x_0"/>
                                                <variable symbol="y_0"/>
                                               </function>
                                              </constantatomformula>
                                             </formulalist>
                                             <formulalist>
                                              <constantatomformula symbol="=">
                                               <function symbol="∩">
                                                <function symbol="∪">
                                                 <variable symbol="x_0"/>
                                                 <variable symbol="y_0"/>
                                                </function>
                                                <variable symbol="z_0"/>
                                               </function>
                                               <function symbol="∪">
                                                <variable symbol="x_0"/>
                                                <variable symbol="y_0"/>
                                               </function>
                                              </constantatomformula>
                                             </formulalist>
                                            </sequent>
                                           </rule>
                                          </rule>
                                         </rule>
                                        </rule>
                                       </rule>
                                      </rule>
                                     </rule>
                                    </rule>
                                   </rule>
                                  </rule>
                                 </rule>
                                </rule>
                               </rule>
                              </rule>
                             </rule>
                            </rule>
                           </rule>
                          </rule>
                         </rule>
                        </rule>
                       </rule>
                      </rule>
                     </rule>
                    </rule>
                   </rule>
                  </rule>
                 </rule>
                </rule>
               </rule>
              </rule>
             </rule>
            </rule>
           </rule>
          </rule>
         </rule>
        </rule>
       </rule>
      </rule>
     </rule>
    </rule>
   </rule>
   <rule type="defl" symbol="d:l" param="();[()];()">
    <sequent>
     <formulalist>
      <constantatomformula symbol="L3" definition="L3"/>
     </formulalist>
     <formulalist>
      <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
     </formulalist>
    </sequent>
    <rule type="contrl" symbol="c:l" param="2">
     <sequent>
      <formulalist>
       <conjunctiveformula type="and">
        <constantatomformula symbol="POSET" definition="POSET"/>
        <conjunctiveformula type="and">
         <constantatomformula symbol="GLB" definition="GLB"/>
         <constantatomformula symbol="LUB" definition="LUB"/>
        </conjunctiveformula>
       </conjunctiveformula>
      </formulalist>
      <formulalist>
       <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
      </formulalist>
     </sequent>
     <rule type="andl2" symbol="\and:l2">
      <sequent>
       <formulalist>
        <conjunctiveformula type="and">
         <constantatomformula symbol="POSET" definition="POSET"/>
         <conjunctiveformula type="and">
          <constantatomformula symbol="GLB" definition="GLB"/>
          <constantatomformula symbol="LUB" definition="LUB"/>
         </conjunctiveformula>
        </conjunctiveformula>
        <conjunctiveformula type="and">
         <constantatomformula symbol="POSET" definition="POSET"/>
         <conjunctiveformula type="and">
          <constantatomformula symbol="GLB" definition="GLB"/>
          <constantatomformula symbol="LUB" definition="LUB"/>
         </conjunctiveformula>
        </conjunctiveformula>
       </formulalist>
       <formulalist>
        <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
       </formulalist>
      </sequent>
      <rule type="contrl" symbol="c:l" param="2,1">
       <sequent>
        <formulalist>
         <conjunctiveformula type="and">
          <constantatomformula symbol="GLB" definition="GLB"/>
          <constantatomformula symbol="LUB" definition="LUB"/>
         </conjunctiveformula>
         <conjunctiveformula type="and">
          <constantatomformula symbol="POSET" definition="POSET"/>
          <conjunctiveformula type="and">
           <constantatomformula symbol="GLB" definition="GLB"/>
           <constantatomformula symbol="LUB" definition="LUB"/>
          </conjunctiveformula>
         </conjunctiveformula>
        </formulalist>
        <formulalist>
         <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
        </formulalist>
       </sequent>
       <rule type="perml" symbol="\pi:l" param="(2 3)">
        <sequent>
         <formulalist>
          <conjunctiveformula type="and">
           <constantatomformula symbol="GLB" definition="GLB"/>
           <constantatomformula symbol="LUB" definition="LUB"/>
          </conjunctiveformula>
          <conjunctiveformula type="and">
           <constantatomformula symbol="GLB" definition="GLB"/>
           <constantatomformula symbol="LUB" definition="LUB"/>
          </conjunctiveformula>
          <conjunctiveformula type="and">
           <constantatomformula symbol="POSET" definition="POSET"/>
           <conjunctiveformula type="and">
            <constantatomformula symbol="GLB" definition="GLB"/>
            <constantatomformula symbol="LUB" definition="LUB"/>
           </conjunctiveformula>
          </conjunctiveformula>
         </formulalist>
         <formulalist>
          <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
         </formulalist>
        </sequent>
        <rule type="andl1" symbol="\and:l1">
         <sequent>
          <formulalist>
           <conjunctiveformula type="and">
            <constantatomformula symbol="GLB" definition="GLB"/>
            <constantatomformula symbol="LUB" definition="LUB"/>
           </conjunctiveformula>
           <conjunctiveformula type="and">
            <constantatomformula symbol="POSET" definition="POSET"/>
            <conjunctiveformula type="and">
             <constantatomformula symbol="GLB" definition="GLB"/>
             <constantatomformula symbol="LUB" definition="LUB"/>
            </conjunctiveformula>
           </conjunctiveformula>
           <conjunctiveformula type="and">
            <constantatomformula symbol="GLB" definition="GLB"/>
            <constantatomformula symbol="LUB" definition="LUB"/>
           </conjunctiveformula>
          </formulalist>
          <formulalist>
           <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
          </formulalist>
         </sequent>
         <rule type="perml" symbol="\pi:l" param="(1 3)">
          <sequent>
           <formulalist>
            <constantatomformula symbol="GLB" definition="GLB"/>
            <conjunctiveformula type="and">
             <constantatomformula symbol="POSET" definition="POSET"/>
             <conjunctiveformula type="and">
              <constantatomformula symbol="GLB" definition="GLB"/>
              <constantatomformula symbol="LUB" definition="LUB"/>
             </conjunctiveformula>
            </conjunctiveformula>
            <conjunctiveformula type="and">
             <constantatomformula symbol="GLB" definition="GLB"/>
             <constantatomformula symbol="LUB" definition="LUB"/>
            </conjunctiveformula>
           </formulalist>
           <formulalist>
            <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
           </formulalist>
          </sequent>
          <rule type="andl2" symbol="\and:l2">
           <sequent>
            <formulalist>
             <conjunctiveformula type="and">
              <constantatomformula symbol="GLB" definition="GLB"/>
              <constantatomformula symbol="LUB" definition="LUB"/>
             </conjunctiveformula>
             <conjunctiveformula type="and">
              <constantatomformula symbol="POSET" definition="POSET"/>
              <conjunctiveformula type="and">
               <constantatomformula symbol="GLB" definition="GLB"/>
               <constantatomformula symbol="LUB" definition="LUB"/>
              </conjunctiveformula>
             </conjunctiveformula>
             <constantatomformula symbol="GLB" definition="GLB"/>
            </formulalist>
            <formulalist>
             <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
            </formulalist>
           </sequent>
           <rule type="perml" symbol="\pi:l" param="(1 2)">
            <sequent>
             <formulalist>
              <constantatomformula symbol="LUB" definition="LUB"/>
              <conjunctiveformula type="and">
               <constantatomformula symbol="POSET" definition="POSET"/>
               <conjunctiveformula type="and">
                <constantatomformula symbol="GLB" definition="GLB"/>
                <constantatomformula symbol="LUB" definition="LUB"/>
               </conjunctiveformula>
              </conjunctiveformula>
              <constantatomformula symbol="GLB" definition="GLB"/>
             </formulalist>
             <formulalist>
              <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
             </formulalist>
            </sequent>
            <rule type="andl1" symbol="\and:l1">
             <sequent>
              <formulalist>
               <conjunctiveformula type="and">
                <constantatomformula symbol="POSET" definition="POSET"/>
                <conjunctiveformula type="and">
                 <constantatomformula symbol="GLB" definition="GLB"/>
                 <constantatomformula symbol="LUB" definition="LUB"/>
                </conjunctiveformula>
               </conjunctiveformula>
               <constantatomformula symbol="LUB" definition="LUB"/>
               <constantatomformula symbol="GLB" definition="GLB"/>
              </formulalist>
              <formulalist>
               <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
              </formulalist>
             </sequent>
             <rule type="defl" symbol="d:l" param="();[()];()">
              <sequent>
               <formulalist>
                <constantatomformula symbol="POSET" definition="POSET"/>
                <constantatomformula symbol="LUB" definition="LUB"/>
                <constantatomformula symbol="GLB" definition="GLB"/>
               </formulalist>
               <formulalist>
                <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
               </formulalist>
              </sequent>
              <rule type="contrl" symbol="c:l" param="2,1,1">
               <sequent>
                <formulalist>
                 <conjunctiveformula type="and">
                  <constantatomformula symbol="R" definition="R"/>
                  <conjunctiveformula type="and">
                   <constantatomformula symbol="AS" definition="AS"/>
                   <constantatomformula symbol="T" definition="T"/>
                  </conjunctiveformula>
                 </conjunctiveformula>
                 <constantatomformula symbol="LUB" definition="LUB"/>
                 <constantatomformula symbol="GLB" definition="GLB"/>
                </formulalist>
                <formulalist>
                 <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                </formulalist>
               </sequent>
               <rule type="andl1" symbol="\and:l1">
                <sequent>
                 <formulalist>
                  <conjunctiveformula type="and">
                   <constantatomformula symbol="R" definition="R"/>
                   <conjunctiveformula type="and">
                    <constantatomformula symbol="AS" definition="AS"/>
                    <constantatomformula symbol="T" definition="T"/>
                   </conjunctiveformula>
                  </conjunctiveformula>
                  <conjunctiveformula type="and">
                   <constantatomformula symbol="R" definition="R"/>
                   <conjunctiveformula type="and">
                    <constantatomformula symbol="AS" definition="AS"/>
                    <constantatomformula symbol="T" definition="T"/>
                   </conjunctiveformula>
                  </conjunctiveformula>
                  <constantatomformula symbol="LUB" definition="LUB"/>
                  <constantatomformula symbol="GLB" definition="GLB"/>
                 </formulalist>
                 <formulalist>
                  <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                 </formulalist>
                </sequent>
                <rule type="perml" symbol="\pi:l" param="(1 2)">
                 <sequent>
                  <formulalist>
                   <constantatomformula symbol="R" definition="R"/>
                   <conjunctiveformula type="and">
                    <constantatomformula symbol="R" definition="R"/>
                    <conjunctiveformula type="and">
                     <constantatomformula symbol="AS" definition="AS"/>
                     <constantatomformula symbol="T" definition="T"/>
                    </conjunctiveformula>
                   </conjunctiveformula>
                   <constantatomformula symbol="LUB" definition="LUB"/>
                   <constantatomformula symbol="GLB" definition="GLB"/>
                  </formulalist>
                  <formulalist>
                   <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                  </formulalist>
                 </sequent>
                 <rule type="andl2" symbol="\and:l2">
                  <sequent>
                   <formulalist>
                    <conjunctiveformula type="and">
                     <constantatomformula symbol="R" definition="R"/>
                     <conjunctiveformula type="and">
                      <constantatomformula symbol="AS" definition="AS"/>
                      <constantatomformula symbol="T" definition="T"/>
                     </conjunctiveformula>
                    </conjunctiveformula>
                    <constantatomformula symbol="R" definition="R"/>
                    <constantatomformula symbol="LUB" definition="LUB"/>
                    <constantatomformula symbol="GLB" definition="GLB"/>
                   </formulalist>
                   <formulalist>
                    <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                   </formulalist>
                  </sequent>
                  <rule type="andl1" symbol="\and:l1">
                   <sequent>
                    <formulalist>
                     <conjunctiveformula type="and">
                      <constantatomformula symbol="AS" definition="AS"/>
                      <constantatomformula symbol="T" definition="T"/>
                     </conjunctiveformula>
                     <constantatomformula symbol="R" definition="R"/>
                     <constantatomformula symbol="LUB" definition="LUB"/>
                     <constantatomformula symbol="GLB" definition="GLB"/>
                    </formulalist>
                    <formulalist>
                     <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                    </formulalist>
                   </sequent>
                   <rule type="defr" symbol="d:r" param="();[()];()">
                    <sequent>
                     <formulalist>
                      <constantatomformula symbol="AS" definition="AS"/>
                      <constantatomformula symbol="R" definition="R"/>
                      <constantatomformula symbol="LUB" definition="LUB"/>
                      <constantatomformula symbol="GLB" definition="GLB"/>
                     </formulalist>
                     <formulalist>
                      <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                     </formulalist>
                    </sequent>
                    <rule type="contrl" symbol="c:l" param="2,2,2,2">
                     <sequent>
                      <formulalist>
                       <constantatomformula symbol="AS" definition="AS"/>
                       <constantatomformula symbol="R" definition="R"/>
                       <constantatomformula symbol="LUB" definition="LUB"/>
                       <constantatomformula symbol="GLB" definition="GLB"/>
                      </formulalist>
                      <formulalist>
                       <conjunctiveformula type="and">
                        <constantatomformula symbol="=">
                         <function symbol="∪">
                          <function symbol="∩">
                           <constant symbol="s_1"/>
                           <constant symbol="s_2"/>
                          </function>
                          <constant symbol="s_1"/>
                         </function>
                         <constant symbol="s_1"/>
                        </constantatomformula>
                        <constantatomformula symbol="=">
                         <function symbol="∩">
                          <function symbol="∪">
                           <constant symbol="s_3"/>
                           <constant symbol="s_4"/>
                          </function>
                          <constant symbol="s_3"/>
                         </function>
                         <constant symbol="s_3"/>
                        </constantatomformula>
                       </conjunctiveformula>
                      </formulalist>
                     </sequent>
                     <rule type="perml" symbol="\pi:l" param="(2 5)(4 8)(6 7)">
                      <sequent>
                       <formulalist>
                        <constantatomformula symbol="AS" definition="AS"/>
                        <constantatomformula symbol="AS" definition="AS"/>
                        <constantatomformula symbol="R" definition="R"/>
                        <constantatomformula symbol="R" definition="R"/>
                        <constantatomformula symbol="LUB" definition="LUB"/>
                        <constantatomformula symbol="LUB" definition="LUB"/>
                        <constantatomformula symbol="GLB" definition="GLB"/>
                        <constantatomformula symbol="GLB" definition="GLB"/>
                       </formulalist>
                       <formulalist>
                        <conjunctiveformula type="and">
                         <constantatomformula symbol="=">
                          <function symbol="∪">
                           <function symbol="∩">
                            <constant symbol="s_1"/>
                            <constant symbol="s_2"/>
                           </function>
                           <constant symbol="s_1"/>
                          </function>
                          <constant symbol="s_1"/>
                         </constantatomformula>
                         <constantatomformula symbol="=">
                          <function symbol="∩">
                           <function symbol="∪">
                            <constant symbol="s_3"/>
                            <constant symbol="s_4"/>
                           </function>
                           <constant symbol="s_3"/>
                          </function>
                          <constant symbol="s_3"/>
                         </constantatomformula>
                        </conjunctiveformula>
                       </formulalist>
                      </sequent>
                      <rule type="andr" symbol="\and:r">
                       <sequent>
                        <formulalist>
                         <constantatomformula symbol="AS" definition="AS"/>
                         <constantatomformula symbol="LUB" definition="LUB"/>
                         <constantatomformula symbol="R" definition="R"/>
                         <constantatomformula symbol="GLB" definition="GLB"/>
                         <constantatomformula symbol="AS" definition="AS"/>
                         <constantatomformula symbol="GLB" definition="GLB"/>
                         <constantatomformula symbol="LUB" definition="LUB"/>
                         <constantatomformula symbol="R" definition="R"/>
                        </formulalist>
                        <formulalist>
                         <conjunctiveformula type="and">
                          <constantatomformula symbol="=">
                           <function symbol="∪">
                            <function symbol="∩">
                             <constant symbol="s_1"/>
                             <constant symbol="s_2"/>
                            </function>
                            <constant symbol="s_1"/>
                           </function>
                           <constant symbol="s_1"/>
                          </constantatomformula>
                          <constantatomformula symbol="=">
                           <function symbol="∩">
                            <function symbol="∪">
                             <constant symbol="s_3"/>
                             <constant symbol="s_4"/>
                            </function>
                            <constant symbol="s_3"/>
                           </function>
                           <constant symbol="s_3"/>
                          </constantatomformula>
                         </conjunctiveformula>
                        </formulalist>
                       </sequent>
                       <rule type="defl" symbol="d:l" param="();[()];()">
                        <sequent>
                         <formulalist>
                          <constantatomformula symbol="AS" definition="AS"/>
                          <constantatomformula symbol="LUB" definition="LUB"/>
                          <constantatomformula symbol="R" definition="R"/>
                          <constantatomformula symbol="GLB" definition="GLB"/>
                         </formulalist>
                         <formulalist>
                          <constantatomformula symbol="=">
                           <function symbol="∪">
                            <function symbol="∩">
                             <constant symbol="s_1"/>
                             <constant symbol="s_2"/>
                            </function>
                            <constant symbol="s_1"/>
                           </function>
                           <constant symbol="s_1"/>
                          </constantatomformula>
                         </formulalist>
                        </sequent>
                        <rule type="foralll" symbol="\all:l">
                         <sequent>
                          <formulalist>
                           <quantifiedformula type="all">
                            <variable symbol="x"/>
                            <quantifiedformula type="all">
                             <variable symbol="y"/>
                             <conjunctiveformula type="impl">
                              <conjunctiveformula type="and">
                               <constantatomformula symbol="≤" definition="≤">
                                <variable symbol="x"/>
                                <variable symbol="y"/>
                               </constantatomformula>
                               <constantatomformula symbol="≤" definition="≤">
                                <variable symbol="y"/>
                                <variable symbol="x"/>
                               </constantatomformula>
                              </conjunctiveformula>
                              <constantatomformula symbol="=">
                               <variable symbol="x"/>
                               <variable symbol="y"/>
                              </constantatomformula>
                             </conjunctiveformula>
                            </quantifiedformula>
                           </quantifiedformula>
                           <constantatomformula symbol="LUB" definition="LUB"/>
                           <constantatomformula symbol="R" definition="R"/>
                           <constantatomformula symbol="GLB" definition="GLB"/>
                          </formulalist>
                          <formulalist>
                           <constantatomformula symbol="=">
                            <function symbol="∪">
                             <function symbol="∩">
                              <constant symbol="s_1"/>
                              <constant symbol="s_2"/>
                             </function>
                             <constant symbol="s_1"/>
                            </function>
                            <constant symbol="s_1"/>
                           </constantatomformula>
                          </formulalist>
                         </sequent>
                         <rule type="foralll" symbol="\all:l">
                          <sequent>
                           <formulalist>
                            <quantifiedformula type="all">
                             <variable symbol="y"/>
                             <conjunctiveformula type="impl">
                              <conjunctiveformula type="and">
                               <constantatomformula symbol="≤" definition="≤">
                                <function symbol="∪">
                                 <function symbol="∩">
                                  <constant symbol="s_1"/>
                                  <constant symbol="s_2"/>
                                 </function>
                                 <constant symbol="s_1"/>
                                </function>
                                <variable symbol="y"/>
                               </constantatomformula>
                               <constantatomformula symbol="≤" definition="≤">
                                <variable symbol="y"/>
                                <function symbol="∪">
                                 <function symbol="∩">
                                  <constant symbol="s_1"/>
                                  <constant symbol="s_2"/>
                                 </function>
                                 <constant symbol="s_1"/>
                                </function>
                               </constantatomformula>
                              </conjunctiveformula>
                              <constantatomformula symbol="=">
                               <function symbol="∪">
                                <function symbol="∩">
                                 <constant symbol="s_1"/>
                                 <constant symbol="s_2"/>
                                </function>
                                <constant symbol="s_1"/>
                               </function>
                               <variable symbol="y"/>
                              </constantatomformula>
                             </conjunctiveformula>
                            </quantifiedformula>
                            <constantatomformula symbol="LUB" definition="LUB"/>
                            <constantatomformula symbol="R" definition="R"/>
                            <constantatomformula symbol="GLB" definition="GLB"/>
                           </formulalist>
                           <formulalist>
                            <constantatomformula symbol="=">
                             <function symbol="∪">
                              <function symbol="∩">
                               <constant symbol="s_1"/>
                               <constant symbol="s_2"/>
                              </function>
                              <constant symbol="s_1"/>
                             </function>
                             <constant symbol="s_1"/>
                            </constantatomformula>
                           </formulalist>
                          </sequent>
                          <rule type="impll" symbol="\impl:l">
                           <sequent>
                            <formulalist>
                             <conjunctiveformula type="impl">
                              <conjunctiveformula type="and">
                               <constantatomformula symbol="≤" definition="≤">
                                <function symbol="∪">
                                 <function symbol="∩">
                                  <constant symbol="s_1"/>
                                  <constant symbol="s_2"/>
                                 </function>
                                 <constant symbol="s_1"/>
                                </function>
                                <constant symbol="s_1"/>
                               </constantatomformula>
                               <constantatomformula symbol="≤" definition="≤">
                                <constant symbol="s_1"/>
                                <function symbol="∪">
                                 <function symbol="∩">
                                  <constant symbol="s_1"/>
                                  <constant symbol="s_2"/>
                                 </function>
                                 <constant symbol="s_1"/>
                                </function>
                               </constantatomformula>
                              </conjunctiveformula>
                              <constantatomformula symbol="=">
                               <function symbol="∪">
                                <function symbol="∩">
                                 <constant symbol="s_1"/>
                                 <constant symbol="s_2"/>
                                </function>
                                <constant symbol="s_1"/>
                               </function>
                               <constant symbol="s_1"/>
                              </constantatomformula>
                             </conjunctiveformula>
                             <constantatomformula symbol="LUB" definition="LUB"/>
                             <constantatomformula symbol="R" definition="R"/>
                             <constantatomformula symbol="GLB" definition="GLB"/>
                            </formulalist>
                            <formulalist>
                             <constantatomformula symbol="=">
                              <function symbol="∪">
                               <function symbol="∩">
                                <constant symbol="s_1"/>
                                <constant symbol="s_2"/>
                               </function>
                               <constant symbol="s_1"/>
                              </function>
                              <constant symbol="s_1"/>
                             </constantatomformula>
                            </formulalist>
                           </sequent>
                           <rule type="defl" symbol="d:l" param="();[()];()">
                            <sequent>
                             <formulalist>
                              <constantatomformula symbol="LUB" definition="LUB"/>
                              <constantatomformula symbol="R" definition="R"/>
                              <constantatomformula symbol="GLB" definition="GLB"/>
                             </formulalist>
                             <formulalist>
                              <conjunctiveformula type="and">
                               <constantatomformula symbol="≤" definition="≤">
                                <function symbol="∪">
                                 <function symbol="∩">
                                  <constant symbol="s_1"/>
                                  <constant symbol="s_2"/>
                                 </function>
                                 <constant symbol="s_1"/>
                                </function>
                                <constant symbol="s_1"/>
                               </constantatomformula>
                               <constantatomformula symbol="≤" definition="≤">
                                <constant symbol="s_1"/>
                                <function symbol="∪">
                                 <function symbol="∩">
                                  <constant symbol="s_1"/>
                                  <constant symbol="s_2"/>
                                 </function>
                                 <constant symbol="s_1"/>
                                </function>
                               </constantatomformula>
                              </conjunctiveformula>
                             </formulalist>
                            </sequent>
                            <rule type="foralll" symbol="\all:l">
                             <sequent>
                              <formulalist>
                               <quantifiedformula type="all">
                                <variable symbol="x"/>
                                <quantifiedformula type="all">
                                 <variable symbol="y"/>
                                 <conjunctiveformula type="and">
                                  <conjunctiveformula type="and">
                                   <constantatomformula symbol="≤" definition="≤">
                                    <variable symbol="x"/>
                                    <function symbol="∪">
                                     <variable symbol="x"/>
                                     <variable symbol="y"/>
                                    </function>
                                   </constantatomformula>
                                   <constantatomformula symbol="≤" definition="≤">
                                    <variable symbol="y"/>
                                    <function symbol="∪">
                                     <variable symbol="x"/>
                                     <variable symbol="y"/>
                                    </function>
                                   </constantatomformula>
                                  </conjunctiveformula>
                                  <quantifiedformula type="all">
                                   <variable symbol="z"/>
                                   <conjunctiveformula type="impl">
                                    <conjunctiveformula type="and">
                                     <constantatomformula symbol="≤" definition="≤">
                                      <variable symbol="x"/>
                                      <variable symbol="z"/>
                                     </constantatomformula>
                                     <constantatomformula symbol="≤" definition="≤">
                                      <variable symbol="y"/>
                                      <variable symbol="z"/>
                                     </constantatomformula>
                                    </conjunctiveformula>
                                    <constantatomformula symbol="≤" definition="≤">
                                     <function symbol="∪">
                                      <variable symbol="x"/>
                                      <variable symbol="y"/>
                                     </function>
                                     <variable symbol="z"/>
                                    </constantatomformula>
                                   </conjunctiveformula>
                                  </quantifiedformula>
                                 </conjunctiveformula>
                                </quantifiedformula>
                               </quantifiedformula>
                               <constantatomformula symbol="R" definition="R"/>
                               <constantatomformula symbol="GLB" definition="GLB"/>
                              </formulalist>
                              <formulalist>
                               <conjunctiveformula type="and">
                                <constantatomformula symbol="≤" definition="≤">
                                 <function symbol="∪">
                                  <function symbol="∩">
                                   <constant symbol="s_1"/>
                                   <constant symbol="s_2"/>
                                  </function>
                                  <constant symbol="s_1"/>
                                 </function>
                                 <constant symbol="s_1"/>
                                </constantatomformula>
                                <constantatomformula symbol="≤" definition="≤">
                                 <constant symbol="s_1"/>
                                 <function symbol="∪">
                                  <function symbol="∩">
                                   <constant symbol="s_1"/>
                                   <constant symbol="s_2"/>
                                  </function>
                                  <constant symbol="s_1"/>
                                 </function>
                                </constantatomformula>
                               </conjunctiveformula>
                              </formulalist>
                             </sequent>
                             <rule type="foralll" symbol="\all:l">
                              <sequent>
                               <formulalist>
                                <quantifiedformula type="all">
                                 <variable symbol="y"/>
                                 <conjunctiveformula type="and">
                                  <conjunctiveformula type="and">
                                   <constantatomformula symbol="≤" definition="≤">
                                    <function symbol="∩">
                                     <constant symbol="s_1"/>
                                     <constant symbol="s_2"/>
                                    </function>
                                    <function symbol="∪">
                                     <function symbol="∩">
                                      <constant symbol="s_1"/>
                                      <constant symbol="s_2"/>
                                     </function>
                                     <variable symbol="y"/>
                                    </function>
                                   </constantatomformula>
                                   <constantatomformula symbol="≤" definition="≤">
                                    <variable symbol="y"/>
                                    <function symbol="∪">
                                     <function symbol="∩">
                                      <constant symbol="s_1"/>
                                      <constant symbol="s_2"/>
                                     </function>
                                     <variable symbol="y"/>
                                    </function>
                                   </constantatomformula>
                                  </conjunctiveformula>
                                  <quantifiedformula type="all">
                                   <variable symbol="z"/>
                                   <conjunctiveformula type="impl">
                                    <conjunctiveformula type="and">
                                     <constantatomformula symbol="≤" definition="≤">
                                      <function symbol="∩">
                                       <constant symbol="s_1"/>
                                       <constant symbol="s_2"/>
                                      </function>
                                      <variable symbol="z"/>
                                     </constantatomformula>
                                     <constantatomformula symbol="≤" definition="≤">
                                      <variable symbol="y"/>
                                      <variable symbol="z"/>
                                     </constantatomformula>
                                    </conjunctiveformula>
                                    <constantatomformula symbol="≤" definition="≤">
                                     <function symbol="∪">
                                      <function symbol="∩">
                                       <constant symbol="s_1"/>
                                       <constant symbol="s_2"/>
                                      </function>
                                      <variable symbol="y"/>
                                     </function>
                                     <variable symbol="z"/>
                                    </constantatomformula>
                                   </conjunctiveformula>
                                  </quantifiedformula>
                                 </conjunctiveformula>
                                </quantifiedformula>
                                <constantatomformula symbol="R" definition="R"/>
                                <constantatomformula symbol="GLB" definition="GLB"/>
                               </formulalist>
                               <formulalist>
                                <conjunctiveformula type="and">
                                 <constantatomformula symbol="≤" definition="≤">
                                  <function symbol="∪">
                                   <function symbol="∩">
                                    <constant symbol="s_1"/>
                                    <constant symbol="s_2"/>
                                   </function>
                                   <constant symbol="s_1"/>
                                  </function>
                                  <constant symbol="s_1"/>
                                 </constantatomformula>
                                 <constantatomformula symbol="≤" definition="≤">
                                  <constant symbol="s_1"/>
                                  <function symbol="∪">
                                   <function symbol="∩">
                                    <constant symbol="s_1"/>
                                    <constant symbol="s_2"/>
                                   </function>
                                   <constant symbol="s_1"/>
                                  </function>
                                 </constantatomformula>
                                </conjunctiveformula>
                               </formulalist>
                              </sequent>
                              <rule type="contrl" symbol="c:l" param="2,1,1">
                               <sequent>
                                <formulalist>
                                 <conjunctiveformula type="and">
                                  <conjunctiveformula type="and">
                                   <constantatomformula symbol="≤" definition="≤">
                                    <function symbol="∩">
                                     <constant symbol="s_1"/>
                                     <constant symbol="s_2"/>
                                    </function>
                                    <function symbol="∪">
                                     <function symbol="∩">
                                      <constant symbol="s_1"/>
                                      <constant symbol="s_2"/>
                                     </function>
                                     <constant symbol="s_1"/>
                                    </function>
                                   </constantatomformula>
                                   <constantatomformula symbol="≤" definition="≤">
                                    <constant symbol="s_1"/>
                                    <function symbol="∪">
                                     <function symbol="∩">
                                      <constant symbol="s_1"/>
                                      <constant symbol="s_2"/>
                                     </function>
                                     <constant symbol="s_1"/>
                                    </function>
                                   </constantatomformula>
                                  </conjunctiveformula>
                                  <quantifiedformula type="all">
                                   <variable symbol="z"/>
                                   <conjunctiveformula type="impl">
                                    <conjunctiveformula type="and">
                                     <constantatomformula symbol="≤" definition="≤">
                                      <function symbol="∩">
                                       <constant symbol="s_1"/>
                                       <constant symbol="s_2"/>
                                      </function>
                                      <variable symbol="z"/>
                                     </constantatomformula>
                                     <constantatomformula symbol="≤" definition="≤">
                                      <constant symbol="s_1"/>
                                      <variable symbol="z"/>
                                     </constantatomformula>
                                    </conjunctiveformula>
                                    <constantatomformula symbol="≤" definition="≤">
                                     <function symbol="∪">
                                      <function symbol="∩">
                                       <constant symbol="s_1"/>
                                       <constant symbol="s_2"/>
                                      </function>
                                      <constant symbol="s_1"/>
                                     </function>
                                     <variable symbol="z"/>
                                    </constantatomformula>
                                   </conjunctiveformula>
                                  </quantifiedformula>
                                 </conjunctiveformula>
                                 <constantatomformula symbol="R" definition="R"/>
                                 <constantatomformula symbol="GLB" definition="GLB"/>
                                </formulalist>
                                <formulalist>
                                 <conjunctiveformula type="and">
                                  <constantatomformula symbol="≤" definition="≤">
                                   <function symbol="∪">
                                    <function symbol="∩">
                                     <constant symbol="s_1"/>
                                     <constant symbol="s_2"/>
                                    </function>
                                    <constant symbol="s_1"/>
                                   </function>
                                   <constant symbol="s_1"/>
                                  </constantatomformula>
                                  <constantatomformula symbol="≤" definition="≤">
                                   <constant symbol="s_1"/>
                                   <function symbol="∪">
                                    <function symbol="∩">
                                     <constant symbol="s_1"/>
                                     <constant symbol="s_2"/>
                                    </function>
                                    <constant symbol="s_1"/>
                                   </function>
                                  </constantatomformula>
                                 </conjunctiveformula>
                                </formulalist>
                               </sequent>
                               <rule type="perml" symbol="\pi:l" param="(2 4)">
                                <sequent>
                                 <formulalist>
                                  <conjunctiveformula type="and">
                                   <conjunctiveformula type="and">
                                    <constantatomformula symbol="≤" definition="≤">
                                     <function symbol="∩">
                                      <constant symbol="s_1"/>
                                      <constant symbol="s_2"/>
                                     </function>
                                     <function symbol="∪">
                                      <function symbol="∩">
                                       <constant symbol="s_1"/>
                                       <constant symbol="s_2"/>
                                      </function>
                                      <constant symbol="s_1"/>
                                     </function>
                                    </constantatomformula>
                                    <constantatomformula symbol="≤" definition="≤">
                                     <constant symbol="s_1"/>
                                     <function symbol="∪">
                                      <function symbol="∩">
                                       <constant symbol="s_1"/>
                                       <constant symbol="s_2"/>
                                      </function>
                                      <constant symbol="s_1"/>
                                     </function>
                                    </constantatomformula>
                                   </conjunctiveformula>
                                   <quantifiedformula type="all">
                                    <variable symbol="z"/>
                                    <conjunctiveformula type="impl">
                                     <conjunctiveformula type="and">
                                      <constantatomformula symbol="≤" definition="≤">
                                       <function symbol="∩">
                                        <constant symbol="s_1"/>
                                        <constant symbol="s_2"/>
                                       </function>
                                       <variable symbol="z"/>
                                      </constantatomformula>
                                      <constantatomformula symbol="≤" definition="≤">
                                       <constant symbol="s_1"/>
                                       <variable symbol="z"/>
                                      </constantatomformula>
                                     </conjunctiveformula>
                                     <constantatomformula symbol="≤" definition="≤">
                                      <function symbol="∪">
                                       <function symbol="∩">
                                        <constant symbol="s_1"/>
                                        <constant symbol="s_2"/>
                                       </function>
                                       <constant symbol="s_1"/>
                                      </function>
                                      <variable symbol="z"/>
                                     </constantatomformula>
                                    </conjunctiveformula>
                                   </quantifiedformula>
                                  </conjunctiveformula>
                                  <conjunctiveformula type="and">
                                   <conjunctiveformula type="and">
                                    <constantatomformula symbol="≤" definition="≤">
                                     <function symbol="∩">
                                      <constant symbol="s_1"/>
                                      <constant symbol="s_2"/>
                                     </function>
                                     <function symbol="∪">
                                      <function symbol="∩">
                                       <constant symbol="s_1"/>
                                       <constant symbol="s_2"/>
                                      </function>
                                      <constant symbol="s_1"/>
                                     </function>
                                    </constantatomformula>
                                    <constantatomformula symbol="≤" definition="≤">
                                     <constant symbol="s_1"/>
                                     <function symbol="∪">
                                      <function symbol="∩">
                                       <constant symbol="s_1"/>
                                       <constant symbol="s_2"/>
                                      </function>
                                      <constant symbol="s_1"/>
                                     </function>
                                    </constantatomformula>
                                   </conjunctiveformula>
                                   <quantifiedformula type="all">
                                    <variable symbol="z"/>
                                    <conjunctiveformula type="impl">
                                     <conjunctiveformula type="and">
                                      <constantatomformula symbol="≤" definition="≤">
                                       <function symbol="∩">
                                        <constant symbol="s_1"/>
                                        <constant symbol="s_2"/>
                                       </function>
                                       <variable symbol="z"/>
                                      </constantatomformula>
                                      <constantatomformula symbol="≤" definition="≤">
                                       <constant symbol="s_1"/>
                                       <variable symbol="z"/>
                                      </constantatomformula>
                                     </conjunctiveformula>
                                     <constantatomformula symbol="≤" definition="≤">
                                      <function symbol="∪">
                                       <function symbol="∩">
                                        <constant symbol="s_1"/>
                                        <constant symbol="s_2"/>
                                       </function>
                                       <constant symbol="s_1"/>
                                      </function>
                                      <variable symbol="z"/>
                                     </constantatomformula>
                                    </conjunctiveformula>
                                   </quantifiedformula>
                                  </conjunctiveformula>
                                  <constantatomformula symbol="R" definition="R"/>
                                  <constantatomformula symbol="GLB" definition="GLB"/>
                                 </formulalist>
                                 <formulalist>
                                  <conjunctiveformula type="and">
                                   <constantatomformula symbol="≤" definition="≤">
                                    <function symbol="∪">
                                     <function symbol="∩">
                                      <constant symbol="s_1"/>
                                      <constant symbol="s_2"/>
                                     </function>
                                     <constant symbol="s_1"/>
                                    </function>
                                    <constant symbol="s_1"/>
                                   </constantatomformula>
                                   <constantatomformula symbol="≤" definition="≤">
                                    <constant symbol="s_1"/>
                                    <function symbol="∪">
                                     <function symbol="∩">
                                      <constant symbol="s_1"/>
                                      <constant symbol="s_2"/>
                                     </function>
                                     <constant symbol="s_1"/>
                                    </function>
                                   </constantatomformula>
                                  </conjunctiveformula>
                                 </formulalist>
                                </sequent>
                                <rule type="andl2" symbol="\and:l2">
                                 <sequent>
                                  <formulalist>
                                   <conjunctiveformula type="and">
                                    <conjunctiveformula type="and">
                                     <constantatomformula symbol="≤" definition="≤">
                                      <function symbol="∩">
                                       <constant symbol="s_1"/>
                                       <constant symbol="s_2"/>
                                      </function>
                                      <function symbol="∪">
                                       <function symbol="∩">
                                        <constant symbol="s_1"/>
                                        <constant symbol="s_2"/>
                                       </function>
                                       <constant symbol="s_1"/>
                                      </function>
                                     </constantatomformula>
                                     <constantatomformula symbol="≤" definition="≤">
                                      <constant symbol="s_1"/>
                                      <function symbol="∪">
                                       <function symbol="∩">
                                        <constant symbol="s_1"/>
                                        <constant symbol="s_2"/>
                                       </function>
                                       <constant symbol="s_1"/>
                                      </function>
                                     </constantatomformula>
                                    </conjunctiveformula>
                                    <quantifiedformula type="all">
                                     <variable symbol="z"/>
                                     <conjunctiveformula type="impl">
                                      <conjunctiveformula type="and">
                                       <constantatomformula symbol="≤" definition="≤">
                                        <function symbol="∩">
                                         <constant symbol="s_1"/>
                                         <constant symbol="s_2"/>
                                        </function>
                                        <variable symbol="z"/>
                                       </constantatomformula>
                                       <constantatomformula symbol="≤" definition="≤">
                                        <constant symbol="s_1"/>
                                        <variable symbol="z"/>
                                       </constantatomformula>
                                      </conjunctiveformula>
                                      <constantatomformula symbol="≤" definition="≤">
                                       <function symbol="∪">
                                        <function symbol="∩">
                                         <constant symbol="s_1"/>
                                         <constant symbol="s_2"/>
                                        </function>
                                        <constant symbol="s_1"/>
                                       </function>
                                       <variable symbol="z"/>
                                      </constantatomformula>
                                     </conjunctiveformula>
                                    </quantifiedformula>
                                   </conjunctiveformula>
                                   <constantatomformula symbol="GLB" definition="GLB"/>
                                   <constantatomformula symbol="R" definition="R"/>
                                   <conjunctiveformula type="and">
                                    <conjunctiveformula type="and">
                                     <constantatomformula symbol="≤" definition="≤">
                                      <function symbol="∩">
                                       <constant symbol="s_1"/>
                                       <constant symbol="s_2"/>
                                      </function>
                                      <function symbol="∪">
                                       <function symbol="∩">
                                        <constant symbol="s_1"/>
                                        <constant symbol="s_2"/>
                                       </function>
                                       <constant symbol="s_1"/>
                                      </function>
                                     </constantatomformula>
                                     <constantatomformula symbol="≤" definition="≤">
                                      <constant symbol="s_1"/>
                                      <function symbol="∪">
                                       <function symbol="∩">
                                        <constant symbol="s_1"/>
                                        <constant symbol="s_2"/>
                                       </function>
                                       <constant symbol="s_1"/>
                                      </function>
                                     </constantatomformula>
                                    </conjunctiveformula>
                                    <quantifiedformula type="all">
                                     <variable symbol="z"/>
                                     <conjunctiveformula type="impl">
                                      <conjunctiveformula type="and">
                                       <constantatomformula symbol="≤" definition="≤">
                                        <function symbol="∩">
                                         <constant symbol="s_1"/>
                                         <constant symbol="s_2"/>
                                        </function>
                                        <variable symbol="z"/>
                                       </constantatomformula>
                                       <constantatomformula symbol="≤" definition="≤">
                                        <constant symbol="s_1"/>
                                        <variable symbol="z"/>
                                       </constantatomformula>
                                      </conjunctiveformula>
                                      <constantatomformula symbol="≤" definition="≤">
                                       <function symbol="∪">
                                        <function symbol="∩">
                                         <constant symbol="s_1"/>
                                         <constant symbol="s_2"/>
                                        </function>
                                        <constant symbol="s_1"/>
                                       </function>
                                       <variable symbol="z"/>
                                      </constantatomformula>
                                     </conjunctiveformula>
                                    </quantifiedformula>
                                   </conjunctiveformula>
                                  </formulalist>
                                  <formulalist>
                                   <conjunctiveformula type="and">
                                    <constantatomformula symbol="≤" definition="≤">
                                     <function symbol="∪">
                                      <function symbol="∩">
                                       <constant symbol="s_1"/>
                                       <constant symbol="s_2"/>
                                      </function>
                                      <constant symbol="s_1"/>
                                     </function>
                                     <constant symbol="s_1"/>
                                    </constantatomformula>
                                    <constantatomformula symbol="≤" definition="≤">
                                     <constant symbol="s_1"/>
                                     <function symbol="∪">
                                      <function symbol="∩">
                                       <constant symbol="s_1"/>
                                       <constant symbol="s_2"/>
                                      </function>
                                      <constant symbol="s_1"/>
                                     </function>
                                    </constantatomformula>
                                   </conjunctiveformula>
                                  </formulalist>
                                 </sequent>
                                 <rule type="perml" symbol="\pi:l" param="(1 4)">
                                  <sequent>
                                   <formulalist>
                                    <quantifiedformula type="all">
                                     <variable symbol="z"/>
                                     <conjunctiveformula type="impl">
                                      <conjunctiveformula type="and">
                                       <constantatomformula symbol="≤" definition="≤">
                                        <function symbol="∩">
                                         <constant symbol="s_1"/>
                                         <constant symbol="s_2"/>
                                        </function>
                                        <variable symbol="z"/>
                                       </constantatomformula>
                                       <constantatomformula symbol="≤" definition="≤">
                                        <constant symbol="s_1"/>
                                        <variable symbol="z"/>
                                       </constantatomformula>
                                      </conjunctiveformula>
                                      <constantatomformula symbol="≤" definition="≤">
                                       <function symbol="∪">
                                        <function symbol="∩">
                                         <constant symbol="s_1"/>
                                         <constant symbol="s_2"/>
                                        </function>
                                        <constant symbol="s_1"/>
                                       </function>
                                       <variable symbol="z"/>
                                      </constantatomformula>
                                     </conjunctiveformula>
                                    </quantifiedformula>
                                    <constantatomformula symbol="GLB" definition="GLB"/>
                                    <constantatomformula symbol="R" definition="R"/>
                                    <conjunctiveformula type="and">
                                     <conjunctiveformula type="and">
                                      <constantatomformula symbol="≤" definition="≤">
                                       <function symbol="∩">
                                        <constant symbol="s_1"/>
                                        <constant symbol="s_2"/>
                                       </function>
                                       <function symbol="∪">
                                        <function symbol="∩">
                                         <constant symbol="s_1"/>
                                         <constant symbol="s_2"/>
                                        </function>
                                        <constant symbol="s_1"/>
                                       </function>
                                      </constantatomformula>
                                      <constantatomformula symbol="≤" definition="≤">
                                       <constant symbol="s_1"/>
                                       <function symbol="∪">
                                        <function symbol="∩">
                                         <constant symbol="s_1"/>
                                         <constant symbol="s_2"/>
                                        </function>
                                        <constant symbol="s_1"/>
                                       </function>
                                      </constantatomformula>
                                     </conjunctiveformula>
                                     <quantifiedformula type="all">
                                      <variable symbol="z"/>
                                      <conjunctiveformula type="impl">
                                       <conjunctiveformula type="and">
                                        <constantatomformula symbol="≤" definition="≤">
                                         <function symbol="∩">
                                          <constant symbol="s_1"/>
                                          <constant symbol="s_2"/>
                                         </function>
                                         <variable symbol="z"/>
                                        </constantatomformula>
                                        <constantatomformula symbol="≤" definition="≤">
                                         <constant symbol="s_1"/>
                                         <variable symbol="z"/>
                                        </constantatomformula>
                                       </conjunctiveformula>
                                       <constantatomformula symbol="≤" definition="≤">
                                        <function symbol="∪">
                                         <function symbol="∩">
                                          <constant symbol="s_1"/>
                                          <constant symbol="s_2"/>
                                         </function>
                                         <constant symbol="s_1"/>
                                        </function>
                                        <variable symbol="z"/>
                                       </constantatomformula>
                                      </conjunctiveformula>
                                     </quantifiedformula>
                                    </conjunctiveformula>
                                   </formulalist>
                                   <formulalist>
                                    <conjunctiveformula type="and">
                                     <constantatomformula symbol="≤" definition="≤">
                                      <function symbol="∪">
                                       <function symbol="∩">
                                        <constant symbol="s_1"/>
                                        <constant symbol="s_2"/>
                                       </function>
                                       <constant symbol="s_1"/>
                                      </function>
                                      <constant symbol="s_1"/>
                                     </constantatomformula>
                                     <constantatomformula symbol="≤" definition="≤">
                                      <constant symbol="s_1"/>
                                      <function symbol="∪">
                                       <function symbol="∩">
                                        <constant symbol="s_1"/>
                                        <constant symbol="s_2"/>
                                       </function>
                                       <constant symbol="s_1"/>
                                      </function>
                                     </constantatomformula>
                                    </conjunctiveformula>
                                   </formulalist>
                                  </sequent>
                                  <rule type="andl1" symbol="\and:l1">
                                   <sequent>
                                    <formulalist>
                                     <conjunctiveformula type="and">
                                      <conjunctiveformula type="and">
                                       <constantatomformula symbol="≤" definition="≤">
                                        <function symbol="∩">
                                         <constant symbol="s_1"/>
                                         <constant symbol="s_2"/>
                                        </function>
                                        <function symbol="∪">
                                         <function symbol="∩">
                                          <constant symbol="s_1"/>
                                          <constant symbol="s_2"/>
                                         </function>
                                         <constant symbol="s_1"/>
                                        </function>
                                       </constantatomformula>
                                       <constantatomformula symbol="≤" definition="≤">
                                        <constant symbol="s_1"/>
                                        <function symbol="∪">
                                         <function symbol="∩">
                                          <constant symbol="s_1"/>
                                          <constant symbol="s_2"/>
                                         </function>
                                         <constant symbol="s_1"/>
                                        </function>
                                       </constantatomformula>
                                      </conjunctiveformula>
                                      <quantifiedformula type="all">
                                       <variable symbol="z"/>
                                       <conjunctiveformula type="impl">
                                        <conjunctiveformula type="and">
                                         <constantatomformula symbol="≤" definition="≤">
                                          <function symbol="∩">
                                           <constant symbol="s_1"/>
                                           <constant symbol="s_2"/>
                                          </function>
                                          <variable symbol="z"/>
                                         </constantatomformula>
                                         <constantatomformula symbol="≤" definition="≤">
                                          <constant symbol="s_1"/>
                                          <variable symbol="z"/>
                                         </constantatomformula>
                                        </conjunctiveformula>
                                        <constantatomformula symbol="≤" definition="≤">
                                         <function symbol="∪">
                                          <function symbol="∩">
                                           <constant symbol="s_1"/>
                                           <constant symbol="s_2"/>
                                          </function>
                                          <constant symbol="s_1"/>
                                         </function>
                                         <variable symbol="z"/>
                                        </constantatomformula>
                                       </conjunctiveformula>
                                      </quantifiedformula>
                                     </conjunctiveformula>
                                     <constantatomformula symbol="GLB" definition="GLB"/>
                                     <constantatomformula symbol="R" definition="R"/>
                                     <quantifiedformula type="all">
                                      <variable symbol="z"/>
                                      <conjunctiveformula type="impl">
                                       <conjunctiveformula type="and">
                                        <constantatomformula symbol="≤" definition="≤">
                                         <function symbol="∩">
                                          <constant symbol="s_1"/>
                                          <constant symbol="s_2"/>
                                         </function>
                                         <variable symbol="z"/>
                                        </constantatomformula>
                                        <constantatomformula symbol="≤" definition="≤">
                                         <constant symbol="s_1"/>
                                         <variable symbol="z"/>
                                        </constantatomformula>
                                       </conjunctiveformula>
                                       <constantatomformula symbol="≤" definition="≤">
                                        <function symbol="∪">
                                         <function symbol="∩">
                                          <constant symbol="s_1"/>
                                          <constant symbol="s_2"/>
                                         </function>
                                         <constant symbol="s_1"/>
                                        </function>
                                        <variable symbol="z"/>
                                       </constantatomformula>
                                      </conjunctiveformula>
                                     </quantifiedformula>
                                    </formulalist>
                                    <formulalist>
                                     <conjunctiveformula type="and">
                                      <constantatomformula symbol="≤" definition="≤">
                                       <function symbol="∪">
                                        <function symbol="∩">
                                         <constant symbol="s_1"/>
                                         <constant symbol="s_2"/>
                                        </function>
                                        <constant symbol="s_1"/>
                                       </function>
                                       <constant symbol="s_1"/>
                                      </constantatomformula>
                                      <constantatomformula symbol="≤" definition="≤">
                                       <constant symbol="s_1"/>
                                       <function symbol="∪">
                                        <function symbol="∩">
                                         <constant symbol="s_1"/>
                                         <constant symbol="s_2"/>
                                        </function>
                                        <constant symbol="s_1"/>
                                       </function>
                                      </constantatomformula>
                                     </conjunctiveformula>
                                    </formulalist>
                                   </sequent>
                                   <rule type="andl2" symbol="\and:l2">
                                    <sequent>
                                     <formulalist>
                                      <conjunctiveformula type="and">
                                       <constantatomformula symbol="≤" definition="≤">
                                        <function symbol="∩">
                                         <constant symbol="s_1"/>
                                         <constant symbol="s_2"/>
                                        </function>
                                        <function symbol="∪">
                                         <function symbol="∩">
                                          <constant symbol="s_1"/>
                                          <constant symbol="s_2"/>
                                         </function>
                                         <constant symbol="s_1"/>
                                        </function>
                                       </constantatomformula>
                                       <constantatomformula symbol="≤" definition="≤">
                                        <constant symbol="s_1"/>
                                        <function symbol="∪">
                                         <function symbol="∩">
                                          <constant symbol="s_1"/>
                                          <constant symbol="s_2"/>
                                         </function>
                                         <constant symbol="s_1"/>
                                        </function>
                                       </constantatomformula>
                                      </conjunctiveformula>
                                      <constantatomformula symbol="GLB" definition="GLB"/>
                                      <constantatomformula symbol="R" definition="R"/>
                                      <quantifiedformula type="all">
                                       <variable symbol="z"/>
                                       <conjunctiveformula type="impl">
                                        <conjunctiveformula type="and">
                                         <constantatomformula symbol="≤" definition="≤">
                                          <function symbol="∩">
                                           <constant symbol="s_1"/>
                                           <constant symbol="s_2"/>
                                          </function>
                                          <variable symbol="z"/>
                                         </constantatomformula>
                                         <constantatomformula symbol="≤" definition="≤">
                                          <constant symbol="s_1"/>
                                          <variable symbol="z"/>
                                         </constantatomformula>
                                        </conjunctiveformula>
                                        <constantatomformula symbol="≤" definition="≤">
                                         <function symbol="∪">
                                          <function symbol="∩">
                                           <constant symbol="s_1"/>
                                           <constant symbol="s_2"/>
                                          </function>
                                          <constant symbol="s_1"/>
                                         </function>
                                         <variable symbol="z"/>
                                        </constantatomformula>
                                       </conjunctiveformula>
                                      </quantifiedformula>
                                     </formulalist>
                                     <formulalist>
                                      <conjunctiveformula type="and">
                                       <constantatomformula symbol="≤" definition="≤">
                                        <function symbol="∪">
                                         <function symbol="∩">
                                          <constant symbol="s_1"/>
                                          <constant symbol="s_2"/>
                                         </function>
                                         <constant symbol="s_1"/>
                                        </function>
                                        <constant symbol="s_1"/>
                                       </constantatomformula>
                                       <constantatomformula symbol="≤" definition="≤">
                                        <constant symbol="s_1"/>
                                        <function symbol="∪">
                                         <function symbol="∩">
                                          <constant symbol="s_1"/>
                                          <constant symbol="s_2"/>
                                         </function>
                                         <constant symbol="s_1"/>
                                        </function>
                                       </constantatomformula>
                                      </conjunctiveformula>
                                     </formulalist>
                                    </sequent>
                                    <rule type="perml" symbol="\pi:l" param="(1 4)">
                                     <sequent>
                                      <formulalist>
                                       <constantatomformula symbol="≤" definition="≤">
                                        <constant symbol="s_1"/>
                                        <function symbol="∪">
                                         <function symbol="∩">
                                          <constant symbol="s_1"/>
                                          <constant symbol="s_2"/>
                                         </function>
                                         <constant symbol="s_1"/>
                                        </function>
                                       </constantatomformula>
                                       <constantatomformula symbol="GLB" definition="GLB"/>
                                       <constantatomformula symbol="R" definition="R"/>
                                       <quantifiedformula type="all">
                                        <variable symbol="z"/>
                                        <conjunctiveformula type="impl">
                                         <conjunctiveformula type="and">
                                          <constantatomformula symbol="≤" definition="≤">
                                           <function symbol="∩">
                                            <constant symbol="s_1"/>
                                            <constant symbol="s_2"/>
                                           </function>
                                           <variable symbol="z"/>
                                          </constantatomformula>
                                          <constantatomformula symbol="≤" definition="≤">
                                           <constant symbol="s_1"/>
                                           <variable symbol="z"/>
                                          </constantatomformula>
                                         </conjunctiveformula>
                                         <constantatomformula symbol="≤" definition="≤">
                                          <function symbol="∪">
                                           <function symbol="∩">
                                            <constant symbol="s_1"/>
                                            <constant symbol="s_2"/>
                                           </function>
                                           <constant symbol="s_1"/>
                                          </function>
                                          <variable symbol="z"/>
                                         </constantatomformula>
                                        </conjunctiveformula>
                                       </quantifiedformula>
                                      </formulalist>
                                      <formulalist>
                                       <conjunctiveformula type="and">
                                        <constantatomformula symbol="≤" definition="≤">
                                         <function symbol="∪">
                                          <function symbol="∩">
                                           <constant symbol="s_1"/>
                                           <constant symbol="s_2"/>
                                          </function>
                                          <constant symbol="s_1"/>
                                         </function>
                                         <constant symbol="s_1"/>
                                        </constantatomformula>
                                        <constantatomformula symbol="≤" definition="≤">
                                         <constant symbol="s_1"/>
                                         <function symbol="∪">
                                          <function symbol="∩">
                                           <constant symbol="s_1"/>
                                           <constant symbol="s_2"/>
                                          </function>
                                          <constant symbol="s_1"/>
                                         </function>
                                        </constantatomformula>
                                       </conjunctiveformula>
                                      </formulalist>
                                     </sequent>
                                     <rule type="andr" symbol="\and:r">
                                      <sequent>
                                       <formulalist>
                                        <quantifiedformula type="all">
                                         <variable symbol="z"/>
                                         <conjunctiveformula type="impl">
                                          <conjunctiveformula type="and">
                                           <constantatomformula symbol="≤" definition="≤">
                                            <function symbol="∩">
                                             <constant symbol="s_1"/>
                                             <constant symbol="s_2"/>
                                            </function>
                                            <variable symbol="z"/>
                                           </constantatomformula>
                                           <constantatomformula symbol="≤" definition="≤">
                                            <constant symbol="s_1"/>
                                            <variable symbol="z"/>
                                           </constantatomformula>
                                          </conjunctiveformula>
                                          <constantatomformula symbol="≤" definition="≤">
                                           <function symbol="∪">
                                            <function symbol="∩">
                                             <constant symbol="s_1"/>
                                             <constant symbol="s_2"/>
                                            </function>
                                            <constant symbol="s_1"/>
                                           </function>
                                           <variable symbol="z"/>
                                          </constantatomformula>
                                         </conjunctiveformula>
                                        </quantifiedformula>
                                        <constantatomformula symbol="GLB" definition="GLB"/>
                                        <constantatomformula symbol="R" definition="R"/>
                                        <constantatomformula symbol="≤" definition="≤">
                                         <constant symbol="s_1"/>
                                         <function symbol="∪">
                                          <function symbol="∩">
                                           <constant symbol="s_1"/>
                                           <constant symbol="s_2"/>
                                          </function>
                                          <constant symbol="s_1"/>
                                         </function>
                                        </constantatomformula>
                                       </formulalist>
                                       <formulalist>
                                        <conjunctiveformula type="and">
                                         <constantatomformula symbol="≤" definition="≤">
                                          <function symbol="∪">
                                           <function symbol="∩">
                                            <constant symbol="s_1"/>
                                            <constant symbol="s_2"/>
                                           </function>
                                           <constant symbol="s_1"/>
                                          </function>
                                          <constant symbol="s_1"/>
                                         </constantatomformula>
                                         <constantatomformula symbol="≤" definition="≤">
                                          <constant symbol="s_1"/>
                                          <function symbol="∪">
                                           <function symbol="∩">
                                            <constant symbol="s_1"/>
                                            <constant symbol="s_2"/>
                                           </function>
                                           <constant symbol="s_1"/>
                                          </function>
                                         </constantatomformula>
                                        </conjunctiveformula>
                                       </formulalist>
                                      </sequent>
                                      <rule type="foralll" symbol="\all:l">
                                       <sequent>
                                        <formulalist>
                                         <quantifiedformula type="all">
                                          <variable symbol="z"/>
                                          <conjunctiveformula type="impl">
                                           <conjunctiveformula type="and">
                                            <constantatomformula symbol="≤" definition="≤">
                                             <function symbol="∩">
                                              <constant symbol="s_1"/>
                                              <constant symbol="s_2"/>
                                             </function>
                                             <variable symbol="z"/>
                                            </constantatomformula>
                                            <constantatomformula symbol="≤" definition="≤">
                                             <constant symbol="s_1"/>
                                             <variable symbol="z"/>
                                            </constantatomformula>
                                           </conjunctiveformula>
                                           <constantatomformula symbol="≤" definition="≤">
                                            <function symbol="∪">
                                             <function symbol="∩">
                                              <constant symbol="s_1"/>
                                              <constant symbol="s_2"/>
                                             </function>
                                             <constant symbol="s_1"/>
                                            </function>
                                            <variable symbol="z"/>
                                           </constantatomformula>
                                          </conjunctiveformula>
                                         </quantifiedformula>
                                         <constantatomformula symbol="GLB" definition="GLB"/>
                                         <constantatomformula symbol="R" definition="R"/>
                                        </formulalist>
                                        <formulalist>
                                         <constantatomformula symbol="≤" definition="≤">
                                          <function symbol="∪">
                                           <function symbol="∩">
                                            <constant symbol="s_1"/>
                                            <constant symbol="s_2"/>
                                           </function>
                                           <constant symbol="s_1"/>
                                          </function>
                                          <constant symbol="s_1"/>
                                         </constantatomformula>
                                        </formulalist>
                                       </sequent>
                                       <rule type="impll" symbol="\impl:l">
                                        <sequent>
                                         <formulalist>
                                          <conjunctiveformula type="impl">
                                           <conjunctiveformula type="and">
                                            <constantatomformula symbol="≤" definition="≤">
                                             <function symbol="∩">
                                              <constant symbol="s_1"/>
                                              <constant symbol="s_2"/>
                                             </function>
                                             <constant symbol="s_1"/>
                                            </constantatomformula>
                                            <constantatomformula symbol="≤" definition="≤">
                                             <constant symbol="s_1"/>
                                             <constant symbol="s_1"/>
                                            </constantatomformula>
                                           </conjunctiveformula>
                                           <constantatomformula symbol="≤" definition="≤">
                                            <function symbol="∪">
                                             <function symbol="∩">
                                              <constant symbol="s_1"/>
                                              <constant symbol="s_2"/>
                                             </function>
                                             <constant symbol="s_1"/>
                                            </function>
                                            <constant symbol="s_1"/>
                                           </constantatomformula>
                                          </conjunctiveformula>
                                          <constantatomformula symbol="GLB" definition="GLB"/>
                                          <constantatomformula symbol="R" definition="R"/>
                                         </formulalist>
                                         <formulalist>
                                          <constantatomformula symbol="≤" definition="≤">
                                           <function symbol="∪">
                                            <function symbol="∩">
                                             <constant symbol="s_1"/>
                                             <constant symbol="s_2"/>
                                            </function>
                                            <constant symbol="s_1"/>
                                           </function>
                                           <constant symbol="s_1"/>
                                          </constantatomformula>
                                         </formulalist>
                                        </sequent>
                                        <rule type="andr" symbol="\and:r">
                                         <sequent>
                                          <formulalist>
                                           <constantatomformula symbol="GLB" definition="GLB"/>
                                           <constantatomformula symbol="R" definition="R"/>
                                          </formulalist>
                                          <formulalist>
                                           <conjunctiveformula type="and">
                                            <constantatomformula symbol="≤" definition="≤">
                                             <function symbol="∩">
                                              <constant symbol="s_1"/>
                                              <constant symbol="s_2"/>
                                             </function>
                                             <constant symbol="s_1"/>
                                            </constantatomformula>
                                            <constantatomformula symbol="≤" definition="≤">
                                             <constant symbol="s_1"/>
                                             <constant symbol="s_1"/>
                                            </constantatomformula>
                                           </conjunctiveformula>
                                          </formulalist>
                                         </sequent>
                                         <rule type="defl" symbol="d:l" param="();[()];()">
                                          <sequent>
                                           <formulalist>
                                            <constantatomformula symbol="GLB" definition="GLB"/>
                                           </formulalist>
                                           <formulalist>
                                            <constantatomformula symbol="≤" definition="≤">
                                             <function symbol="∩">
                                              <constant symbol="s_1"/>
                                              <constant symbol="s_2"/>
                                             </function>
                                             <constant symbol="s_1"/>
                                            </constantatomformula>
                                           </formulalist>
                                          </sequent>
                                          <rule type="foralll" symbol="\all:l">
                                           <sequent>
                                            <formulalist>
                                             <quantifiedformula type="all">
                                              <variable symbol="x"/>
                                              <quantifiedformula type="all">
                                               <variable symbol="y"/>
                                               <conjunctiveformula type="and">
                                                <conjunctiveformula type="and">
                                                 <constantatomformula symbol="≤" definition="≤">
                                                  <function symbol="∩">
                                                   <variable symbol="x"/>
                                                   <variable symbol="y"/>
                                                  </function>
                                                  <variable symbol="x"/>
                                                 </constantatomformula>
                                                 <constantatomformula symbol="≤" definition="≤">
                                                  <function symbol="∩">
                                                   <variable symbol="x"/>
                                                   <variable symbol="y"/>
                                                  </function>
                                                  <variable symbol="y"/>
                                                 </constantatomformula>
                                                </conjunctiveformula>
                                                <quantifiedformula type="all">
                                                 <variable symbol="z"/>
                                                 <conjunctiveformula type="impl">
                                                  <conjunctiveformula type="and">
                                                   <constantatomformula symbol="≤" definition="≤">
                                                    <variable symbol="z"/>
                                                    <variable symbol="x"/>
                                                   </constantatomformula>
                                                   <constantatomformula symbol="≤" definition="≤">
                                                    <variable symbol="z"/>
                                                    <variable symbol="y"/>
                                                   </constantatomformula>
                                                  </conjunctiveformula>
                                                  <constantatomformula symbol="≤" definition="≤">
                                                   <variable symbol="z"/>
                                                   <function symbol="∩">
                                                    <variable symbol="x"/>
                                                    <variable symbol="y"/>
                                                   </function>
                                                  </constantatomformula>
                                                 </conjunctiveformula>
                                                </quantifiedformula>
                                               </conjunctiveformula>
                                              </quantifiedformula>
                                             </quantifiedformula>
                                            </formulalist>
                                            <formulalist>
                                             <constantatomformula symbol="≤" definition="≤">
                                              <function symbol="∩">
                                               <constant symbol="s_1"/>
                                               <constant symbol="s_2"/>
                                              </function>
                                              <constant symbol="s_1"/>
                                             </constantatomformula>
                                            </formulalist>
                                           </sequent>
                                           <rule type="foralll" symbol="\all:l">
                                            <sequent>
                                             <formulalist>
                                              <quantifiedformula type="all">
                                               <variable symbol="y"/>
                                               <conjunctiveformula type="and">
                                                <conjunctiveformula type="and">
                                                 <constantatomformula symbol="≤" definition="≤">
                                                  <function symbol="∩">
                                                   <constant symbol="s_1"/>
                                                   <variable symbol="y"/>
                                                  </function>
                                                  <constant symbol="s_1"/>
                                                 </constantatomformula>
                                                 <constantatomformula symbol="≤" definition="≤">
                                                  <function symbol="∩">
                                                   <constant symbol="s_1"/>
                                                   <variable symbol="y"/>
                                                  </function>
                                                  <variable symbol="y"/>
                                                 </constantatomformula>
                                                </conjunctiveformula>
                                                <quantifiedformula type="all">
                                                 <variable symbol="z"/>
                                                 <conjunctiveformula type="impl">
                                                  <conjunctiveformula type="and">
                                                   <constantatomformula symbol="≤" definition="≤">
                                                    <variable symbol="z"/>
                                                    <constant symbol="s_1"/>
                                                   </constantatomformula>
                                                   <constantatomformula symbol="≤" definition="≤">
                                                    <variable symbol="z"/>
                                                    <variable symbol="y"/>
                                                   </constantatomformula>
                                                  </conjunctiveformula>
                                                  <constantatomformula symbol="≤" definition="≤">
                                                   <variable symbol="z"/>
                                                   <function symbol="∩">
                                                    <constant symbol="s_1"/>
                                                    <variable symbol="y"/>
                                                   </function>
                                                  </constantatomformula>
                                                 </conjunctiveformula>
                                                </quantifiedformula>
                                               </conjunctiveformula>
                                              </quantifiedformula>
                                             </formulalist>
                                             <formulalist>
                                              <constantatomformula symbol="≤" definition="≤">
                                               <function symbol="∩">
                                                <constant symbol="s_1"/>
                                                <constant symbol="s_2"/>
                                               </function>
                                               <constant symbol="s_1"/>
                                              </constantatomformula>
                                             </formulalist>
                                            </sequent>
                                            <rule type="andl1" symbol="\and:l1">
                                             <sequent>
                                              <formulalist>
                                               <conjunctiveformula type="and">
                                                <conjunctiveformula type="and">
                                                 <constantatomformula symbol="≤" definition="≤">
                                                  <function symbol="∩">
                                                   <constant symbol="s_1"/>
                                                   <constant symbol="s_2"/>
                                                  </function>
                                                  <constant symbol="s_1"/>
                                                 </constantatomformula>
                                                 <constantatomformula symbol="≤" definition="≤">
                                                  <function symbol="∩">
                                                   <constant symbol="s_1"/>
                                                   <constant symbol="s_2"/>
                                                  </function>
                                                  <constant symbol="s_2"/>
                                                 </constantatomformula>
                                                </conjunctiveformula>
                                                <quantifiedformula type="all">
                                                 <variable symbol="z"/>
                                                 <conjunctiveformula type="impl">
                                                  <conjunctiveformula type="and">
                                                   <constantatomformula symbol="≤" definition="≤">
                                                    <variable symbol="z"/>
                                                    <constant symbol="s_1"/>
                                                   </constantatomformula>
                                                   <constantatomformula symbol="≤" definition="≤">
                                                    <variable symbol="z"/>
                                                    <constant symbol="s_2"/>
                                                   </constantatomformula>
                                                  </conjunctiveformula>
                                                  <constantatomformula symbol="≤" definition="≤">
                                                   <variable symbol="z"/>
                                                   <function symbol="∩">
                                                    <constant symbol="s_1"/>
                                                    <constant symbol="s_2"/>
                                                   </function>
                                                  </constantatomformula>
                                                 </conjunctiveformula>
                                                </quantifiedformula>
                                               </conjunctiveformula>
                                              </formulalist>
                                              <formulalist>
                                               <constantatomformula symbol="≤" definition="≤">
                                                <function symbol="∩">
                                                 <constant symbol="s_1"/>
                                                 <constant symbol="s_2"/>
                                                </function>
                                                <constant symbol="s_1"/>
                                               </constantatomformula>
                                              </formulalist>
                                             </sequent>
                                             <rule type="andl1" symbol="\and:l1">
                                              <sequent>
                                               <formulalist>
                                                <conjunctiveformula type="and">
                                                 <constantatomformula symbol="≤" definition="≤">
                                                  <function symbol="∩">
                                                   <constant symbol="s_1"/>
                                                   <constant symbol="s_2"/>
                                                  </function>
                                                  <constant symbol="s_1"/>
                                                 </constantatomformula>
                                                 <constantatomformula symbol="≤" definition="≤">
                                                  <function symbol="∩">
                                                   <constant symbol="s_1"/>
                                                   <constant symbol="s_2"/>
                                                  </function>
                                                  <constant symbol="s_2"/>
                                                 </constantatomformula>
                                                </conjunctiveformula>
                                               </formulalist>
                                               <formulalist>
                                                <constantatomformula symbol="≤" definition="≤">
                                                 <function symbol="∩">
                                                  <constant symbol="s_1"/>
                                                  <constant symbol="s_2"/>
                                                 </function>
                                                 <constant symbol="s_1"/>
                                                </constantatomformula>
                                               </formulalist>
                                              </sequent>
                                              <rule type="defl" symbol="d:l" param="();[(1)(1 1)(2)][(2)(1 2)];()">
                                               <sequent>
                                                <formulalist>
                                                 <constantatomformula symbol="≤" definition="≤">
                                                  <function symbol="∩">
                                                   <constant symbol="s_1"/>
                                                   <constant symbol="s_2"/>
                                                  </function>
                                                  <constant symbol="s_1"/>
                                                 </constantatomformula>
                                                </formulalist>
                                                <formulalist>
                                                 <constantatomformula symbol="≤" definition="≤">
                                                  <function symbol="∩">
                                                   <constant symbol="s_1"/>
                                                   <constant symbol="s_2"/>
                                                  </function>
                                                  <constant symbol="s_1"/>
                                                 </constantatomformula>
                                                </formulalist>
                                               </sequent>
                                               <rule type="defr" symbol="d:r" param="();[(1)(1 1)(2)][(2)(1 2)];()">
                                                <sequent>
                                                 <formulalist>
                                                  <constantatomformula symbol="=">
                                                   <function symbol="∩">
                                                    <function symbol="∩">
                                                     <constant symbol="s_1"/>
                                                     <constant symbol="s_2"/>
                                                    </function>
                                                    <constant symbol="s_1"/>
                                                   </function>
                                                   <function symbol="∩">
                                                    <constant symbol="s_1"/>
                                                    <constant symbol="s_2"/>
                                                   </function>
                                                  </constantatomformula>
                                                 </formulalist>
                                                 <formulalist>
                                                  <constantatomformula symbol="≤" definition="≤">
                                                   <function symbol="∩">
                                                    <constant symbol="s_1"/>
                                                    <constant symbol="s_2"/>
                                                   </function>
                                                   <constant symbol="s_1"/>
                                                  </constantatomformula>
                                                 </formulalist>
                                                </sequent>
                                                <rule type="axiom" symbol="axiom">
                                                 <sequent>
                                                  <formulalist>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="∩">
                                                     <function symbol="∩">
                                                      <constant symbol="s_1"/>
                                                      <constant symbol="s_2"/>
                                                     </function>
                                                     <constant symbol="s_1"/>
                                                    </function>
                                                    <function symbol="∩">
                                                     <constant symbol="s_1"/>
                                                     <constant symbol="s_2"/>
                                                    </function>
                                                   </constantatomformula>
                                                  </formulalist>
                                                  <formulalist>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="∩">
                                                     <function symbol="∩">
                                                      <constant symbol="s_1"/>
                                                      <constant symbol="s_2"/>
                                                     </function>
                                                     <constant symbol="s_1"/>
                                                    </function>
                                                    <function symbol="∩">
                                                     <constant symbol="s_1"/>
                                                     <constant symbol="s_2"/>
                                                    </function>
                                                   </constantatomformula>
                                                  </formulalist>
                                                 </sequent>
                                                </rule>
                                               </rule>
                                              </rule>
                                             </rule>
                                            </rule>
                                           </rule>
                                          </rule>
                                         </rule>
                                         <rule type="defl" symbol="d:l" param="();[()];()">
                                          <sequent>
                                           <formulalist>
                                            <constantatomformula symbol="R" definition="R"/>
                                           </formulalist>
                                           <formulalist>
                                            <constantatomformula symbol="≤" definition="≤">
                                             <constant symbol="s_1"/>
                                             <constant symbol="s_1"/>
                                            </constantatomformula>
                                           </formulalist>
                                          </sequent>
                                          <rule type="foralll" symbol="\all:l">
                                           <sequent>
                                            <formulalist>
                                             <quantifiedformula type="all">
                                              <variable symbol="x"/>
                                              <constantatomformula symbol="≤" definition="≤">
                                               <variable symbol="x"/>
                                               <variable symbol="x"/>
                                              </constantatomformula>
                                             </quantifiedformula>
                                            </formulalist>
                                            <formulalist>
                                             <constantatomformula symbol="≤" definition="≤">
                                              <constant symbol="s_1"/>
                                              <constant symbol="s_1"/>
                                             </constantatomformula>
                                            </formulalist>
                                           </sequent>
                                           <rule type="defl" symbol="d:l" param="();[(1)(1 1)(2)][(2)(1 2)];()">
                                            <sequent>
                                             <formulalist>
                                              <constantatomformula symbol="≤" definition="≤">
                                               <constant symbol="s_1"/>
                                               <constant symbol="s_1"/>
                                              </constantatomformula>
                                             </formulalist>
                                             <formulalist>
                                              <constantatomformula symbol="≤" definition="≤">
                                               <constant symbol="s_1"/>
                                               <constant symbol="s_1"/>
                                              </constantatomformula>
                                             </formulalist>
                                            </sequent>
                                            <rule type="defr" symbol="d:r" param="();[(1)(1 1)(2)][(2)(1 2)];()">
                                             <sequent>
                                              <formulalist>
                                               <constantatomformula symbol="=">
                                                <function symbol="∩">
                                                 <constant symbol="s_1"/>
                                                 <constant symbol="s_1"/>
                                                </function>
                                                <constant symbol="s_1"/>
                                               </constantatomformula>
                                              </formulalist>
                                              <formulalist>
                                               <constantatomformula symbol="≤" definition="≤">
                                                <constant symbol="s_1"/>
                                                <constant symbol="s_1"/>
                                               </constantatomformula>
                                              </formulalist>
                                             </sequent>
                                             <rule type="axiom" symbol="axiom">
                                              <sequent>
                                               <formulalist>
                                                <constantatomformula symbol="=">
                                                 <function symbol="∩">
                                                  <constant symbol="s_1"/>
                                                  <constant symbol="s_1"/>
                                                 </function>
                                                 <constant symbol="s_1"/>
                                                </constantatomformula>
                                               </formulalist>
                                               <formulalist>
                                                <constantatomformula symbol="=">
                                                 <function symbol="∩">
                                                  <constant symbol="s_1"/>
                                                  <constant symbol="s_1"/>
                                                 </function>
                                                 <constant symbol="s_1"/>
                                                </constantatomformula>
                                               </formulalist>
                                              </sequent>
                                             </rule>
                                            </rule>
                                           </rule>
                                          </rule>
                                         </rule>
                                        </rule>
                                        <rule type="defl" symbol="d:l" param="();[(1)(1 1)(2)][(2)(1 2)];()">
                                         <sequent>
                                          <formulalist>
                                           <constantatomformula symbol="≤" definition="≤">
                                            <function symbol="∪">
                                             <function symbol="∩">
                                              <constant symbol="s_1"/>
                                              <constant symbol="s_2"/>
                                             </function>
                                             <constant symbol="s_1"/>
                                            </function>
                                            <constant symbol="s_1"/>
                                           </constantatomformula>
                                          </formulalist>
                                          <formulalist>
                                           <constantatomformula symbol="≤" definition="≤">
                                            <function symbol="∪">
                                             <function symbol="∩">
                                              <constant symbol="s_1"/>
                                              <constant symbol="s_2"/>
                                             </function>
                                             <constant symbol="s_1"/>
                                            </function>
                                            <constant symbol="s_1"/>
                                           </constantatomformula>
                                          </formulalist>
                                         </sequent>
                                         <rule type="defr" symbol="d:r" param="();[(1)(1 1)(2)][(2)(1 2)];()">
                                          <sequent>
                                           <formulalist>
                                            <constantatomformula symbol="=">
                                             <function symbol="∩">
                                              <function symbol="∪">
                                               <function symbol="∩">
                                                <constant symbol="s_1"/>
                                                <constant symbol="s_2"/>
                                               </function>
                                               <constant symbol="s_1"/>
                                              </function>
                                              <constant symbol="s_1"/>
                                             </function>
                                             <function symbol="∪">
                                              <function symbol="∩">
                                               <constant symbol="s_1"/>
                                               <constant symbol="s_2"/>
                                              </function>
                                              <constant symbol="s_1"/>
                                             </function>
                                            </constantatomformula>
                                           </formulalist>
                                           <formulalist>
                                            <constantatomformula symbol="≤" definition="≤">
                                             <function symbol="∪">
                                              <function symbol="∩">
                                               <constant symbol="s_1"/>
                                               <constant symbol="s_2"/>
                                              </function>
                                              <constant symbol="s_1"/>
                                             </function>
                                             <constant symbol="s_1"/>
                                            </constantatomformula>
                                           </formulalist>
                                          </sequent>
                                          <rule type="axiom" symbol="axiom">
                                           <sequent>
                                            <formulalist>
                                             <constantatomformula symbol="=">
                                              <function symbol="∩">
                                               <function symbol="∪">
                                                <function symbol="∩">
                                                 <constant symbol="s_1"/>
                                                 <constant symbol="s_2"/>
                                                </function>
                                                <constant symbol="s_1"/>
                                               </function>
                                               <constant symbol="s_1"/>
                                              </function>
                                              <function symbol="∪">
                                               <function symbol="∩">
                                                <constant symbol="s_1"/>
                                                <constant symbol="s_2"/>
                                               </function>
                                               <constant symbol="s_1"/>
                                              </function>
                                             </constantatomformula>
                                            </formulalist>
                                            <formulalist>
                                             <constantatomformula symbol="=">
                                              <function symbol="∩">
                                               <function symbol="∪">
                                                <function symbol="∩">
                                                 <constant symbol="s_1"/>
                                                 <constant symbol="s_2"/>
                                                </function>
                                                <constant symbol="s_1"/>
                                               </function>
                                               <constant symbol="s_1"/>
                                              </function>
                                              <function symbol="∪">
                                               <function symbol="∩">
                                                <constant symbol="s_1"/>
                                                <constant symbol="s_2"/>
                                               </function>
                                               <constant symbol="s_1"/>
                                              </function>
                                             </constantatomformula>
                                            </formulalist>
                                           </sequent>
                                          </rule>
                                         </rule>
                                        </rule>
                                       </rule>
                                      </rule>
                                      <rule type="defl" symbol="d:l" param="();[(1)(1 1)(2)][(2)(1 2)];()">
                                       <sequent>
                                        <formulalist>
                                         <constantatomformula symbol="≤" definition="≤">
                                          <constant symbol="s_1"/>
                                          <function symbol="∪">
                                           <function symbol="∩">
                                            <constant symbol="s_1"/>
                                            <constant symbol="s_2"/>
                                           </function>
                                           <constant symbol="s_1"/>
                                          </function>
                                         </constantatomformula>
                                        </formulalist>
                                        <formulalist>
                                         <constantatomformula symbol="≤" definition="≤">
                                          <constant symbol="s_1"/>
                                          <function symbol="∪">
                                           <function symbol="∩">
                                            <constant symbol="s_1"/>
                                            <constant symbol="s_2"/>
                                           </function>
                                           <constant symbol="s_1"/>
                                          </function>
                                         </constantatomformula>
                                        </formulalist>
                                       </sequent>
                                       <rule type="defr" symbol="d:r" param="();[(1)(1 1)(2)][(2)(1 2)];()">
                                        <sequent>
                                         <formulalist>
                                          <constantatomformula symbol="=">
                                           <function symbol="∩">
                                            <constant symbol="s_1"/>
                                            <function symbol="∪">
                                             <function symbol="∩">
                                              <constant symbol="s_1"/>
                                              <constant symbol="s_2"/>
                                             </function>
                                             <constant symbol="s_1"/>
                                            </function>
                                           </function>
                                           <constant symbol="s_1"/>
                                          </constantatomformula>
                                         </formulalist>
                                         <formulalist>
                                          <constantatomformula symbol="≤" definition="≤">
                                           <constant symbol="s_1"/>
                                           <function symbol="∪">
                                            <function symbol="∩">
                                             <constant symbol="s_1"/>
                                             <constant symbol="s_2"/>
                                            </function>
                                            <constant symbol="s_1"/>
                                           </function>
                                          </constantatomformula>
                                         </formulalist>
                                        </sequent>
                                        <rule type="axiom" symbol="axiom">
                                         <sequent>
                                          <formulalist>
                                           <constantatomformula symbol="=">
                                            <function symbol="∩">
                                             <constant symbol="s_1"/>
                                             <function symbol="∪">
                                              <function symbol="∩">
                                               <constant symbol="s_1"/>
                                               <constant symbol="s_2"/>
                                              </function>
                                              <constant symbol="s_1"/>
                                             </function>
                                            </function>
                                            <constant symbol="s_1"/>
                                           </constantatomformula>
                                          </formulalist>
                                          <formulalist>
                                           <constantatomformula symbol="=">
                                            <function symbol="∩">
                                             <constant symbol="s_1"/>
                                             <function symbol="∪">
                                              <function symbol="∩">
                                               <constant symbol="s_1"/>
                                               <constant symbol="s_2"/>
                                              </function>
                                              <constant symbol="s_1"/>
                                             </function>
                                            </function>
                                            <constant symbol="s_1"/>
                                           </constantatomformula>
                                          </formulalist>
                                         </sequent>
                                        </rule>
                                       </rule>
                                      </rule>
                                     </rule>
                                    </rule>
                                   </rule>
                                  </rule>
                                 </rule>
                                </rule>
                               </rule>
                              </rule>
                             </rule>
                            </rule>
                           </rule>
                           <rule type="axiom" symbol="axiom">
                            <sequent>
                             <formulalist>
                              <constantatomformula symbol="=">
                               <function symbol="∪">
                                <function symbol="∩">
                                 <constant symbol="s_1"/>
                                 <constant symbol="s_2"/>
                                </function>
                                <constant symbol="s_1"/>
                               </function>
                               <constant symbol="s_1"/>
                              </constantatomformula>
                             </formulalist>
                             <formulalist>
                              <constantatomformula symbol="=">
                               <function symbol="∪">
                                <function symbol="∩">
                                 <constant symbol="s_1"/>
                                 <constant symbol="s_2"/>
                                </function>
                                <constant symbol="s_1"/>
                               </function>
                               <constant symbol="s_1"/>
                              </constantatomformula>
                             </formulalist>
                            </sequent>
                           </rule>
                          </rule>
                         </rule>
                        </rule>
                       </rule>
                       <rule type="defl" symbol="d:l" param="();[()];()">
                        <sequent>
                         <formulalist>
                          <constantatomformula symbol="AS" definition="AS"/>
                          <constantatomformula symbol="GLB" definition="GLB"/>
                          <constantatomformula symbol="LUB" definition="LUB"/>
                          <constantatomformula symbol="R" definition="R"/>
                         </formulalist>
                         <formulalist>
                          <constantatomformula symbol="=">
                           <function symbol="∩">
                            <function symbol="∪">
                             <constant symbol="s_3"/>
                             <constant symbol="s_4"/>
                            </function>
                            <constant symbol="s_3"/>
                           </function>
                           <constant symbol="s_3"/>
                          </constantatomformula>
                         </formulalist>
                        </sequent>
                        <rule type="foralll" symbol="\all:l">
                         <sequent>
                          <formulalist>
                           <quantifiedformula type="all">
                            <variable symbol="x"/>
                            <quantifiedformula type="all">
                             <variable symbol="y"/>
                             <conjunctiveformula type="impl">
                              <conjunctiveformula type="and">
                               <constantatomformula symbol="≤" definition="≤">
                                <variable symbol="x"/>
                                <variable symbol="y"/>
                               </constantatomformula>
                               <constantatomformula symbol="≤" definition="≤">
                                <variable symbol="y"/>
                                <variable symbol="x"/>
                               </constantatomformula>
                              </conjunctiveformula>
                              <constantatomformula symbol="=">
                               <variable symbol="x"/>
                               <variable symbol="y"/>
                              </constantatomformula>
                             </conjunctiveformula>
                            </quantifiedformula>
                           </quantifiedformula>
                           <constantatomformula symbol="GLB" definition="GLB"/>
                           <constantatomformula symbol="LUB" definition="LUB"/>
                           <constantatomformula symbol="R" definition="R"/>
                          </formulalist>
                          <formulalist>
                           <constantatomformula symbol="=">
                            <function symbol="∩">
                             <function symbol="∪">
                              <constant symbol="s_3"/>
                              <constant symbol="s_4"/>
                             </function>
                             <constant symbol="s_3"/>
                            </function>
                            <constant symbol="s_3"/>
                           </constantatomformula>
                          </formulalist>
                         </sequent>
                         <rule type="foralll" symbol="\all:l">
                          <sequent>
                           <formulalist>
                            <quantifiedformula type="all">
                             <variable symbol="y"/>
                             <conjunctiveformula type="impl">
                              <conjunctiveformula type="and">
                               <constantatomformula symbol="≤" definition="≤">
                                <function symbol="∩">
                                 <function symbol="∪">
                                  <constant symbol="s_3"/>
                                  <constant symbol="s_4"/>
                                 </function>
                                 <constant symbol="s_3"/>
                                </function>
                                <variable symbol="y"/>
                               </constantatomformula>
                               <constantatomformula symbol="≤" definition="≤">
                                <variable symbol="y"/>
                                <function symbol="∩">
                                 <function symbol="∪">
                                  <constant symbol="s_3"/>
                                  <constant symbol="s_4"/>
                                 </function>
                                 <constant symbol="s_3"/>
                                </function>
                               </constantatomformula>
                              </conjunctiveformula>
                              <constantatomformula symbol="=">
                               <function symbol="∩">
                                <function symbol="∪">
                                 <constant symbol="s_3"/>
                                 <constant symbol="s_4"/>
                                </function>
                                <constant symbol="s_3"/>
                               </function>
                               <variable symbol="y"/>
                              </constantatomformula>
                             </conjunctiveformula>
                            </quantifiedformula>
                            <constantatomformula symbol="GLB" definition="GLB"/>
                            <constantatomformula symbol="LUB" definition="LUB"/>
                            <constantatomformula symbol="R" definition="R"/>
                           </formulalist>
                           <formulalist>
                            <constantatomformula symbol="=">
                             <function symbol="∩">
                              <function symbol="∪">
                               <constant symbol="s_3"/>
                               <constant symbol="s_4"/>
                              </function>
                              <constant symbol="s_3"/>
                             </function>
                             <constant symbol="s_3"/>
                            </constantatomformula>
                           </formulalist>
                          </sequent>
                          <rule type="impll" symbol="\impl:l">
                           <sequent>
                            <formulalist>
                             <conjunctiveformula type="impl">
                              <conjunctiveformula type="and">
                               <constantatomformula symbol="≤" definition="≤">
                                <function symbol="∩">
                                 <function symbol="∪">
                                  <constant symbol="s_3"/>
                                  <constant symbol="s_4"/>
                                 </function>
                                 <constant symbol="s_3"/>
                                </function>
                                <constant symbol="s_3"/>
                               </constantatomformula>
                               <constantatomformula symbol="≤" definition="≤">
                                <constant symbol="s_3"/>
                                <function symbol="∩">
                                 <function symbol="∪">
                                  <constant symbol="s_3"/>
                                  <constant symbol="s_4"/>
                                 </function>
                                 <constant symbol="s_3"/>
                                </function>
                               </constantatomformula>
                              </conjunctiveformula>
                              <constantatomformula symbol="=">
                               <function symbol="∩">
                                <function symbol="∪">
                                 <constant symbol="s_3"/>
                                 <constant symbol="s_4"/>
                                </function>
                                <constant symbol="s_3"/>
                               </function>
                               <constant symbol="s_3"/>
                              </constantatomformula>
                             </conjunctiveformula>
                             <constantatomformula symbol="GLB" definition="GLB"/>
                             <constantatomformula symbol="LUB" definition="LUB"/>
                             <constantatomformula symbol="R" definition="R"/>
                            </formulalist>
                            <formulalist>
                             <constantatomformula symbol="=">
                              <function symbol="∩">
                               <function symbol="∪">
                                <constant symbol="s_3"/>
                                <constant symbol="s_4"/>
                               </function>
                               <constant symbol="s_3"/>
                              </function>
                              <constant symbol="s_3"/>
                             </constantatomformula>
                            </formulalist>
                           </sequent>
                           <rule type="defl" symbol="d:l" param="();[()];()">
                            <sequent>
                             <formulalist>
                              <constantatomformula symbol="GLB" definition="GLB"/>
                              <constantatomformula symbol="LUB" definition="LUB"/>
                              <constantatomformula symbol="R" definition="R"/>
                             </formulalist>
                             <formulalist>
                              <conjunctiveformula type="and">
                               <constantatomformula symbol="≤" definition="≤">
                                <function symbol="∩">
                                 <function symbol="∪">
                                  <constant symbol="s_3"/>
                                  <constant symbol="s_4"/>
                                 </function>
                                 <constant symbol="s_3"/>
                                </function>
                                <constant symbol="s_3"/>
                               </constantatomformula>
                               <constantatomformula symbol="≤" definition="≤">
                                <constant symbol="s_3"/>
                                <function symbol="∩">
                                 <function symbol="∪">
                                  <constant symbol="s_3"/>
                                  <constant symbol="s_4"/>
                                 </function>
                                 <constant symbol="s_3"/>
                                </function>
                               </constantatomformula>
                              </conjunctiveformula>
                             </formulalist>
                            </sequent>
                            <rule type="foralll" symbol="\all:l">
                             <sequent>
                              <formulalist>
                               <quantifiedformula type="all">
                                <variable symbol="x"/>
                                <quantifiedformula type="all">
                                 <variable symbol="y"/>
                                 <conjunctiveformula type="and">
                                  <conjunctiveformula type="and">
                                   <constantatomformula symbol="≤" definition="≤">
                                    <function symbol="∩">
                                     <variable symbol="x"/>
                                     <variable symbol="y"/>
                                    </function>
                                    <variable symbol="x"/>
                                   </constantatomformula>
                                   <constantatomformula symbol="≤" definition="≤">
                                    <function symbol="∩">
                                     <variable symbol="x"/>
                                     <variable symbol="y"/>
                                    </function>
                                    <variable symbol="y"/>
                                   </constantatomformula>
                                  </conjunctiveformula>
                                  <quantifiedformula type="all">
                                   <variable symbol="z"/>
                                   <conjunctiveformula type="impl">
                                    <conjunctiveformula type="and">
                                     <constantatomformula symbol="≤" definition="≤">
                                      <variable symbol="z"/>
                                      <variable symbol="x"/>
                                     </constantatomformula>
                                     <constantatomformula symbol="≤" definition="≤">
                                      <variable symbol="z"/>
                                      <variable symbol="y"/>
                                     </constantatomformula>
                                    </conjunctiveformula>
                                    <constantatomformula symbol="≤" definition="≤">
                                     <variable symbol="z"/>
                                     <function symbol="∩">
                                      <variable symbol="x"/>
                                      <variable symbol="y"/>
                                     </function>
                                    </constantatomformula>
                                   </conjunctiveformula>
                                  </quantifiedformula>
                                 </conjunctiveformula>
                                </quantifiedformula>
                               </quantifiedformula>
                               <constantatomformula symbol="LUB" definition="LUB"/>
                               <constantatomformula symbol="R" definition="R"/>
                              </formulalist>
                              <formulalist>
                               <conjunctiveformula type="and">
                                <constantatomformula symbol="≤" definition="≤">
                                 <function symbol="∩">
                                  <function symbol="∪">
                                   <constant symbol="s_3"/>
                                   <constant symbol="s_4"/>
                                  </function>
                                  <constant symbol="s_3"/>
                                 </function>
                                 <constant symbol="s_3"/>
                                </constantatomformula>
                                <constantatomformula symbol="≤" definition="≤">
                                 <constant symbol="s_3"/>
                                 <function symbol="∩">
                                  <function symbol="∪">
                                   <constant symbol="s_3"/>
                                   <constant symbol="s_4"/>
                                  </function>
                                  <constant symbol="s_3"/>
                                 </function>
                                </constantatomformula>
                               </conjunctiveformula>
                              </formulalist>
                             </sequent>
                             <rule type="foralll" symbol="\all:l">
                              <sequent>
                               <formulalist>
                                <quantifiedformula type="all">
                                 <variable symbol="y"/>
                                 <conjunctiveformula type="and">
                                  <conjunctiveformula type="and">
                                   <constantatomformula symbol="≤" definition="≤">
                                    <function symbol="∩">
                                     <function symbol="∪">
                                      <constant symbol="s_3"/>
                                      <constant symbol="s_4"/>
                                     </function>
                                     <variable symbol="y"/>
                                    </function>
                                    <function symbol="∪">
                                     <constant symbol="s_3"/>
                                     <constant symbol="s_4"/>
                                    </function>
                                   </constantatomformula>
                                   <constantatomformula symbol="≤" definition="≤">
                                    <function symbol="∩">
                                     <function symbol="∪">
                                      <constant symbol="s_3"/>
                                      <constant symbol="s_4"/>
                                     </function>
                                     <variable symbol="y"/>
                                    </function>
                                    <variable symbol="y"/>
                                   </constantatomformula>
                                  </conjunctiveformula>
                                  <quantifiedformula type="all">
                                   <variable symbol="z"/>
                                   <conjunctiveformula type="impl">
                                    <conjunctiveformula type="and">
                                     <constantatomformula symbol="≤" definition="≤">
                                      <variable symbol="z"/>
                                      <function symbol="∪">
                                       <constant symbol="s_3"/>
                                       <constant symbol="s_4"/>
                                      </function>
                                     </constantatomformula>
                                     <constantatomformula symbol="≤" definition="≤">
                                      <variable symbol="z"/>
                                      <variable symbol="y"/>
                                     </constantatomformula>
                                    </conjunctiveformula>
                                    <constantatomformula symbol="≤" definition="≤">
                                     <variable symbol="z"/>
                                     <function symbol="∩">
                                      <function symbol="∪">
                                       <constant symbol="s_3"/>
                                       <constant symbol="s_4"/>
                                      </function>
                                      <variable symbol="y"/>
                                     </function>
                                    </constantatomformula>
                                   </conjunctiveformula>
                                  </quantifiedformula>
                                 </conjunctiveformula>
                                </quantifiedformula>
                                <constantatomformula symbol="LUB" definition="LUB"/>
                                <constantatomformula symbol="R" definition="R"/>
                               </formulalist>
                               <formulalist>
                                <conjunctiveformula type="and">
                                 <constantatomformula symbol="≤" definition="≤">
                                  <function symbol="∩">
                                   <function symbol="∪">
                                    <constant symbol="s_3"/>
                                    <constant symbol="s_4"/>
                                   </function>
                                   <constant symbol="s_3"/>
                                  </function>
                                  <constant symbol="s_3"/>
                                 </constantatomformula>
                                 <constantatomformula symbol="≤" definition="≤">
                                  <constant symbol="s_3"/>
                                  <function symbol="∩">
                                   <function symbol="∪">
                                    <constant symbol="s_3"/>
                                    <constant symbol="s_4"/>
                                   </function>
                                   <constant symbol="s_3"/>
                                  </function>
                                 </constantatomformula>
                                </conjunctiveformula>
                               </formulalist>
                              </sequent>
                              <rule type="contrl" symbol="c:l" param="2,1,1">
                               <sequent>
                                <formulalist>
                                 <conjunctiveformula type="and">
                                  <conjunctiveformula type="and">
                                   <constantatomformula symbol="≤" definition="≤">
                                    <function symbol="∩">
                                     <function symbol="∪">
                                      <constant symbol="s_3"/>
                                      <constant symbol="s_4"/>
                                     </function>
                                     <constant symbol="s_3"/>
                                    </function>
                                    <function symbol="∪">
                                     <constant symbol="s_3"/>
                                     <constant symbol="s_4"/>
                                    </function>
                                   </constantatomformula>
                                   <constantatomformula symbol="≤" definition="≤">
                                    <function symbol="∩">
                                     <function symbol="∪">
                                      <constant symbol="s_3"/>
                                      <constant symbol="s_4"/>
                                     </function>
                                     <constant symbol="s_3"/>
                                    </function>
                                    <constant symbol="s_3"/>
                                   </constantatomformula>
                                  </conjunctiveformula>
                                  <quantifiedformula type="all">
                                   <variable symbol="z"/>
                                   <conjunctiveformula type="impl">
                                    <conjunctiveformula type="and">
                                     <constantatomformula symbol="≤" definition="≤">
                                      <variable symbol="z"/>
                                      <function symbol="∪">
                                       <constant symbol="s_3"/>
                                       <constant symbol="s_4"/>
                                      </function>
                                     </constantatomformula>
                                     <constantatomformula symbol="≤" definition="≤">
                                      <variable symbol="z"/>
                                      <constant symbol="s_3"/>
                                     </constantatomformula>
                                    </conjunctiveformula>
                                    <constantatomformula symbol="≤" definition="≤">
                                     <variable symbol="z"/>
                                     <function symbol="∩">
                                      <function symbol="∪">
                                       <constant symbol="s_3"/>
                                       <constant symbol="s_4"/>
                                      </function>
                                      <constant symbol="s_3"/>
                                     </function>
                                    </constantatomformula>
                                   </conjunctiveformula>
                                  </quantifiedformula>
                                 </conjunctiveformula>
                                 <constantatomformula symbol="LUB" definition="LUB"/>
                                 <constantatomformula symbol="R" definition="R"/>
                                </formulalist>
                                <formulalist>
                                 <conjunctiveformula type="and">
                                  <constantatomformula symbol="≤" definition="≤">
                                   <function symbol="∩">
                                    <function symbol="∪">
                                     <constant symbol="s_3"/>
                                     <constant symbol="s_4"/>
                                    </function>
                                    <constant symbol="s_3"/>
                                   </function>
                                   <constant symbol="s_3"/>
                                  </constantatomformula>
                                  <constantatomformula symbol="≤" definition="≤">
                                   <constant symbol="s_3"/>
                                   <function symbol="∩">
                                    <function symbol="∪">
                                     <constant symbol="s_3"/>
                                     <constant symbol="s_4"/>
                                    </function>
                                    <constant symbol="s_3"/>
                                   </function>
                                  </constantatomformula>
                                 </conjunctiveformula>
                                </formulalist>
                               </sequent>
                               <rule type="andl2" symbol="\and:l2">
                                <sequent>
                                 <formulalist>
                                  <conjunctiveformula type="and">
                                   <conjunctiveformula type="and">
                                    <constantatomformula symbol="≤" definition="≤">
                                     <function symbol="∩">
                                      <function symbol="∪">
                                       <constant symbol="s_3"/>
                                       <constant symbol="s_4"/>
                                      </function>
                                      <constant symbol="s_3"/>
                                     </function>
                                     <function symbol="∪">
                                      <constant symbol="s_3"/>
                                      <constant symbol="s_4"/>
                                     </function>
                                    </constantatomformula>
                                    <constantatomformula symbol="≤" definition="≤">
                                     <function symbol="∩">
                                      <function symbol="∪">
                                       <constant symbol="s_3"/>
                                       <constant symbol="s_4"/>
                                      </function>
                                      <constant symbol="s_3"/>
                                     </function>
                                     <constant symbol="s_3"/>
                                    </constantatomformula>
                                   </conjunctiveformula>
                                   <quantifiedformula type="all">
                                    <variable symbol="z"/>
                                    <conjunctiveformula type="impl">
                                     <conjunctiveformula type="and">
                                      <constantatomformula symbol="≤" definition="≤">
                                       <variable symbol="z"/>
                                       <function symbol="∪">
                                        <constant symbol="s_3"/>
                                        <constant symbol="s_4"/>
                                       </function>
                                      </constantatomformula>
                                      <constantatomformula symbol="≤" definition="≤">
                                       <variable symbol="z"/>
                                       <constant symbol="s_3"/>
                                      </constantatomformula>
                                     </conjunctiveformula>
                                     <constantatomformula symbol="≤" definition="≤">
                                      <variable symbol="z"/>
                                      <function symbol="∩">
                                       <function symbol="∪">
                                        <constant symbol="s_3"/>
                                        <constant symbol="s_4"/>
                                       </function>
                                       <constant symbol="s_3"/>
                                      </function>
                                     </constantatomformula>
                                    </conjunctiveformula>
                                   </quantifiedformula>
                                  </conjunctiveformula>
                                  <conjunctiveformula type="and">
                                   <conjunctiveformula type="and">
                                    <constantatomformula symbol="≤" definition="≤">
                                     <function symbol="∩">
                                      <function symbol="∪">
                                       <constant symbol="s_3"/>
                                       <constant symbol="s_4"/>
                                      </function>
                                      <constant symbol="s_3"/>
                                     </function>
                                     <function symbol="∪">
                                      <constant symbol="s_3"/>
                                      <constant symbol="s_4"/>
                                     </function>
                                    </constantatomformula>
                                    <constantatomformula symbol="≤" definition="≤">
                                     <function symbol="∩">
                                      <function symbol="∪">
                                       <constant symbol="s_3"/>
                                       <constant symbol="s_4"/>
                                      </function>
                                      <constant symbol="s_3"/>
                                     </function>
                                     <constant symbol="s_3"/>
                                    </constantatomformula>
                                   </conjunctiveformula>
                                   <quantifiedformula type="all">
                                    <variable symbol="z"/>
                                    <conjunctiveformula type="impl">
                                     <conjunctiveformula type="and">
                                      <constantatomformula symbol="≤" definition="≤">
                                       <variable symbol="z"/>
                                       <function symbol="∪">
                                        <constant symbol="s_3"/>
                                        <constant symbol="s_4"/>
                                       </function>
                                      </constantatomformula>
                                      <constantatomformula symbol="≤" definition="≤">
                                       <variable symbol="z"/>
                                       <constant symbol="s_3"/>
                                      </constantatomformula>
                                     </conjunctiveformula>
                                     <constantatomformula symbol="≤" definition="≤">
                                      <variable symbol="z"/>
                                      <function symbol="∩">
                                       <function symbol="∪">
                                        <constant symbol="s_3"/>
                                        <constant symbol="s_4"/>
                                       </function>
                                       <constant symbol="s_3"/>
                                      </function>
                                     </constantatomformula>
                                    </conjunctiveformula>
                                   </quantifiedformula>
                                  </conjunctiveformula>
                                  <constantatomformula symbol="LUB" definition="LUB"/>
                                  <constantatomformula symbol="R" definition="R"/>
                                 </formulalist>
                                 <formulalist>
                                  <conjunctiveformula type="and">
                                   <constantatomformula symbol="≤" definition="≤">
                                    <function symbol="∩">
                                     <function symbol="∪">
                                      <constant symbol="s_3"/>
                                      <constant symbol="s_4"/>
                                     </function>
                                     <constant symbol="s_3"/>
                                    </function>
                                    <constant symbol="s_3"/>
                                   </constantatomformula>
                                   <constantatomformula symbol="≤" definition="≤">
                                    <constant symbol="s_3"/>
                                    <function symbol="∩">
                                     <function symbol="∪">
                                      <constant symbol="s_3"/>
                                      <constant symbol="s_4"/>
                                     </function>
                                     <constant symbol="s_3"/>
                                    </function>
                                   </constantatomformula>
                                  </conjunctiveformula>
                                 </formulalist>
                                </sequent>
                                <rule type="perml" symbol="\pi:l" param="(1 2)">
                                 <sequent>
                                  <formulalist>
                                   <quantifiedformula type="all">
                                    <variable symbol="z"/>
                                    <conjunctiveformula type="impl">
                                     <conjunctiveformula type="and">
                                      <constantatomformula symbol="≤" definition="≤">
                                       <variable symbol="z"/>
                                       <function symbol="∪">
                                        <constant symbol="s_3"/>
                                        <constant symbol="s_4"/>
                                       </function>
                                      </constantatomformula>
                                      <constantatomformula symbol="≤" definition="≤">
                                       <variable symbol="z"/>
                                       <constant symbol="s_3"/>
                                      </constantatomformula>
                                     </conjunctiveformula>
                                     <constantatomformula symbol="≤" definition="≤">
                                      <variable symbol="z"/>
                                      <function symbol="∩">
                                       <function symbol="∪">
                                        <constant symbol="s_3"/>
                                        <constant symbol="s_4"/>
                                       </function>
                                       <constant symbol="s_3"/>
                                      </function>
                                     </constantatomformula>
                                    </conjunctiveformula>
                                   </quantifiedformula>
                                   <conjunctiveformula type="and">
                                    <conjunctiveformula type="and">
                                     <constantatomformula symbol="≤" definition="≤">
                                      <function symbol="∩">
                                       <function symbol="∪">
                                        <constant symbol="s_3"/>
                                        <constant symbol="s_4"/>
                                       </function>
                                       <constant symbol="s_3"/>
                                      </function>
                                      <function symbol="∪">
                                       <constant symbol="s_3"/>
                                       <constant symbol="s_4"/>
                                      </function>
                                     </constantatomformula>
                                     <constantatomformula symbol="≤" definition="≤">
                                      <function symbol="∩">
                                       <function symbol="∪">
                                        <constant symbol="s_3"/>
                                        <constant symbol="s_4"/>
                                       </function>
                                       <constant symbol="s_3"/>
                                      </function>
                                      <constant symbol="s_3"/>
                                     </constantatomformula>
                                    </conjunctiveformula>
                                    <quantifiedformula type="all">
                                     <variable symbol="z"/>
                                     <conjunctiveformula type="impl">
                                      <conjunctiveformula type="and">
                                       <constantatomformula symbol="≤" definition="≤">
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                                        <function symbol="∪">
                                         <constant symbol="s_3"/>
                                         <constant symbol="s_4"/>
                                        </function>
                                       </constantatomformula>
                                       <constantatomformula symbol="≤" definition="≤">
                                        <variable symbol="z"/>
                                        <constant symbol="s_3"/>
                                       </constantatomformula>
                                      </conjunctiveformula>
                                      <constantatomformula symbol="≤" definition="≤">
                                       <variable symbol="z"/>
                                       <function symbol="∩">
                                        <function symbol="∪">
                                         <constant symbol="s_3"/>
                                         <constant symbol="s_4"/>
                                        </function>
                                        <constant symbol="s_3"/>
                                       </function>
                                      </constantatomformula>
                                     </conjunctiveformula>
                                    </quantifiedformula>
                                   </conjunctiveformula>
                                   <constantatomformula symbol="LUB" definition="LUB"/>
                                   <constantatomformula symbol="R" definition="R"/>
                                  </formulalist>
                                  <formulalist>
                                   <conjunctiveformula type="and">
                                    <constantatomformula symbol="≤" definition="≤">
                                     <function symbol="∩">
                                      <function symbol="∪">
                                       <constant symbol="s_3"/>
                                       <constant symbol="s_4"/>
                                      </function>
                                      <constant symbol="s_3"/>
                                     </function>
                                     <constant symbol="s_3"/>
                                    </constantatomformula>
                                    <constantatomformula symbol="≤" definition="≤">
                                     <constant symbol="s_3"/>
                                     <function symbol="∩">
                                      <function symbol="∪">
                                       <constant symbol="s_3"/>
                                       <constant symbol="s_4"/>
                                      </function>
                                      <constant symbol="s_3"/>
                                     </function>
                                    </constantatomformula>
                                   </conjunctiveformula>
                                  </formulalist>
                                 </sequent>
                                 <rule type="andl1" symbol="\and:l1">
                                  <sequent>
                                   <formulalist>
                                    <conjunctiveformula type="and">
                                     <conjunctiveformula type="and">
                                      <constantatomformula symbol="≤" definition="≤">
                                       <function symbol="∩">
                                        <function symbol="∪">
                                         <constant symbol="s_3"/>
                                         <constant symbol="s_4"/>
                                        </function>
                                        <constant symbol="s_3"/>
                                       </function>
                                       <function symbol="∪">
                                        <constant symbol="s_3"/>
                                        <constant symbol="s_4"/>
                                       </function>
                                      </constantatomformula>
                                      <constantatomformula symbol="≤" definition="≤">
                                       <function symbol="∩">
                                        <function symbol="∪">
                                         <constant symbol="s_3"/>
                                         <constant symbol="s_4"/>
                                        </function>
                                        <constant symbol="s_3"/>
                                       </function>
                                       <constant symbol="s_3"/>
                                      </constantatomformula>
                                     </conjunctiveformula>
                                     <quantifiedformula type="all">
                                      <variable symbol="z"/>
                                      <conjunctiveformula type="impl">
                                       <conjunctiveformula type="and">
                                        <constantatomformula symbol="≤" definition="≤">
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                                         <function symbol="∪">
                                          <constant symbol="s_3"/>
                                          <constant symbol="s_4"/>
                                         </function>
                                        </constantatomformula>
                                        <constantatomformula symbol="≤" definition="≤">
                                         <variable symbol="z"/>
                                         <constant symbol="s_3"/>
                                        </constantatomformula>
                                       </conjunctiveformula>
                                       <constantatomformula symbol="≤" definition="≤">
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                                        <function symbol="∩">
                                         <function symbol="∪">
                                          <constant symbol="s_3"/>
                                          <constant symbol="s_4"/>
                                         </function>
                                         <constant symbol="s_3"/>
                                        </function>
                                       </constantatomformula>
                                      </conjunctiveformula>
                                     </quantifiedformula>
                                    </conjunctiveformula>
                                    <quantifiedformula type="all">
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                                      <conjunctiveformula type="and">
                                       <constantatomformula symbol="≤" definition="≤">
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                                        </function>
                                       </constantatomformula>
                                       <constantatomformula symbol="≤" definition="≤">
                                        <variable symbol="z"/>
                                        <constant symbol="s_3"/>
                                       </constantatomformula>
                                      </conjunctiveformula>
                                      <constantatomformula symbol="≤" definition="≤">
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                                       <function symbol="∩">
                                        <function symbol="∪">
                                         <constant symbol="s_3"/>
                                         <constant symbol="s_4"/>
                                        </function>
                                        <constant symbol="s_3"/>
                                       </function>
                                      </constantatomformula>
                                     </conjunctiveformula>
                                    </quantifiedformula>
                                    <constantatomformula symbol="LUB" definition="LUB"/>
                                    <constantatomformula symbol="R" definition="R"/>
                                   </formulalist>
                                   <formulalist>
                                    <conjunctiveformula type="and">
                                     <constantatomformula symbol="≤" definition="≤">
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                                       <function symbol="∪">
                                        <constant symbol="s_3"/>
                                        <constant symbol="s_4"/>
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                                       <constant symbol="s_3"/>
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                                      <constant symbol="s_3"/>
                                     </constantatomformula>
                                     <constantatomformula symbol="≤" definition="≤">
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                                       <function symbol="∪">
                                        <constant symbol="s_3"/>
                                        <constant symbol="s_4"/>
                                       </function>
                                       <constant symbol="s_3"/>
                                      </function>
                                     </constantatomformula>
                                    </conjunctiveformula>
                                   </formulalist>
                                  </sequent>
                                  <rule type="andl2" symbol="\and:l2">
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                                    <formulalist>
                                     <conjunctiveformula type="and">
                                      <constantatomformula symbol="≤" definition="≤">
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                                        <function symbol="∪">
                                         <constant symbol="s_3"/>
                                         <constant symbol="s_4"/>
                                        </function>
                                        <constant symbol="s_3"/>
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                                       <function symbol="∪">
                                        <constant symbol="s_3"/>
                                        <constant symbol="s_4"/>
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                                      <constantatomformula symbol="≤" definition="≤">
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                                        <function symbol="∪">
                                         <constant symbol="s_3"/>
                                         <constant symbol="s_4"/>
                                        </function>
                                        <constant symbol="s_3"/>
                                       </function>
                                       <constant symbol="s_3"/>
                                      </constantatomformula>
                                     </conjunctiveformula>
                                     <quantifiedformula type="all">
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                                      <conjunctiveformula type="impl">
                                       <conjunctiveformula type="and">
                                        <constantatomformula symbol="≤" definition="≤">
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                                          <constant symbol="s_3"/>
                                          <constant symbol="s_4"/>
                                         </function>
                                        </constantatomformula>
                                        <constantatomformula symbol="≤" definition="≤">
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                                         <constant symbol="s_3"/>
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                                       </conjunctiveformula>
                                       <constantatomformula symbol="≤" definition="≤">
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                                          <constant symbol="s_4"/>
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                                         <constant symbol="s_3"/>
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                                       </constantatomformula>
                                      </conjunctiveformula>
                                     </quantifiedformula>
                                     <constantatomformula symbol="LUB" definition="LUB"/>
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                                    </formulalist>
                                    <formulalist>
                                     <conjunctiveformula type="and">
                                      <constantatomformula symbol="≤" definition="≤">
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                                         <constant symbol="s_4"/>
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                                       <constant symbol="s_3"/>
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                                      <constantatomformula symbol="≤" definition="≤">
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                                         <constant symbol="s_4"/>
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                                        <constant symbol="s_3"/>
                                       </function>
                                      </constantatomformula>
                                     </conjunctiveformula>
                                    </formulalist>
                                   </sequent>
                                   <rule type="andr" symbol="\and:r">
                                    <sequent>
                                     <formulalist>
                                      <constantatomformula symbol="≤" definition="≤">
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                                        <function symbol="∪">
                                         <constant symbol="s_3"/>
                                         <constant symbol="s_4"/>
                                        </function>
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                                       </function>
                                       <constant symbol="s_3"/>
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                                      <quantifiedformula type="all">
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                                        <conjunctiveformula type="and">
                                         <constantatomformula symbol="≤" definition="≤">
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                                          <function symbol="∪">
                                           <constant symbol="s_3"/>
                                           <constant symbol="s_4"/>
                                          </function>
                                         </constantatomformula>
                                         <constantatomformula symbol="≤" definition="≤">
                                          <variable symbol="z"/>
                                          <constant symbol="s_3"/>
                                         </constantatomformula>
                                        </conjunctiveformula>
                                        <constantatomformula symbol="≤" definition="≤">
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                                         <function symbol="∩">
                                          <function symbol="∪">
                                           <constant symbol="s_3"/>
                                           <constant symbol="s_4"/>
                                          </function>
                                          <constant symbol="s_3"/>
                                         </function>
                                        </constantatomformula>
                                       </conjunctiveformula>
                                      </quantifiedformula>
                                      <constantatomformula symbol="LUB" definition="LUB"/>
                                      <constantatomformula symbol="R" definition="R"/>
                                     </formulalist>
                                     <formulalist>
                                      <conjunctiveformula type="and">
                                       <constantatomformula symbol="≤" definition="≤">
                                        <function symbol="∩">
                                         <function symbol="∪">
                                          <constant symbol="s_3"/>
                                          <constant symbol="s_4"/>
                                         </function>
                                         <constant symbol="s_3"/>
                                        </function>
                                        <constant symbol="s_3"/>
                                       </constantatomformula>
                                       <constantatomformula symbol="≤" definition="≤">
                                        <constant symbol="s_3"/>
                                        <function symbol="∩">
                                         <function symbol="∪">
                                          <constant symbol="s_3"/>
                                          <constant symbol="s_4"/>
                                         </function>
                                         <constant symbol="s_3"/>
                                        </function>
                                       </constantatomformula>
                                      </conjunctiveformula>
                                     </formulalist>
                                    </sequent>
                                    <rule type="defl" symbol="d:l" param="();[(1)(1 1)(2)][(2)(1 2)];()">
                                     <sequent>
                                      <formulalist>
                                       <constantatomformula symbol="≤" definition="≤">
                                        <function symbol="∩">
                                         <function symbol="∪">
                                          <constant symbol="s_3"/>
                                          <constant symbol="s_4"/>
                                         </function>
                                         <constant symbol="s_3"/>
                                        </function>
                                        <constant symbol="s_3"/>
                                       </constantatomformula>
                                      </formulalist>
                                      <formulalist>
                                       <constantatomformula symbol="≤" definition="≤">
                                        <function symbol="∩">
                                         <function symbol="∪">
                                          <constant symbol="s_3"/>
                                          <constant symbol="s_4"/>
                                         </function>
                                         <constant symbol="s_3"/>
                                        </function>
                                        <constant symbol="s_3"/>
                                       </constantatomformula>
                                      </formulalist>
                                     </sequent>
                                     <rule type="defr" symbol="d:r" param="();[(1)(1 1)(2)][(2)(1 2)];()">
                                      <sequent>
                                       <formulalist>
                                        <constantatomformula symbol="=">
                                         <function symbol="∩">
                                          <function symbol="∩">
                                           <function symbol="∪">
                                            <constant symbol="s_3"/>
                                            <constant symbol="s_4"/>
                                           </function>
                                           <constant symbol="s_3"/>
                                          </function>
                                          <constant symbol="s_3"/>
                                         </function>
                                         <function symbol="∩">
                                          <function symbol="∪">
                                           <constant symbol="s_3"/>
                                           <constant symbol="s_4"/>
                                          </function>
                                          <constant symbol="s_3"/>
                                         </function>
                                        </constantatomformula>
                                       </formulalist>
                                       <formulalist>
                                        <constantatomformula symbol="≤" definition="≤">
                                         <function symbol="∩">
                                          <function symbol="∪">
                                           <constant symbol="s_3"/>
                                           <constant symbol="s_4"/>
                                          </function>
                                          <constant symbol="s_3"/>
                                         </function>
                                         <constant symbol="s_3"/>
                                        </constantatomformula>
                                       </formulalist>
                                      </sequent>
                                      <rule type="axiom" symbol="axiom">
                                       <sequent>
                                        <formulalist>
                                         <constantatomformula symbol="=">
                                          <function symbol="∩">
                                           <function symbol="∩">
                                            <function symbol="∪">
                                             <constant symbol="s_3"/>
                                             <constant symbol="s_4"/>
                                            </function>
                                            <constant symbol="s_3"/>
                                           </function>
                                           <constant symbol="s_3"/>
                                          </function>
                                          <function symbol="∩">
                                           <function symbol="∪">
                                            <constant symbol="s_3"/>
                                            <constant symbol="s_4"/>
                                           </function>
                                           <constant symbol="s_3"/>
                                          </function>
                                         </constantatomformula>
                                        </formulalist>
                                        <formulalist>
                                         <constantatomformula symbol="=">
                                          <function symbol="∩">
                                           <function symbol="∩">
                                            <function symbol="∪">
                                             <constant symbol="s_3"/>
                                             <constant symbol="s_4"/>
                                            </function>
                                            <constant symbol="s_3"/>
                                           </function>
                                           <constant symbol="s_3"/>
                                          </function>
                                          <function symbol="∩">
                                           <function symbol="∪">
                                            <constant symbol="s_3"/>
                                            <constant symbol="s_4"/>
                                           </function>
                                           <constant symbol="s_3"/>
                                          </function>
                                         </constantatomformula>
                                        </formulalist>
                                       </sequent>
                                      </rule>
                                     </rule>
                                    </rule>
                                    <rule type="foralll" symbol="\all:l">
                                     <sequent>
                                      <formulalist>
                                       <quantifiedformula type="all">
                                        <variable symbol="z"/>
                                        <conjunctiveformula type="impl">
                                         <conjunctiveformula type="and">
                                          <constantatomformula symbol="≤" definition="≤">
                                           <variable symbol="z"/>
                                           <function symbol="∪">
                                            <constant symbol="s_3"/>
                                            <constant symbol="s_4"/>
                                           </function>
                                          </constantatomformula>
                                          <constantatomformula symbol="≤" definition="≤">
                                           <variable symbol="z"/>
                                           <constant symbol="s_3"/>
                                          </constantatomformula>
                                         </conjunctiveformula>
                                         <constantatomformula symbol="≤" definition="≤">
                                          <variable symbol="z"/>
                                          <function symbol="∩">
                                           <function symbol="∪">
                                            <constant symbol="s_3"/>
                                            <constant symbol="s_4"/>
                                           </function>
                                           <constant symbol="s_3"/>
                                          </function>
                                         </constantatomformula>
                                        </conjunctiveformula>
                                       </quantifiedformula>
                                       <constantatomformula symbol="LUB" definition="LUB"/>
                                       <constantatomformula symbol="R" definition="R"/>
                                      </formulalist>
                                      <formulalist>
                                       <constantatomformula symbol="≤" definition="≤">
                                        <constant symbol="s_3"/>
                                        <function symbol="∩">
                                         <function symbol="∪">
                                          <constant symbol="s_3"/>
                                          <constant symbol="s_4"/>
                                         </function>
                                         <constant symbol="s_3"/>
                                        </function>
                                       </constantatomformula>
                                      </formulalist>
                                     </sequent>
                                     <rule type="impll" symbol="\impl:l">
                                      <sequent>
                                       <formulalist>
                                        <conjunctiveformula type="impl">
                                         <conjunctiveformula type="and">
                                          <constantatomformula symbol="≤" definition="≤">
                                           <constant symbol="s_3"/>
                                           <function symbol="∪">
                                            <constant symbol="s_3"/>
                                            <constant symbol="s_4"/>
                                           </function>
                                          </constantatomformula>
                                          <constantatomformula symbol="≤" definition="≤">
                                           <constant symbol="s_3"/>
                                           <constant symbol="s_3"/>
                                          </constantatomformula>
                                         </conjunctiveformula>
                                         <constantatomformula symbol="≤" definition="≤">
                                          <constant symbol="s_3"/>
                                          <function symbol="∩">
                                           <function symbol="∪">
                                            <constant symbol="s_3"/>
                                            <constant symbol="s_4"/>
                                           </function>
                                           <constant symbol="s_3"/>
                                          </function>
                                         </constantatomformula>
                                        </conjunctiveformula>
                                        <constantatomformula symbol="LUB" definition="LUB"/>
                                        <constantatomformula symbol="R" definition="R"/>
                                       </formulalist>
                                       <formulalist>
                                        <constantatomformula symbol="≤" definition="≤">
                                         <constant symbol="s_3"/>
                                         <function symbol="∩">
                                          <function symbol="∪">
                                           <constant symbol="s_3"/>
                                           <constant symbol="s_4"/>
                                          </function>
                                          <constant symbol="s_3"/>
                                         </function>
                                        </constantatomformula>
                                       </formulalist>
                                      </sequent>
                                      <rule type="andr" symbol="\and:r">
                                       <sequent>
                                        <formulalist>
                                         <constantatomformula symbol="LUB" definition="LUB"/>
                                         <constantatomformula symbol="R" definition="R"/>
                                        </formulalist>
                                        <formulalist>
                                         <conjunctiveformula type="and">
                                          <constantatomformula symbol="≤" definition="≤">
                                           <constant symbol="s_3"/>
                                           <function symbol="∪">
                                            <constant symbol="s_3"/>
                                            <constant symbol="s_4"/>
                                           </function>
                                          </constantatomformula>
                                          <constantatomformula symbol="≤" definition="≤">
                                           <constant symbol="s_3"/>
                                           <constant symbol="s_3"/>
                                          </constantatomformula>
                                         </conjunctiveformula>
                                        </formulalist>
                                       </sequent>
                                       <rule type="defl" symbol="d:l" param="();[()];()">
                                        <sequent>
                                         <formulalist>
                                          <constantatomformula symbol="LUB" definition="LUB"/>
                                         </formulalist>
                                         <formulalist>
                                          <constantatomformula symbol="≤" definition="≤">
                                           <constant symbol="s_3"/>
                                           <function symbol="∪">
                                            <constant symbol="s_3"/>
                                            <constant symbol="s_4"/>
                                           </function>
                                          </constantatomformula>
                                         </formulalist>
                                        </sequent>
                                        <rule type="foralll" symbol="\all:l">
                                         <sequent>
                                          <formulalist>
                                           <quantifiedformula type="all">
                                            <variable symbol="x"/>
                                            <quantifiedformula type="all">
                                             <variable symbol="y"/>
                                             <conjunctiveformula type="and">
                                              <conjunctiveformula type="and">
                                               <constantatomformula symbol="≤" definition="≤">
                                                <variable symbol="x"/>
                                                <function symbol="∪">
                                                 <variable symbol="x"/>
                                                 <variable symbol="y"/>
                                                </function>
                                               </constantatomformula>
                                               <constantatomformula symbol="≤" definition="≤">
                                                <variable symbol="y"/>
                                                <function symbol="∪">
                                                 <variable symbol="x"/>
                                                 <variable symbol="y"/>
                                                </function>
                                               </constantatomformula>
                                              </conjunctiveformula>
                                              <quantifiedformula type="all">
                                               <variable symbol="z"/>
                                               <conjunctiveformula type="impl">
                                                <conjunctiveformula type="and">
                                                 <constantatomformula symbol="≤" definition="≤">
                                                  <variable symbol="x"/>
                                                  <variable symbol="z"/>
                                                 </constantatomformula>
                                                 <constantatomformula symbol="≤" definition="≤">
                                                  <variable symbol="y"/>
                                                  <variable symbol="z"/>
                                                 </constantatomformula>
                                                </conjunctiveformula>
                                                <constantatomformula symbol="≤" definition="≤">
                                                 <function symbol="∪">
                                                  <variable symbol="x"/>
                                                  <variable symbol="y"/>
                                                 </function>
                                                 <variable symbol="z"/>
                                                </constantatomformula>
                                               </conjunctiveformula>
                                              </quantifiedformula>
                                             </conjunctiveformula>
                                            </quantifiedformula>
                                           </quantifiedformula>
                                          </formulalist>
                                          <formulalist>
                                           <constantatomformula symbol="≤" definition="≤">
                                            <constant symbol="s_3"/>
                                            <function symbol="∪">
                                             <constant symbol="s_3"/>
                                             <constant symbol="s_4"/>
                                            </function>
                                           </constantatomformula>
                                          </formulalist>
                                         </sequent>
                                         <rule type="foralll" symbol="\all:l">
                                          <sequent>
                                           <formulalist>
                                            <quantifiedformula type="all">
                                             <variable symbol="y"/>
                                             <conjunctiveformula type="and">
                                              <conjunctiveformula type="and">
                                               <constantatomformula symbol="≤" definition="≤">
                                                <constant symbol="s_3"/>
                                                <function symbol="∪">
                                                 <constant symbol="s_3"/>
                                                 <variable symbol="y"/>
                                                </function>
                                               </constantatomformula>
                                               <constantatomformula symbol="≤" definition="≤">
                                                <variable symbol="y"/>
                                                <function symbol="∪">
                                                 <constant symbol="s_3"/>
                                                 <variable symbol="y"/>
                                                </function>
                                               </constantatomformula>
                                              </conjunctiveformula>
                                              <quantifiedformula type="all">
                                               <variable symbol="z"/>
                                               <conjunctiveformula type="impl">
                                                <conjunctiveformula type="and">
                                                 <constantatomformula symbol="≤" definition="≤">
                                                  <constant symbol="s_3"/>
                                                  <variable symbol="z"/>
                                                 </constantatomformula>
                                                 <constantatomformula symbol="≤" definition="≤">
                                                  <variable symbol="y"/>
                                                  <variable symbol="z"/>
                                                 </constantatomformula>
                                                </conjunctiveformula>
                                                <constantatomformula symbol="≤" definition="≤">
                                                 <function symbol="∪">
                                                  <constant symbol="s_3"/>
                                                  <variable symbol="y"/>
                                                 </function>
                                                 <variable symbol="z"/>
                                                </constantatomformula>
                                               </conjunctiveformula>
                                              </quantifiedformula>
                                             </conjunctiveformula>
                                            </quantifiedformula>
                                           </formulalist>
                                           <formulalist>
                                            <constantatomformula symbol="≤" definition="≤">
                                             <constant symbol="s_3"/>
                                             <function symbol="∪">
                                              <constant symbol="s_3"/>
                                              <constant symbol="s_4"/>
                                             </function>
                                            </constantatomformula>
                                           </formulalist>
                                          </sequent>
                                          <rule type="andl1" symbol="\and:l1">
                                           <sequent>
                                            <formulalist>
                                             <conjunctiveformula type="and">
                                              <conjunctiveformula type="and">
                                               <constantatomformula symbol="≤" definition="≤">
                                                <constant symbol="s_3"/>
                                                <function symbol="∪">
                                                 <constant symbol="s_3"/>
                                                 <constant symbol="s_4"/>
                                                </function>
                                               </constantatomformula>
                                               <constantatomformula symbol="≤" definition="≤">
                                                <constant symbol="s_4"/>
                                                <function symbol="∪">
                                                 <constant symbol="s_3"/>
                                                 <constant symbol="s_4"/>
                                                </function>
                                               </constantatomformula>
                                              </conjunctiveformula>
                                              <quantifiedformula type="all">
                                               <variable symbol="z"/>
                                               <conjunctiveformula type="impl">
                                                <conjunctiveformula type="and">
                                                 <constantatomformula symbol="≤" definition="≤">
                                                  <constant symbol="s_3"/>
                                                  <variable symbol="z"/>
                                                 </constantatomformula>
                                                 <constantatomformula symbol="≤" definition="≤">
                                                  <constant symbol="s_4"/>
                                                  <variable symbol="z"/>
                                                 </constantatomformula>
                                                </conjunctiveformula>
                                                <constantatomformula symbol="≤" definition="≤">
                                                 <function symbol="∪">
                                                  <constant symbol="s_3"/>
                                                  <constant symbol="s_4"/>
                                                 </function>
                                                 <variable symbol="z"/>
                                                </constantatomformula>
                                               </conjunctiveformula>
                                              </quantifiedformula>
                                             </conjunctiveformula>
                                            </formulalist>
                                            <formulalist>
                                             <constantatomformula symbol="≤" definition="≤">
                                              <constant symbol="s_3"/>
                                              <function symbol="∪">
                                               <constant symbol="s_3"/>
                                               <constant symbol="s_4"/>
                                              </function>
                                             </constantatomformula>
                                            </formulalist>
                                           </sequent>
                                           <rule type="andl1" symbol="\and:l1">
                                            <sequent>
                                             <formulalist>
                                              <conjunctiveformula type="and">
                                               <constantatomformula symbol="≤" definition="≤">
                                                <constant symbol="s_3"/>
                                                <function symbol="∪">
                                                 <constant symbol="s_3"/>
                                                 <constant symbol="s_4"/>
                                                </function>
                                               </constantatomformula>
                                               <constantatomformula symbol="≤" definition="≤">
                                                <constant symbol="s_4"/>
                                                <function symbol="∪">
                                                 <constant symbol="s_3"/>
                                                 <constant symbol="s_4"/>
                                                </function>
                                               </constantatomformula>
                                              </conjunctiveformula>
                                             </formulalist>
                                             <formulalist>
                                              <constantatomformula symbol="≤" definition="≤">
                                               <constant symbol="s_3"/>
                                               <function symbol="∪">
                                                <constant symbol="s_3"/>
                                                <constant symbol="s_4"/>
                                               </function>
                                              </constantatomformula>
                                             </formulalist>
                                            </sequent>
                                            <rule type="defl" symbol="d:l" param="();[(1)(1 1)(2)][(2)(1 2)];()">
                                             <sequent>
                                              <formulalist>
                                               <constantatomformula symbol="≤" definition="≤">
                                                <constant symbol="s_3"/>
                                                <function symbol="∪">
                                                 <constant symbol="s_3"/>
                                                 <constant symbol="s_4"/>
                                                </function>
                                               </constantatomformula>
                                              </formulalist>
                                              <formulalist>
                                               <constantatomformula symbol="≤" definition="≤">
                                                <constant symbol="s_3"/>
                                                <function symbol="∪">
                                                 <constant symbol="s_3"/>
                                                 <constant symbol="s_4"/>
                                                </function>
                                               </constantatomformula>
                                              </formulalist>
                                             </sequent>
                                             <rule type="defr" symbol="d:r" param="();[(1)(1 1)(2)][(2)(1 2)];()">
                                              <sequent>
                                               <formulalist>
                                                <constantatomformula symbol="=">
                                                 <function symbol="∩">
                                                  <constant symbol="s_3"/>
                                                  <function symbol="∪">
                                                   <constant symbol="s_3"/>
                                                   <constant symbol="s_4"/>
                                                  </function>
                                                 </function>
                                                 <constant symbol="s_3"/>
                                                </constantatomformula>
                                               </formulalist>
                                               <formulalist>
                                                <constantatomformula symbol="≤" definition="≤">
                                                 <constant symbol="s_3"/>
                                                 <function symbol="∪">
                                                  <constant symbol="s_3"/>
                                                  <constant symbol="s_4"/>
                                                 </function>
                                                </constantatomformula>
                                               </formulalist>
                                              </sequent>
                                              <rule type="axiom" symbol="axiom">
                                               <sequent>
                                                <formulalist>
                                                 <constantatomformula symbol="=">
                                                  <function symbol="∩">
                                                   <constant symbol="s_3"/>
                                                   <function symbol="∪">
                                                    <constant symbol="s_3"/>
                                                    <constant symbol="s_4"/>
                                                   </function>
                                                  </function>
                                                  <constant symbol="s_3"/>
                                                 </constantatomformula>
                                                </formulalist>
                                                <formulalist>
                                                 <constantatomformula symbol="=">
                                                  <function symbol="∩">
                                                   <constant symbol="s_3"/>
                                                   <function symbol="∪">
                                                    <constant symbol="s_3"/>
                                                    <constant symbol="s_4"/>
                                                   </function>
                                                  </function>
                                                  <constant symbol="s_3"/>
                                                 </constantatomformula>
                                                </formulalist>
                                               </sequent>
                                              </rule>
                                             </rule>
                                            </rule>
                                           </rule>
                                          </rule>
                                         </rule>
                                        </rule>
                                       </rule>
                                       <rule type="defl" symbol="d:l" param="();[()];()">
                                        <sequent>
                                         <formulalist>
                                          <constantatomformula symbol="R" definition="R"/>
                                         </formulalist>
                                         <formulalist>
                                          <constantatomformula symbol="≤" definition="≤">
                                           <constant symbol="s_3"/>
                                           <constant symbol="s_3"/>
                                          </constantatomformula>
                                         </formulalist>
                                        </sequent>
                                        <rule type="foralll" symbol="\all:l">
                                         <sequent>
                                          <formulalist>
                                           <quantifiedformula type="all">
                                            <variable symbol="x"/>
                                            <constantatomformula symbol="≤" definition="≤">
                                             <variable symbol="x"/>
                                             <variable symbol="x"/>
                                            </constantatomformula>
                                           </quantifiedformula>
                                          </formulalist>
                                          <formulalist>
                                           <constantatomformula symbol="≤" definition="≤">
                                            <constant symbol="s_3"/>
                                            <constant symbol="s_3"/>
                                           </constantatomformula>
                                          </formulalist>
                                         </sequent>
                                         <rule type="defl" symbol="d:l" param="();[(1)(1 1)(2)][(2)(1 2)];()">
                                          <sequent>
                                           <formulalist>
                                            <constantatomformula symbol="≤" definition="≤">
                                             <constant symbol="s_3"/>
                                             <constant symbol="s_3"/>
                                            </constantatomformula>
                                           </formulalist>
                                           <formulalist>
                                            <constantatomformula symbol="≤" definition="≤">
                                             <constant symbol="s_3"/>
                                             <constant symbol="s_3"/>
                                            </constantatomformula>
                                           </formulalist>
                                          </sequent>
                                          <rule type="defr" symbol="d:r" param="();[(1)(1 1)(2)][(2)(1 2)];()">
                                           <sequent>
                                            <formulalist>
                                             <constantatomformula symbol="=">
                                              <function symbol="∩">
                                               <constant symbol="s_3"/>
                                               <constant symbol="s_3"/>
                                              </function>
                                              <constant symbol="s_3"/>
                                             </constantatomformula>
                                            </formulalist>
                                            <formulalist>
                                             <constantatomformula symbol="≤" definition="≤">
                                              <constant symbol="s_3"/>
                                              <constant symbol="s_3"/>
                                             </constantatomformula>
                                            </formulalist>
                                           </sequent>
                                           <rule type="axiom" symbol="axiom">
                                            <sequent>
                                             <formulalist>
                                              <constantatomformula symbol="=">
                                               <function symbol="∩">
                                                <constant symbol="s_3"/>
                                                <constant symbol="s_3"/>
                                               </function>
                                               <constant symbol="s_3"/>
                                              </constantatomformula>
                                             </formulalist>
                                             <formulalist>
                                              <constantatomformula symbol="=">
                                               <function symbol="∩">
                                                <constant symbol="s_3"/>
                                                <constant symbol="s_3"/>
                                               </function>
                                               <constant symbol="s_3"/>
                                              </constantatomformula>
                                             </formulalist>
                                            </sequent>
                                           </rule>
                                          </rule>
                                         </rule>
                                        </rule>
                                       </rule>
                                      </rule>
                                      <rule type="defl" symbol="d:l" param="();[(1)(1 1)(2)][(2)(1 2)];()">
                                       <sequent>
                                        <formulalist>
                                         <constantatomformula symbol="≤" definition="≤">
                                          <constant symbol="s_3"/>
                                          <function symbol="∩">
                                           <function symbol="∪">
                                            <constant symbol="s_3"/>
                                            <constant symbol="s_4"/>
                                           </function>
                                           <constant symbol="s_3"/>
                                          </function>
                                         </constantatomformula>
                                        </formulalist>
                                        <formulalist>
                                         <constantatomformula symbol="≤" definition="≤">
                                          <constant symbol="s_3"/>
                                          <function symbol="∩">
                                           <function symbol="∪">
                                            <constant symbol="s_3"/>
                                            <constant symbol="s_4"/>
                                           </function>
                                           <constant symbol="s_3"/>
                                          </function>
                                         </constantatomformula>
                                        </formulalist>
                                       </sequent>
                                       <rule type="defr" symbol="d:r" param="();[(1)(1 1)(2)][(2)(1 2)];()">
                                        <sequent>
                                         <formulalist>
                                          <constantatomformula symbol="=">
                                           <function symbol="∩">
                                            <constant symbol="s_3"/>
                                            <function symbol="∩">
                                             <function symbol="∪">
                                              <constant symbol="s_3"/>
                                              <constant symbol="s_4"/>
                                             </function>
                                             <constant symbol="s_3"/>
                                            </function>
                                           </function>
                                           <constant symbol="s_3"/>
                                          </constantatomformula>
                                         </formulalist>
                                         <formulalist>
                                          <constantatomformula symbol="≤" definition="≤">
                                           <constant symbol="s_3"/>
                                           <function symbol="∩">
                                            <function symbol="∪">
                                             <constant symbol="s_3"/>
                                             <constant symbol="s_4"/>
                                            </function>
                                            <constant symbol="s_3"/>
                                           </function>
                                          </constantatomformula>
                                         </formulalist>
                                        </sequent>
                                        <rule type="axiom" symbol="axiom">
                                         <sequent>
                                          <formulalist>
                                           <constantatomformula symbol="=">
                                            <function symbol="∩">
                                             <constant symbol="s_3"/>
                                             <function symbol="∩">
                                              <function symbol="∪">
                                               <constant symbol="s_3"/>
                                               <constant symbol="s_4"/>
                                              </function>
                                              <constant symbol="s_3"/>
                                             </function>
                                            </function>
                                            <constant symbol="s_3"/>
                                           </constantatomformula>
                                          </formulalist>
                                          <formulalist>
                                           <constantatomformula symbol="=">
                                            <function symbol="∩">
                                             <constant symbol="s_3"/>
                                             <function symbol="∩">
                                              <function symbol="∪">
                                               <constant symbol="s_3"/>
                                               <constant symbol="s_4"/>
                                              </function>
                                              <constant symbol="s_3"/>
                                             </function>
                                            </function>
                                            <constant symbol="s_3"/>
                                           </constantatomformula>
                                          </formulalist>
                                         </sequent>
                                        </rule>
                                       </rule>
                                      </rule>
                                     </rule>
                                    </rule>
                                   </rule>
                                  </rule>
                                 </rule>
                                </rule>
                               </rule>
                              </rule>
                             </rule>
                            </rule>
                           </rule>
                           <rule type="axiom" symbol="axiom">
                            <sequent>
                             <formulalist>
                              <constantatomformula symbol="=">
                               <function symbol="∩">
                                <function symbol="∪">
                                 <constant symbol="s_3"/>
                                 <constant symbol="s_4"/>
                                </function>
                                <constant symbol="s_3"/>
                               </function>
                               <constant symbol="s_3"/>
                              </constantatomformula>
                             </formulalist>
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                           </rule>
                          </rule>
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                        </rule>
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                    </rule>
                   </rule>
                  </rule>
                 </rule>
                </rule>
               </rule>
              </rule>
             </rule>
            </rule>
           </rule>
          </rule>
         </rule>
        </rule>
       </rule>
      </rule>
     </rule>
    </rule>
   </rule>
  </rule>
 </proof>
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                                                  <function symbol="∩">
                                                   <variable symbol="y_0"/>
                                                   <variable symbol="x_0"/>
                                                  </function>
                                                 </constantatomformula>
                                                </formulalist>
                                               </sequent>
                                               <rule type="cllist">
                                                <sequent>
                                                 <formulalist/>
                                                 <formulalist>
                                                  <constantatomformula symbol="=">
                                                   <function symbol="∩">
                                                    <variable symbol="x_0"/>
                                                    <variable symbol="x_0"/>
                                                   </function>
                                                   <variable symbol="x_0"/>
                                                  </constantatomformula>
                                                 </formulalist>
                                                </sequent>
                                               </rule>
                                              </rule>
                                             </rule>
                                            </rule>
                                           </rule>
                                          </rule>
                                         </rule>
                                        </rule>
                                       </rule>
                                      </rule>
                                     </rule>
                                    </rule>
                                   </rule>
                                  </rule>
                                 </rule>
                                </rule>
                               </rule>
                              </rule>
                             </rule>
                            </rule>
                           </rule>
                          </rule>
                         </rule>
                        </rule>
                       </rule>
                      </rule>
                     </rule>
                    </rule>
                   </rule>
                  </rule>
                 </rule>
                </rule>
               </rule>
              </rule>
             </rule>
            </rule>
           </rule>
          </rule>
         </rule>
        </rule>
       </rule>
      </rule>
     </rule>
    </rule>
   </rule>
  </rule>
 </proof>
 <proof symbol="\psi_{1}" calculus="LK">
  <rule type="weakl" symbol="w:l">
   <sequent>
    <formulalist>
     <constantatomformula symbol="L1" definition="L1"/>
     <constantatomformula symbol="=">
      <function symbol="∪">
       <function symbol="∩">
        <constant symbol="s_1"/>
        <constant symbol="s_2"/>
       </function>
       <constant symbol="s_1"/>
      </function>
      <constant symbol="s_1"/>
     </constantatomformula>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <function symbol="∪">
        <constant symbol="s_3"/>
        <constant symbol="s_4"/>
       </function>
       <constant symbol="s_3"/>
      </function>
      <constant symbol="s_3"/>
     </constantatomformula>
    </formulalist>
    <formulalist>
     <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
    </formulalist>
   </sequent>
   <rule type="defr" symbol="d:r" param="();[()];()">
    <sequent>
     <formulalist>
      <constantatomformula symbol="=">
       <function symbol="∪">
        <function symbol="∩">
         <constant symbol="s_1"/>
         <constant symbol="s_2"/>
        </function>
        <constant symbol="s_1"/>
       </function>
       <constant symbol="s_1"/>
      </constantatomformula>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <function symbol="∪">
         <constant symbol="s_3"/>
         <constant symbol="s_4"/>
        </function>
        <constant symbol="s_3"/>
       </function>
       <constant symbol="s_3"/>
      </constantatomformula>
     </formulalist>
     <formulalist>
      <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
     </formulalist>
    </sequent>
    <rule type="andr" symbol="\and:r">
     <sequent>
      <formulalist>
       <constantatomformula symbol="=">
        <function symbol="∪">
         <function symbol="∩">
          <constant symbol="s_1"/>
          <constant symbol="s_2"/>
         </function>
         <constant symbol="s_1"/>
        </function>
        <constant symbol="s_1"/>
       </constantatomformula>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <function symbol="∪">
          <constant symbol="s_3"/>
          <constant symbol="s_4"/>
         </function>
         <constant symbol="s_3"/>
        </function>
        <constant symbol="s_3"/>
       </constantatomformula>
      </formulalist>
      <formulalist>
       <conjunctiveformula type="and">
        <constantatomformula symbol="=">
         <function symbol="∪">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
         <constant symbol="s_1"/>
        </constantatomformula>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
          <constant symbol="s_3"/>
         </function>
         <constant symbol="s_3"/>
        </constantatomformula>
       </conjunctiveformula>
      </formulalist>
     </sequent>
     <rule type="axiom" symbol="axiom">
      <sequent>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∪">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
         <constant symbol="s_1"/>
        </constantatomformula>
       </formulalist>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∪">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
         <constant symbol="s_1"/>
        </constantatomformula>
       </formulalist>
      </sequent>
     </rule>
     <rule type="axiom" symbol="axiom">
      <sequent>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
          <constant symbol="s_3"/>
         </function>
         <constant symbol="s_3"/>
        </constantatomformula>
       </formulalist>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
          <constant symbol="s_3"/>
         </function>
         <constant symbol="s_3"/>
        </constantatomformula>
       </formulalist>
      </sequent>
     </rule>
    </rule>
   </rule>
  </rule>
 </proof>
 <proof symbol="\psi_{2}" calculus="LK">
  <rule type="weakl" symbol="w:l">
   <sequent>
    <formulalist>
     <constantatomformula symbol="L1" definition="L1"/>
     <constantatomformula symbol="=">
      <function symbol="∪">
       <function symbol="∩">
        <constant symbol="s_1"/>
        <constant symbol="s_2"/>
       </function>
       <constant symbol="s_1"/>
      </function>
      <constant symbol="s_1"/>
     </constantatomformula>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <constant symbol="s_3"/>
       <function symbol="∩">
        <function symbol="∪">
         <constant symbol="s_3"/>
         <constant symbol="s_4"/>
        </function>
        <constant symbol="s_3"/>
       </function>
      </function>
      <constant symbol="s_3"/>
     </constantatomformula>
    </formulalist>
    <formulalist>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <constant symbol="s_3"/>
       <function symbol="∩">
        <function symbol="∪">
         <constant symbol="s_3"/>
         <constant symbol="s_4"/>
        </function>
        <constant symbol="s_3"/>
       </function>
      </function>
      <constant symbol="s_3"/>
     </constantatomformula>
     <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
    </formulalist>
   </sequent>
   <rule type="defr" symbol="d:r" param="();[()];()">
    <sequent>
     <formulalist>
      <constantatomformula symbol="=">
       <function symbol="∪">
        <function symbol="∩">
         <constant symbol="s_1"/>
         <constant symbol="s_2"/>
        </function>
        <constant symbol="s_1"/>
       </function>
       <constant symbol="s_1"/>
      </constantatomformula>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <constant symbol="s_3"/>
        <function symbol="∩">
         <function symbol="∪">
          <constant symbol="s_3"/>
          <constant symbol="s_4"/>
         </function>
         <constant symbol="s_3"/>
        </function>
       </function>
       <constant symbol="s_3"/>
      </constantatomformula>
     </formulalist>
     <formulalist>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <constant symbol="s_3"/>
        <function symbol="∩">
         <function symbol="∪">
          <constant symbol="s_3"/>
          <constant symbol="s_4"/>
         </function>
         <constant symbol="s_3"/>
        </function>
       </function>
       <constant symbol="s_3"/>
      </constantatomformula>
      <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
     </formulalist>
    </sequent>
    <rule type="andr" symbol="\and:r">
     <sequent>
      <formulalist>
       <constantatomformula symbol="=">
        <function symbol="∪">
         <function symbol="∩">
          <constant symbol="s_1"/>
          <constant symbol="s_2"/>
         </function>
         <constant symbol="s_1"/>
        </function>
        <constant symbol="s_1"/>
       </constantatomformula>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <constant symbol="s_3"/>
         <function symbol="∩">
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
          <constant symbol="s_3"/>
         </function>
        </function>
        <constant symbol="s_3"/>
       </constantatomformula>
      </formulalist>
      <formulalist>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <constant symbol="s_3"/>
         <function symbol="∩">
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
          <constant symbol="s_3"/>
         </function>
        </function>
        <constant symbol="s_3"/>
       </constantatomformula>
       <conjunctiveformula type="and">
        <constantatomformula symbol="=">
         <function symbol="∪">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
         <constant symbol="s_1"/>
        </constantatomformula>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
          <constant symbol="s_3"/>
         </function>
         <constant symbol="s_3"/>
        </constantatomformula>
       </conjunctiveformula>
      </formulalist>
     </sequent>
     <rule type="axiom" symbol="axiom">
      <sequent>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∪">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
         <constant symbol="s_1"/>
        </constantatomformula>
       </formulalist>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∪">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
         <constant symbol="s_1"/>
        </constantatomformula>
       </formulalist>
      </sequent>
     </rule>
     <rule type="weakr" symbol="w:r">
      <sequent>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <constant symbol="s_3"/>
          <function symbol="∩">
           <function symbol="∪">
            <constant symbol="s_3"/>
            <constant symbol="s_4"/>
           </function>
           <constant symbol="s_3"/>
          </function>
         </function>
         <constant symbol="s_3"/>
        </constantatomformula>
       </formulalist>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <constant symbol="s_3"/>
          <function symbol="∩">
           <function symbol="∪">
            <constant symbol="s_3"/>
            <constant symbol="s_4"/>
           </function>
           <constant symbol="s_3"/>
          </function>
         </function>
         <constant symbol="s_3"/>
        </constantatomformula>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
          <constant symbol="s_3"/>
         </function>
         <constant symbol="s_3"/>
        </constantatomformula>
       </formulalist>
      </sequent>
      <rule type="axiom" symbol="axiom">
       <sequent>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∩">
           <constant symbol="s_3"/>
           <function symbol="∩">
            <function symbol="∪">
             <constant symbol="s_3"/>
             <constant symbol="s_4"/>
            </function>
            <constant symbol="s_3"/>
           </function>
          </function>
          <constant symbol="s_3"/>
         </constantatomformula>
        </formulalist>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∩">
           <constant symbol="s_3"/>
           <function symbol="∩">
            <function symbol="∪">
             <constant symbol="s_3"/>
             <constant symbol="s_4"/>
            </function>
            <constant symbol="s_3"/>
           </function>
          </function>
          <constant symbol="s_3"/>
         </constantatomformula>
        </formulalist>
       </sequent>
      </rule>
     </rule>
    </rule>
   </rule>
  </rule>
 </proof>
 <proof symbol="\psi_{3}" calculus="LK">
  <rule type="weakl" symbol="w:l">
   <sequent>
    <formulalist>
     <constantatomformula symbol="L1" definition="L1"/>
     <constantatomformula symbol="=">
      <function symbol="∪">
       <function symbol="∩">
        <constant symbol="s_1"/>
        <constant symbol="s_2"/>
       </function>
       <constant symbol="s_1"/>
      </function>
      <constant symbol="s_1"/>
     </constantatomformula>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <constant symbol="s_3"/>
       <constant symbol="s_3"/>
      </function>
      <constant symbol="s_3"/>
     </constantatomformula>
    </formulalist>
    <formulalist>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <constant symbol="s_3"/>
       <constant symbol="s_3"/>
      </function>
      <constant symbol="s_3"/>
     </constantatomformula>
     <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
    </formulalist>
   </sequent>
   <rule type="defr" symbol="d:r" param="();[()];()">
    <sequent>
     <formulalist>
      <constantatomformula symbol="=">
       <function symbol="∪">
        <function symbol="∩">
         <constant symbol="s_1"/>
         <constant symbol="s_2"/>
        </function>
        <constant symbol="s_1"/>
       </function>
       <constant symbol="s_1"/>
      </constantatomformula>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <constant symbol="s_3"/>
        <constant symbol="s_3"/>
       </function>
       <constant symbol="s_3"/>
      </constantatomformula>
     </formulalist>
     <formulalist>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <constant symbol="s_3"/>
        <constant symbol="s_3"/>
       </function>
       <constant symbol="s_3"/>
      </constantatomformula>
      <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
     </formulalist>
    </sequent>
    <rule type="andr" symbol="\and:r">
     <sequent>
      <formulalist>
       <constantatomformula symbol="=">
        <function symbol="∪">
         <function symbol="∩">
          <constant symbol="s_1"/>
          <constant symbol="s_2"/>
         </function>
         <constant symbol="s_1"/>
        </function>
        <constant symbol="s_1"/>
       </constantatomformula>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <constant symbol="s_3"/>
         <constant symbol="s_3"/>
        </function>
        <constant symbol="s_3"/>
       </constantatomformula>
      </formulalist>
      <formulalist>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <constant symbol="s_3"/>
         <constant symbol="s_3"/>
        </function>
        <constant symbol="s_3"/>
       </constantatomformula>
       <conjunctiveformula type="and">
        <constantatomformula symbol="=">
         <function symbol="∪">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
         <constant symbol="s_1"/>
        </constantatomformula>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
          <constant symbol="s_3"/>
         </function>
         <constant symbol="s_3"/>
        </constantatomformula>
       </conjunctiveformula>
      </formulalist>
     </sequent>
     <rule type="axiom" symbol="axiom">
      <sequent>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∪">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
         <constant symbol="s_1"/>
        </constantatomformula>
       </formulalist>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∪">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
         <constant symbol="s_1"/>
        </constantatomformula>
       </formulalist>
      </sequent>
     </rule>
     <rule type="weakr" symbol="w:r">
      <sequent>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <constant symbol="s_3"/>
          <constant symbol="s_3"/>
         </function>
         <constant symbol="s_3"/>
        </constantatomformula>
       </formulalist>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <constant symbol="s_3"/>
          <constant symbol="s_3"/>
         </function>
         <constant symbol="s_3"/>
        </constantatomformula>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
          <constant symbol="s_3"/>
         </function>
         <constant symbol="s_3"/>
        </constantatomformula>
       </formulalist>
      </sequent>
      <rule type="axiom" symbol="axiom">
       <sequent>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∩">
           <constant symbol="s_3"/>
           <constant symbol="s_3"/>
          </function>
          <constant symbol="s_3"/>
         </constantatomformula>
        </formulalist>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∩">
           <constant symbol="s_3"/>
           <constant symbol="s_3"/>
          </function>
          <constant symbol="s_3"/>
         </constantatomformula>
        </formulalist>
       </sequent>
      </rule>
     </rule>
    </rule>
   </rule>
  </rule>
 </proof>
 <proof symbol="\psi_{4}" calculus="LK">
  <rule type="weakl" symbol="w:l">
   <sequent>
    <formulalist>
     <constantatomformula symbol="L1" definition="L1"/>
     <constantatomformula symbol="=">
      <function symbol="∪">
       <function symbol="∩">
        <constant symbol="s_1"/>
        <constant symbol="s_2"/>
       </function>
       <constant symbol="s_1"/>
      </function>
      <constant symbol="s_1"/>
     </constantatomformula>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <constant symbol="s_3"/>
       <function symbol="∪">
        <constant symbol="s_3"/>
        <constant symbol="s_4"/>
       </function>
      </function>
      <constant symbol="s_3"/>
     </constantatomformula>
    </formulalist>
    <formulalist>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <constant symbol="s_3"/>
       <function symbol="∪">
        <constant symbol="s_3"/>
        <constant symbol="s_4"/>
       </function>
      </function>
      <constant symbol="s_3"/>
     </constantatomformula>
     <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
    </formulalist>
   </sequent>
   <rule type="defr" symbol="d:r" param="();[()];()">
    <sequent>
     <formulalist>
      <constantatomformula symbol="=">
       <function symbol="∪">
        <function symbol="∩">
         <constant symbol="s_1"/>
         <constant symbol="s_2"/>
        </function>
        <constant symbol="s_1"/>
       </function>
       <constant symbol="s_1"/>
      </constantatomformula>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <constant symbol="s_3"/>
        <function symbol="∪">
         <constant symbol="s_3"/>
         <constant symbol="s_4"/>
        </function>
       </function>
       <constant symbol="s_3"/>
      </constantatomformula>
     </formulalist>
     <formulalist>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <constant symbol="s_3"/>
        <function symbol="∪">
         <constant symbol="s_3"/>
         <constant symbol="s_4"/>
        </function>
       </function>
       <constant symbol="s_3"/>
      </constantatomformula>
      <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
     </formulalist>
    </sequent>
    <rule type="andr" symbol="\and:r">
     <sequent>
      <formulalist>
       <constantatomformula symbol="=">
        <function symbol="∪">
         <function symbol="∩">
          <constant symbol="s_1"/>
          <constant symbol="s_2"/>
         </function>
         <constant symbol="s_1"/>
        </function>
        <constant symbol="s_1"/>
       </constantatomformula>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <constant symbol="s_3"/>
         <function symbol="∪">
          <constant symbol="s_3"/>
          <constant symbol="s_4"/>
         </function>
        </function>
        <constant symbol="s_3"/>
       </constantatomformula>
      </formulalist>
      <formulalist>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <constant symbol="s_3"/>
         <function symbol="∪">
          <constant symbol="s_3"/>
          <constant symbol="s_4"/>
         </function>
        </function>
        <constant symbol="s_3"/>
       </constantatomformula>
       <conjunctiveformula type="and">
        <constantatomformula symbol="=">
         <function symbol="∪">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
         <constant symbol="s_1"/>
        </constantatomformula>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
          <constant symbol="s_3"/>
         </function>
         <constant symbol="s_3"/>
        </constantatomformula>
       </conjunctiveformula>
      </formulalist>
     </sequent>
     <rule type="axiom" symbol="axiom">
      <sequent>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∪">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
         <constant symbol="s_1"/>
        </constantatomformula>
       </formulalist>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∪">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
         <constant symbol="s_1"/>
        </constantatomformula>
       </formulalist>
      </sequent>
     </rule>
     <rule type="weakr" symbol="w:r">
      <sequent>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <constant symbol="s_3"/>
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
         </function>
         <constant symbol="s_3"/>
        </constantatomformula>
       </formulalist>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <constant symbol="s_3"/>
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
         </function>
         <constant symbol="s_3"/>
        </constantatomformula>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
          <constant symbol="s_3"/>
         </function>
         <constant symbol="s_3"/>
        </constantatomformula>
       </formulalist>
      </sequent>
      <rule type="axiom" symbol="axiom">
       <sequent>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∩">
           <constant symbol="s_3"/>
           <function symbol="∪">
            <constant symbol="s_3"/>
            <constant symbol="s_4"/>
           </function>
          </function>
          <constant symbol="s_3"/>
         </constantatomformula>
        </formulalist>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∩">
           <constant symbol="s_3"/>
           <function symbol="∪">
            <constant symbol="s_3"/>
            <constant symbol="s_4"/>
           </function>
          </function>
          <constant symbol="s_3"/>
         </constantatomformula>
        </formulalist>
       </sequent>
      </rule>
     </rule>
    </rule>
   </rule>
  </rule>
 </proof>
 <proof symbol="\psi_{5}" calculus="LK">
  <rule type="weakl" symbol="w:l">
   <sequent>
    <formulalist>
     <constantatomformula symbol="L1" definition="L1"/>
     <constantatomformula symbol="=">
      <function symbol="∪">
       <function symbol="∩">
        <constant symbol="s_1"/>
        <constant symbol="s_2"/>
       </function>
       <constant symbol="s_1"/>
      </function>
      <constant symbol="s_1"/>
     </constantatomformula>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <function symbol="∩">
        <function symbol="∪">
         <constant symbol="s_3"/>
         <constant symbol="s_4"/>
        </function>
        <constant symbol="s_3"/>
       </function>
       <constant symbol="s_3"/>
      </function>
      <function symbol="∩">
       <function symbol="∪">
        <constant symbol="s_3"/>
        <constant symbol="s_4"/>
       </function>
       <constant symbol="s_3"/>
      </function>
     </constantatomformula>
    </formulalist>
    <formulalist>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <function symbol="∩">
        <function symbol="∪">
         <constant symbol="s_3"/>
         <constant symbol="s_4"/>
        </function>
        <constant symbol="s_3"/>
       </function>
       <constant symbol="s_3"/>
      </function>
      <function symbol="∩">
       <function symbol="∪">
        <constant symbol="s_3"/>
        <constant symbol="s_4"/>
       </function>
       <constant symbol="s_3"/>
      </function>
     </constantatomformula>
     <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
    </formulalist>
   </sequent>
   <rule type="defr" symbol="d:r" param="();[()];()">
    <sequent>
     <formulalist>
      <constantatomformula symbol="=">
       <function symbol="∪">
        <function symbol="∩">
         <constant symbol="s_1"/>
         <constant symbol="s_2"/>
        </function>
        <constant symbol="s_1"/>
       </function>
       <constant symbol="s_1"/>
      </constantatomformula>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <function symbol="∩">
         <function symbol="∪">
          <constant symbol="s_3"/>
          <constant symbol="s_4"/>
         </function>
         <constant symbol="s_3"/>
        </function>
        <constant symbol="s_3"/>
       </function>
       <function symbol="∩">
        <function symbol="∪">
         <constant symbol="s_3"/>
         <constant symbol="s_4"/>
        </function>
        <constant symbol="s_3"/>
       </function>
      </constantatomformula>
     </formulalist>
     <formulalist>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <function symbol="∩">
         <function symbol="∪">
          <constant symbol="s_3"/>
          <constant symbol="s_4"/>
         </function>
         <constant symbol="s_3"/>
        </function>
        <constant symbol="s_3"/>
       </function>
       <function symbol="∩">
        <function symbol="∪">
         <constant symbol="s_3"/>
         <constant symbol="s_4"/>
        </function>
        <constant symbol="s_3"/>
       </function>
      </constantatomformula>
      <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
     </formulalist>
    </sequent>
    <rule type="andr" symbol="\and:r">
     <sequent>
      <formulalist>
       <constantatomformula symbol="=">
        <function symbol="∪">
         <function symbol="∩">
          <constant symbol="s_1"/>
          <constant symbol="s_2"/>
         </function>
         <constant symbol="s_1"/>
        </function>
        <constant symbol="s_1"/>
       </constantatomformula>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <function symbol="∩">
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
          <constant symbol="s_3"/>
         </function>
         <constant symbol="s_3"/>
        </function>
        <function symbol="∩">
         <function symbol="∪">
          <constant symbol="s_3"/>
          <constant symbol="s_4"/>
         </function>
         <constant symbol="s_3"/>
        </function>
       </constantatomformula>
      </formulalist>
      <formulalist>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <function symbol="∩">
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
          <constant symbol="s_3"/>
         </function>
         <constant symbol="s_3"/>
        </function>
        <function symbol="∩">
         <function symbol="∪">
          <constant symbol="s_3"/>
          <constant symbol="s_4"/>
         </function>
         <constant symbol="s_3"/>
        </function>
       </constantatomformula>
       <conjunctiveformula type="and">
        <constantatomformula symbol="=">
         <function symbol="∪">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
         <constant symbol="s_1"/>
        </constantatomformula>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
          <constant symbol="s_3"/>
         </function>
         <constant symbol="s_3"/>
        </constantatomformula>
       </conjunctiveformula>
      </formulalist>
     </sequent>
     <rule type="axiom" symbol="axiom">
      <sequent>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∪">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
         <constant symbol="s_1"/>
        </constantatomformula>
       </formulalist>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∪">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
         <constant symbol="s_1"/>
        </constantatomformula>
       </formulalist>
      </sequent>
     </rule>
     <rule type="weakr" symbol="w:r">
      <sequent>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <function symbol="∩">
           <function symbol="∪">
            <constant symbol="s_3"/>
            <constant symbol="s_4"/>
           </function>
           <constant symbol="s_3"/>
          </function>
          <constant symbol="s_3"/>
         </function>
         <function symbol="∩">
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
          <constant symbol="s_3"/>
         </function>
        </constantatomformula>
       </formulalist>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <function symbol="∩">
           <function symbol="∪">
            <constant symbol="s_3"/>
            <constant symbol="s_4"/>
           </function>
           <constant symbol="s_3"/>
          </function>
          <constant symbol="s_3"/>
         </function>
         <function symbol="∩">
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
          <constant symbol="s_3"/>
         </function>
        </constantatomformula>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
          <constant symbol="s_3"/>
         </function>
         <constant symbol="s_3"/>
        </constantatomformula>
       </formulalist>
      </sequent>
      <rule type="axiom" symbol="axiom">
       <sequent>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∩">
           <function symbol="∩">
            <function symbol="∪">
             <constant symbol="s_3"/>
             <constant symbol="s_4"/>
            </function>
            <constant symbol="s_3"/>
           </function>
           <constant symbol="s_3"/>
          </function>
          <function symbol="∩">
           <function symbol="∪">
            <constant symbol="s_3"/>
            <constant symbol="s_4"/>
           </function>
           <constant symbol="s_3"/>
          </function>
         </constantatomformula>
        </formulalist>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∩">
           <function symbol="∩">
            <function symbol="∪">
             <constant symbol="s_3"/>
             <constant symbol="s_4"/>
            </function>
            <constant symbol="s_3"/>
           </function>
           <constant symbol="s_3"/>
          </function>
          <function symbol="∩">
           <function symbol="∪">
            <constant symbol="s_3"/>
            <constant symbol="s_4"/>
           </function>
           <constant symbol="s_3"/>
          </function>
         </constantatomformula>
        </formulalist>
       </sequent>
      </rule>
     </rule>
    </rule>
   </rule>
  </rule>
 </proof>
 <proof symbol="\psi_{6}" calculus="LK">
  <rule type="weakl" symbol="w:l">
   <sequent>
    <formulalist>
     <constantatomformula symbol="L1" definition="L1"/>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <constant symbol="s_1"/>
       <function symbol="∪">
        <function symbol="∩">
         <constant symbol="s_1"/>
         <constant symbol="s_2"/>
        </function>
        <constant symbol="s_1"/>
       </function>
      </function>
      <constant symbol="s_1"/>
     </constantatomformula>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <function symbol="∪">
        <constant symbol="s_3"/>
        <constant symbol="s_4"/>
       </function>
       <constant symbol="s_3"/>
      </function>
      <constant symbol="s_3"/>
     </constantatomformula>
    </formulalist>
    <formulalist>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <constant symbol="s_1"/>
       <function symbol="∪">
        <function symbol="∩">
         <constant symbol="s_1"/>
         <constant symbol="s_2"/>
        </function>
        <constant symbol="s_1"/>
       </function>
      </function>
      <constant symbol="s_1"/>
     </constantatomformula>
     <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
    </formulalist>
   </sequent>
   <rule type="defr" symbol="d:r" param="();[()];()">
    <sequent>
     <formulalist>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <constant symbol="s_1"/>
        <function symbol="∪">
         <function symbol="∩">
          <constant symbol="s_1"/>
          <constant symbol="s_2"/>
         </function>
         <constant symbol="s_1"/>
        </function>
       </function>
       <constant symbol="s_1"/>
      </constantatomformula>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <function symbol="∪">
         <constant symbol="s_3"/>
         <constant symbol="s_4"/>
        </function>
        <constant symbol="s_3"/>
       </function>
       <constant symbol="s_3"/>
      </constantatomformula>
     </formulalist>
     <formulalist>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <constant symbol="s_1"/>
        <function symbol="∪">
         <function symbol="∩">
          <constant symbol="s_1"/>
          <constant symbol="s_2"/>
         </function>
         <constant symbol="s_1"/>
        </function>
       </function>
       <constant symbol="s_1"/>
      </constantatomformula>
      <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
     </formulalist>
    </sequent>
    <rule type="andr" symbol="\and:r">
     <sequent>
      <formulalist>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <constant symbol="s_1"/>
         <function symbol="∪">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
        </function>
        <constant symbol="s_1"/>
       </constantatomformula>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <function symbol="∪">
          <constant symbol="s_3"/>
          <constant symbol="s_4"/>
         </function>
         <constant symbol="s_3"/>
        </function>
        <constant symbol="s_3"/>
       </constantatomformula>
      </formulalist>
      <formulalist>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <constant symbol="s_1"/>
         <function symbol="∪">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
        </function>
        <constant symbol="s_1"/>
       </constantatomformula>
       <conjunctiveformula type="and">
        <constantatomformula symbol="=">
         <function symbol="∪">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
         <constant symbol="s_1"/>
        </constantatomformula>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
          <constant symbol="s_3"/>
         </function>
         <constant symbol="s_3"/>
        </constantatomformula>
       </conjunctiveformula>
      </formulalist>
     </sequent>
     <rule type="weakr" symbol="w:r">
      <sequent>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <constant symbol="s_1"/>
          <function symbol="∪">
           <function symbol="∩">
            <constant symbol="s_1"/>
            <constant symbol="s_2"/>
           </function>
           <constant symbol="s_1"/>
          </function>
         </function>
         <constant symbol="s_1"/>
        </constantatomformula>
       </formulalist>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <constant symbol="s_1"/>
          <function symbol="∪">
           <function symbol="∩">
            <constant symbol="s_1"/>
            <constant symbol="s_2"/>
           </function>
           <constant symbol="s_1"/>
          </function>
         </function>
         <constant symbol="s_1"/>
        </constantatomformula>
        <constantatomformula symbol="=">
         <function symbol="∪">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
         <constant symbol="s_1"/>
        </constantatomformula>
       </formulalist>
      </sequent>
      <rule type="axiom" symbol="axiom">
       <sequent>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <function symbol="∪">
            <function symbol="∩">
             <constant symbol="s_1"/>
             <constant symbol="s_2"/>
            </function>
            <constant symbol="s_1"/>
           </function>
          </function>
          <constant symbol="s_1"/>
         </constantatomformula>
        </formulalist>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <function symbol="∪">
            <function symbol="∩">
             <constant symbol="s_1"/>
             <constant symbol="s_2"/>
            </function>
            <constant symbol="s_1"/>
           </function>
          </function>
          <constant symbol="s_1"/>
         </constantatomformula>
        </formulalist>
       </sequent>
      </rule>
     </rule>
     <rule type="axiom" symbol="axiom">
      <sequent>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
          <constant symbol="s_3"/>
         </function>
         <constant symbol="s_3"/>
        </constantatomformula>
       </formulalist>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
          <constant symbol="s_3"/>
         </function>
         <constant symbol="s_3"/>
        </constantatomformula>
       </formulalist>
      </sequent>
     </rule>
    </rule>
   </rule>
  </rule>
 </proof>
 <proof symbol="\psi_{7}" calculus="LK">
  <rule type="weakl" symbol="w:l">
   <sequent>
    <formulalist>
     <constantatomformula symbol="L1" definition="L1"/>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <constant symbol="s_1"/>
       <function symbol="∪">
        <function symbol="∩">
         <constant symbol="s_1"/>
         <constant symbol="s_2"/>
        </function>
        <constant symbol="s_1"/>
       </function>
      </function>
      <constant symbol="s_1"/>
     </constantatomformula>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <constant symbol="s_3"/>
       <function symbol="∩">
        <function symbol="∪">
         <constant symbol="s_3"/>
         <constant symbol="s_4"/>
        </function>
        <constant symbol="s_3"/>
       </function>
      </function>
      <constant symbol="s_3"/>
     </constantatomformula>
    </formulalist>
    <formulalist>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <constant symbol="s_1"/>
       <function symbol="∪">
        <function symbol="∩">
         <constant symbol="s_1"/>
         <constant symbol="s_2"/>
        </function>
        <constant symbol="s_1"/>
       </function>
      </function>
      <constant symbol="s_1"/>
     </constantatomformula>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <constant symbol="s_3"/>
       <function symbol="∩">
        <function symbol="∪">
         <constant symbol="s_3"/>
         <constant symbol="s_4"/>
        </function>
        <constant symbol="s_3"/>
       </function>
      </function>
      <constant symbol="s_3"/>
     </constantatomformula>
     <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
    </formulalist>
   </sequent>
   <rule type="defr" symbol="d:r" param="();[()];()">
    <sequent>
     <formulalist>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <constant symbol="s_1"/>
        <function symbol="∪">
         <function symbol="∩">
          <constant symbol="s_1"/>
          <constant symbol="s_2"/>
         </function>
         <constant symbol="s_1"/>
        </function>
       </function>
       <constant symbol="s_1"/>
      </constantatomformula>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <constant symbol="s_3"/>
        <function symbol="∩">
         <function symbol="∪">
          <constant symbol="s_3"/>
          <constant symbol="s_4"/>
         </function>
         <constant symbol="s_3"/>
        </function>
       </function>
       <constant symbol="s_3"/>
      </constantatomformula>
     </formulalist>
     <formulalist>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <constant symbol="s_1"/>
        <function symbol="∪">
         <function symbol="∩">
          <constant symbol="s_1"/>
          <constant symbol="s_2"/>
         </function>
         <constant symbol="s_1"/>
        </function>
       </function>
       <constant symbol="s_1"/>
      </constantatomformula>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <constant symbol="s_3"/>
        <function symbol="∩">
         <function symbol="∪">
          <constant symbol="s_3"/>
          <constant symbol="s_4"/>
         </function>
         <constant symbol="s_3"/>
        </function>
       </function>
       <constant symbol="s_3"/>
      </constantatomformula>
      <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
     </formulalist>
    </sequent>
    <rule type="andr" symbol="\and:r">
     <sequent>
      <formulalist>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <constant symbol="s_1"/>
         <function symbol="∪">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
        </function>
        <constant symbol="s_1"/>
       </constantatomformula>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <constant symbol="s_3"/>
         <function symbol="∩">
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
          <constant symbol="s_3"/>
         </function>
        </function>
        <constant symbol="s_3"/>
       </constantatomformula>
      </formulalist>
      <formulalist>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <constant symbol="s_1"/>
         <function symbol="∪">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
        </function>
        <constant symbol="s_1"/>
       </constantatomformula>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <constant symbol="s_3"/>
         <function symbol="∩">
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
          <constant symbol="s_3"/>
         </function>
        </function>
        <constant symbol="s_3"/>
       </constantatomformula>
       <conjunctiveformula type="and">
        <constantatomformula symbol="=">
         <function symbol="∪">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
         <constant symbol="s_1"/>
        </constantatomformula>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
          <constant symbol="s_3"/>
         </function>
         <constant symbol="s_3"/>
        </constantatomformula>
       </conjunctiveformula>
      </formulalist>
     </sequent>
     <rule type="weakr" symbol="w:r">
      <sequent>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <constant symbol="s_1"/>
          <function symbol="∪">
           <function symbol="∩">
            <constant symbol="s_1"/>
            <constant symbol="s_2"/>
           </function>
           <constant symbol="s_1"/>
          </function>
         </function>
         <constant symbol="s_1"/>
        </constantatomformula>
       </formulalist>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <constant symbol="s_1"/>
          <function symbol="∪">
           <function symbol="∩">
            <constant symbol="s_1"/>
            <constant symbol="s_2"/>
           </function>
           <constant symbol="s_1"/>
          </function>
         </function>
         <constant symbol="s_1"/>
        </constantatomformula>
        <constantatomformula symbol="=">
         <function symbol="∪">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
         <constant symbol="s_1"/>
        </constantatomformula>
       </formulalist>
      </sequent>
      <rule type="axiom" symbol="axiom">
       <sequent>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <function symbol="∪">
            <function symbol="∩">
             <constant symbol="s_1"/>
             <constant symbol="s_2"/>
            </function>
            <constant symbol="s_1"/>
           </function>
          </function>
          <constant symbol="s_1"/>
         </constantatomformula>
        </formulalist>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <function symbol="∪">
            <function symbol="∩">
             <constant symbol="s_1"/>
             <constant symbol="s_2"/>
            </function>
            <constant symbol="s_1"/>
           </function>
          </function>
          <constant symbol="s_1"/>
         </constantatomformula>
        </formulalist>
       </sequent>
      </rule>
     </rule>
     <rule type="weakr" symbol="w:r">
      <sequent>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <constant symbol="s_3"/>
          <function symbol="∩">
           <function symbol="∪">
            <constant symbol="s_3"/>
            <constant symbol="s_4"/>
           </function>
           <constant symbol="s_3"/>
          </function>
         </function>
         <constant symbol="s_3"/>
        </constantatomformula>
       </formulalist>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <constant symbol="s_3"/>
          <function symbol="∩">
           <function symbol="∪">
            <constant symbol="s_3"/>
            <constant symbol="s_4"/>
           </function>
           <constant symbol="s_3"/>
          </function>
         </function>
         <constant symbol="s_3"/>
        </constantatomformula>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
          <constant symbol="s_3"/>
         </function>
         <constant symbol="s_3"/>
        </constantatomformula>
       </formulalist>
      </sequent>
      <rule type="axiom" symbol="axiom">
       <sequent>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∩">
           <constant symbol="s_3"/>
           <function symbol="∩">
            <function symbol="∪">
             <constant symbol="s_3"/>
             <constant symbol="s_4"/>
            </function>
            <constant symbol="s_3"/>
           </function>
          </function>
          <constant symbol="s_3"/>
         </constantatomformula>
        </formulalist>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∩">
           <constant symbol="s_3"/>
           <function symbol="∩">
            <function symbol="∪">
             <constant symbol="s_3"/>
             <constant symbol="s_4"/>
            </function>
            <constant symbol="s_3"/>
           </function>
          </function>
          <constant symbol="s_3"/>
         </constantatomformula>
        </formulalist>
       </sequent>
      </rule>
     </rule>
    </rule>
   </rule>
  </rule>
 </proof>
 <proof symbol="\psi_{8}" calculus="LK">
  <rule type="weakl" symbol="w:l">
   <sequent>
    <formulalist>
     <constantatomformula symbol="L1" definition="L1"/>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <constant symbol="s_1"/>
       <function symbol="∪">
        <function symbol="∩">
         <constant symbol="s_1"/>
         <constant symbol="s_2"/>
        </function>
        <constant symbol="s_1"/>
       </function>
      </function>
      <constant symbol="s_1"/>
     </constantatomformula>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <constant symbol="s_3"/>
       <constant symbol="s_3"/>
      </function>
      <constant symbol="s_3"/>
     </constantatomformula>
    </formulalist>
    <formulalist>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <constant symbol="s_1"/>
       <function symbol="∪">
        <function symbol="∩">
         <constant symbol="s_1"/>
         <constant symbol="s_2"/>
        </function>
        <constant symbol="s_1"/>
       </function>
      </function>
      <constant symbol="s_1"/>
     </constantatomformula>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <constant symbol="s_3"/>
       <constant symbol="s_3"/>
      </function>
      <constant symbol="s_3"/>
     </constantatomformula>
     <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
    </formulalist>
   </sequent>
   <rule type="defr" symbol="d:r" param="();[()];()">
    <sequent>
     <formulalist>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <constant symbol="s_1"/>
        <function symbol="∪">
         <function symbol="∩">
          <constant symbol="s_1"/>
          <constant symbol="s_2"/>
         </function>
         <constant symbol="s_1"/>
        </function>
       </function>
       <constant symbol="s_1"/>
      </constantatomformula>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <constant symbol="s_3"/>
        <constant symbol="s_3"/>
       </function>
       <constant symbol="s_3"/>
      </constantatomformula>
     </formulalist>
     <formulalist>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <constant symbol="s_1"/>
        <function symbol="∪">
         <function symbol="∩">
          <constant symbol="s_1"/>
          <constant symbol="s_2"/>
         </function>
         <constant symbol="s_1"/>
        </function>
       </function>
       <constant symbol="s_1"/>
      </constantatomformula>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <constant symbol="s_3"/>
        <constant symbol="s_3"/>
       </function>
       <constant symbol="s_3"/>
      </constantatomformula>
      <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
     </formulalist>
    </sequent>
    <rule type="andr" symbol="\and:r">
     <sequent>
      <formulalist>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <constant symbol="s_1"/>
         <function symbol="∪">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
        </function>
        <constant symbol="s_1"/>
       </constantatomformula>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <constant symbol="s_3"/>
         <constant symbol="s_3"/>
        </function>
        <constant symbol="s_3"/>
       </constantatomformula>
      </formulalist>
      <formulalist>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <constant symbol="s_1"/>
         <function symbol="∪">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
        </function>
        <constant symbol="s_1"/>
       </constantatomformula>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <constant symbol="s_3"/>
         <constant symbol="s_3"/>
        </function>
        <constant symbol="s_3"/>
       </constantatomformula>
       <conjunctiveformula type="and">
        <constantatomformula symbol="=">
         <function symbol="∪">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
         <constant symbol="s_1"/>
        </constantatomformula>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
          <constant symbol="s_3"/>
         </function>
         <constant symbol="s_3"/>
        </constantatomformula>
       </conjunctiveformula>
      </formulalist>
     </sequent>
     <rule type="weakr" symbol="w:r">
      <sequent>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <constant symbol="s_1"/>
          <function symbol="∪">
           <function symbol="∩">
            <constant symbol="s_1"/>
            <constant symbol="s_2"/>
           </function>
           <constant symbol="s_1"/>
          </function>
         </function>
         <constant symbol="s_1"/>
        </constantatomformula>
       </formulalist>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <constant symbol="s_1"/>
          <function symbol="∪">
           <function symbol="∩">
            <constant symbol="s_1"/>
            <constant symbol="s_2"/>
           </function>
           <constant symbol="s_1"/>
          </function>
         </function>
         <constant symbol="s_1"/>
        </constantatomformula>
        <constantatomformula symbol="=">
         <function symbol="∪">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
         <constant symbol="s_1"/>
        </constantatomformula>
       </formulalist>
      </sequent>
      <rule type="axiom" symbol="axiom">
       <sequent>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <function symbol="∪">
            <function symbol="∩">
             <constant symbol="s_1"/>
             <constant symbol="s_2"/>
            </function>
            <constant symbol="s_1"/>
           </function>
          </function>
          <constant symbol="s_1"/>
         </constantatomformula>
        </formulalist>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <function symbol="∪">
            <function symbol="∩">
             <constant symbol="s_1"/>
             <constant symbol="s_2"/>
            </function>
            <constant symbol="s_1"/>
           </function>
          </function>
          <constant symbol="s_1"/>
         </constantatomformula>
        </formulalist>
       </sequent>
      </rule>
     </rule>
     <rule type="weakr" symbol="w:r">
      <sequent>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <constant symbol="s_3"/>
          <constant symbol="s_3"/>
         </function>
         <constant symbol="s_3"/>
        </constantatomformula>
       </formulalist>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <constant symbol="s_3"/>
          <constant symbol="s_3"/>
         </function>
         <constant symbol="s_3"/>
        </constantatomformula>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
          <constant symbol="s_3"/>
         </function>
         <constant symbol="s_3"/>
        </constantatomformula>
       </formulalist>
      </sequent>
      <rule type="axiom" symbol="axiom">
       <sequent>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∩">
           <constant symbol="s_3"/>
           <constant symbol="s_3"/>
          </function>
          <constant symbol="s_3"/>
         </constantatomformula>
        </formulalist>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∩">
           <constant symbol="s_3"/>
           <constant symbol="s_3"/>
          </function>
          <constant symbol="s_3"/>
         </constantatomformula>
        </formulalist>
       </sequent>
      </rule>
     </rule>
    </rule>
   </rule>
  </rule>
 </proof>
 <proof symbol="\psi_{9}" calculus="LK">
  <rule type="weakl" symbol="w:l">
   <sequent>
    <formulalist>
     <constantatomformula symbol="L1" definition="L1"/>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <constant symbol="s_1"/>
       <function symbol="∪">
        <function symbol="∩">
         <constant symbol="s_1"/>
         <constant symbol="s_2"/>
        </function>
        <constant symbol="s_1"/>
       </function>
      </function>
      <constant symbol="s_1"/>
     </constantatomformula>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <constant symbol="s_3"/>
       <function symbol="∪">
        <constant symbol="s_3"/>
        <constant symbol="s_4"/>
       </function>
      </function>
      <constant symbol="s_3"/>
     </constantatomformula>
    </formulalist>
    <formulalist>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <constant symbol="s_1"/>
       <function symbol="∪">
        <function symbol="∩">
         <constant symbol="s_1"/>
         <constant symbol="s_2"/>
        </function>
        <constant symbol="s_1"/>
       </function>
      </function>
      <constant symbol="s_1"/>
     </constantatomformula>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <constant symbol="s_3"/>
       <function symbol="∪">
        <constant symbol="s_3"/>
        <constant symbol="s_4"/>
       </function>
      </function>
      <constant symbol="s_3"/>
     </constantatomformula>
     <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
    </formulalist>
   </sequent>
   <rule type="defr" symbol="d:r" param="();[()];()">
    <sequent>
     <formulalist>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <constant symbol="s_1"/>
        <function symbol="∪">
         <function symbol="∩">
          <constant symbol="s_1"/>
          <constant symbol="s_2"/>
         </function>
         <constant symbol="s_1"/>
        </function>
       </function>
       <constant symbol="s_1"/>
      </constantatomformula>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <constant symbol="s_3"/>
        <function symbol="∪">
         <constant symbol="s_3"/>
         <constant symbol="s_4"/>
        </function>
       </function>
       <constant symbol="s_3"/>
      </constantatomformula>
     </formulalist>
     <formulalist>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <constant symbol="s_1"/>
        <function symbol="∪">
         <function symbol="∩">
          <constant symbol="s_1"/>
          <constant symbol="s_2"/>
         </function>
         <constant symbol="s_1"/>
        </function>
       </function>
       <constant symbol="s_1"/>
      </constantatomformula>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <constant symbol="s_3"/>
        <function symbol="∪">
         <constant symbol="s_3"/>
         <constant symbol="s_4"/>
        </function>
       </function>
       <constant symbol="s_3"/>
      </constantatomformula>
      <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
     </formulalist>
    </sequent>
    <rule type="andr" symbol="\and:r">
     <sequent>
      <formulalist>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <constant symbol="s_1"/>
         <function symbol="∪">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
        </function>
        <constant symbol="s_1"/>
       </constantatomformula>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <constant symbol="s_3"/>
         <function symbol="∪">
          <constant symbol="s_3"/>
          <constant symbol="s_4"/>
         </function>
        </function>
        <constant symbol="s_3"/>
       </constantatomformula>
      </formulalist>
      <formulalist>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <constant symbol="s_1"/>
         <function symbol="∪">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
        </function>
        <constant symbol="s_1"/>
       </constantatomformula>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <constant symbol="s_3"/>
         <function symbol="∪">
          <constant symbol="s_3"/>
          <constant symbol="s_4"/>
         </function>
        </function>
        <constant symbol="s_3"/>
       </constantatomformula>
       <conjunctiveformula type="and">
        <constantatomformula symbol="=">
         <function symbol="∪">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
         <constant symbol="s_1"/>
        </constantatomformula>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
          <constant symbol="s_3"/>
         </function>
         <constant symbol="s_3"/>
        </constantatomformula>
       </conjunctiveformula>
      </formulalist>
     </sequent>
     <rule type="weakr" symbol="w:r">
      <sequent>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <constant symbol="s_1"/>
          <function symbol="∪">
           <function symbol="∩">
            <constant symbol="s_1"/>
            <constant symbol="s_2"/>
           </function>
           <constant symbol="s_1"/>
          </function>
         </function>
         <constant symbol="s_1"/>
        </constantatomformula>
       </formulalist>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <constant symbol="s_1"/>
          <function symbol="∪">
           <function symbol="∩">
            <constant symbol="s_1"/>
            <constant symbol="s_2"/>
           </function>
           <constant symbol="s_1"/>
          </function>
         </function>
         <constant symbol="s_1"/>
        </constantatomformula>
        <constantatomformula symbol="=">
         <function symbol="∪">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
         <constant symbol="s_1"/>
        </constantatomformula>
       </formulalist>
      </sequent>
      <rule type="axiom" symbol="axiom">
       <sequent>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <function symbol="∪">
            <function symbol="∩">
             <constant symbol="s_1"/>
             <constant symbol="s_2"/>
            </function>
            <constant symbol="s_1"/>
           </function>
          </function>
          <constant symbol="s_1"/>
         </constantatomformula>
        </formulalist>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <function symbol="∪">
            <function symbol="∩">
             <constant symbol="s_1"/>
             <constant symbol="s_2"/>
            </function>
            <constant symbol="s_1"/>
           </function>
          </function>
          <constant symbol="s_1"/>
         </constantatomformula>
        </formulalist>
       </sequent>
      </rule>
     </rule>
     <rule type="weakr" symbol="w:r">
      <sequent>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <constant symbol="s_3"/>
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
         </function>
         <constant symbol="s_3"/>
        </constantatomformula>
       </formulalist>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <constant symbol="s_3"/>
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
         </function>
         <constant symbol="s_3"/>
        </constantatomformula>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
          <constant symbol="s_3"/>
         </function>
         <constant symbol="s_3"/>
        </constantatomformula>
       </formulalist>
      </sequent>
      <rule type="axiom" symbol="axiom">
       <sequent>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∩">
           <constant symbol="s_3"/>
           <function symbol="∪">
            <constant symbol="s_3"/>
            <constant symbol="s_4"/>
           </function>
          </function>
          <constant symbol="s_3"/>
         </constantatomformula>
        </formulalist>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∩">
           <constant symbol="s_3"/>
           <function symbol="∪">
            <constant symbol="s_3"/>
            <constant symbol="s_4"/>
           </function>
          </function>
          <constant symbol="s_3"/>
         </constantatomformula>
        </formulalist>
       </sequent>
      </rule>
     </rule>
    </rule>
   </rule>
  </rule>
 </proof>
 <proof symbol="\psi_{10}" calculus="LK">
  <rule type="weakl" symbol="w:l">
   <sequent>
    <formulalist>
     <constantatomformula symbol="L1" definition="L1"/>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <constant symbol="s_1"/>
       <function symbol="∪">
        <function symbol="∩">
         <constant symbol="s_1"/>
         <constant symbol="s_2"/>
        </function>
        <constant symbol="s_1"/>
       </function>
      </function>
      <constant symbol="s_1"/>
     </constantatomformula>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <function symbol="∩">
        <function symbol="∪">
         <constant symbol="s_3"/>
         <constant symbol="s_4"/>
        </function>
        <constant symbol="s_3"/>
       </function>
       <constant symbol="s_3"/>
      </function>
      <function symbol="∩">
       <function symbol="∪">
        <constant symbol="s_3"/>
        <constant symbol="s_4"/>
       </function>
       <constant symbol="s_3"/>
      </function>
     </constantatomformula>
    </formulalist>
    <formulalist>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <constant symbol="s_1"/>
       <function symbol="∪">
        <function symbol="∩">
         <constant symbol="s_1"/>
         <constant symbol="s_2"/>
        </function>
        <constant symbol="s_1"/>
       </function>
      </function>
      <constant symbol="s_1"/>
     </constantatomformula>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <function symbol="∩">
        <function symbol="∪">
         <constant symbol="s_3"/>
         <constant symbol="s_4"/>
        </function>
        <constant symbol="s_3"/>
       </function>
       <constant symbol="s_3"/>
      </function>
      <function symbol="∩">
       <function symbol="∪">
        <constant symbol="s_3"/>
        <constant symbol="s_4"/>
       </function>
       <constant symbol="s_3"/>
      </function>
     </constantatomformula>
     <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
    </formulalist>
   </sequent>
   <rule type="defr" symbol="d:r" param="();[()];()">
    <sequent>
     <formulalist>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <constant symbol="s_1"/>
        <function symbol="∪">
         <function symbol="∩">
          <constant symbol="s_1"/>
          <constant symbol="s_2"/>
         </function>
         <constant symbol="s_1"/>
        </function>
       </function>
       <constant symbol="s_1"/>
      </constantatomformula>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <function symbol="∩">
         <function symbol="∪">
          <constant symbol="s_3"/>
          <constant symbol="s_4"/>
         </function>
         <constant symbol="s_3"/>
        </function>
        <constant symbol="s_3"/>
       </function>
       <function symbol="∩">
        <function symbol="∪">
         <constant symbol="s_3"/>
         <constant symbol="s_4"/>
        </function>
        <constant symbol="s_3"/>
       </function>
      </constantatomformula>
     </formulalist>
     <formulalist>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <constant symbol="s_1"/>
        <function symbol="∪">
         <function symbol="∩">
          <constant symbol="s_1"/>
          <constant symbol="s_2"/>
         </function>
         <constant symbol="s_1"/>
        </function>
       </function>
       <constant symbol="s_1"/>
      </constantatomformula>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <function symbol="∩">
         <function symbol="∪">
          <constant symbol="s_3"/>
          <constant symbol="s_4"/>
         </function>
         <constant symbol="s_3"/>
        </function>
        <constant symbol="s_3"/>
       </function>
       <function symbol="∩">
        <function symbol="∪">
         <constant symbol="s_3"/>
         <constant symbol="s_4"/>
        </function>
        <constant symbol="s_3"/>
       </function>
      </constantatomformula>
      <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
     </formulalist>
    </sequent>
    <rule type="andr" symbol="\and:r">
     <sequent>
      <formulalist>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <constant symbol="s_1"/>
         <function symbol="∪">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
        </function>
        <constant symbol="s_1"/>
       </constantatomformula>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <function symbol="∩">
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
          <constant symbol="s_3"/>
         </function>
         <constant symbol="s_3"/>
        </function>
        <function symbol="∩">
         <function symbol="∪">
          <constant symbol="s_3"/>
          <constant symbol="s_4"/>
         </function>
         <constant symbol="s_3"/>
        </function>
       </constantatomformula>
      </formulalist>
      <formulalist>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <constant symbol="s_1"/>
         <function symbol="∪">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
        </function>
        <constant symbol="s_1"/>
       </constantatomformula>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <function symbol="∩">
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
          <constant symbol="s_3"/>
         </function>
         <constant symbol="s_3"/>
        </function>
        <function symbol="∩">
         <function symbol="∪">
          <constant symbol="s_3"/>
          <constant symbol="s_4"/>
         </function>
         <constant symbol="s_3"/>
        </function>
       </constantatomformula>
       <conjunctiveformula type="and">
        <constantatomformula symbol="=">
         <function symbol="∪">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
         <constant symbol="s_1"/>
        </constantatomformula>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
          <constant symbol="s_3"/>
         </function>
         <constant symbol="s_3"/>
        </constantatomformula>
       </conjunctiveformula>
      </formulalist>
     </sequent>
     <rule type="weakr" symbol="w:r">
      <sequent>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <constant symbol="s_1"/>
          <function symbol="∪">
           <function symbol="∩">
            <constant symbol="s_1"/>
            <constant symbol="s_2"/>
           </function>
           <constant symbol="s_1"/>
          </function>
         </function>
         <constant symbol="s_1"/>
        </constantatomformula>
       </formulalist>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <constant symbol="s_1"/>
          <function symbol="∪">
           <function symbol="∩">
            <constant symbol="s_1"/>
            <constant symbol="s_2"/>
           </function>
           <constant symbol="s_1"/>
          </function>
         </function>
         <constant symbol="s_1"/>
        </constantatomformula>
        <constantatomformula symbol="=">
         <function symbol="∪">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
         <constant symbol="s_1"/>
        </constantatomformula>
       </formulalist>
      </sequent>
      <rule type="axiom" symbol="axiom">
       <sequent>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <function symbol="∪">
            <function symbol="∩">
             <constant symbol="s_1"/>
             <constant symbol="s_2"/>
            </function>
            <constant symbol="s_1"/>
           </function>
          </function>
          <constant symbol="s_1"/>
         </constantatomformula>
        </formulalist>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <function symbol="∪">
            <function symbol="∩">
             <constant symbol="s_1"/>
             <constant symbol="s_2"/>
            </function>
            <constant symbol="s_1"/>
           </function>
          </function>
          <constant symbol="s_1"/>
         </constantatomformula>
        </formulalist>
       </sequent>
      </rule>
     </rule>
     <rule type="weakr" symbol="w:r">
      <sequent>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <function symbol="∩">
           <function symbol="∪">
            <constant symbol="s_3"/>
            <constant symbol="s_4"/>
           </function>
           <constant symbol="s_3"/>
          </function>
          <constant symbol="s_3"/>
         </function>
         <function symbol="∩">
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
          <constant symbol="s_3"/>
         </function>
        </constantatomformula>
       </formulalist>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <function symbol="∩">
           <function symbol="∪">
            <constant symbol="s_3"/>
            <constant symbol="s_4"/>
           </function>
           <constant symbol="s_3"/>
          </function>
          <constant symbol="s_3"/>
         </function>
         <function symbol="∩">
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
          <constant symbol="s_3"/>
         </function>
        </constantatomformula>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
          <constant symbol="s_3"/>
         </function>
         <constant symbol="s_3"/>
        </constantatomformula>
       </formulalist>
      </sequent>
      <rule type="axiom" symbol="axiom">
       <sequent>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∩">
           <function symbol="∩">
            <function symbol="∪">
             <constant symbol="s_3"/>
             <constant symbol="s_4"/>
            </function>
            <constant symbol="s_3"/>
           </function>
           <constant symbol="s_3"/>
          </function>
          <function symbol="∩">
           <function symbol="∪">
            <constant symbol="s_3"/>
            <constant symbol="s_4"/>
           </function>
           <constant symbol="s_3"/>
          </function>
         </constantatomformula>
        </formulalist>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∩">
           <function symbol="∩">
            <function symbol="∪">
             <constant symbol="s_3"/>
             <constant symbol="s_4"/>
            </function>
            <constant symbol="s_3"/>
           </function>
           <constant symbol="s_3"/>
          </function>
          <function symbol="∩">
           <function symbol="∪">
            <constant symbol="s_3"/>
            <constant symbol="s_4"/>
           </function>
           <constant symbol="s_3"/>
          </function>
         </constantatomformula>
        </formulalist>
       </sequent>
      </rule>
     </rule>
    </rule>
   </rule>
  </rule>
 </proof>
 <proof symbol="\psi_{11}" calculus="LK">
  <rule type="weakl" symbol="w:l">
   <sequent>
    <formulalist>
     <constantatomformula symbol="L1" definition="L1"/>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <function symbol="∪">
        <function symbol="∩">
         <constant symbol="s_1"/>
         <constant symbol="s_2"/>
        </function>
        <constant symbol="s_1"/>
       </function>
       <constant symbol="s_1"/>
      </function>
      <function symbol="∪">
       <function symbol="∩">
        <constant symbol="s_1"/>
        <constant symbol="s_2"/>
       </function>
       <constant symbol="s_1"/>
      </function>
     </constantatomformula>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <function symbol="∪">
        <constant symbol="s_3"/>
        <constant symbol="s_4"/>
       </function>
       <constant symbol="s_3"/>
      </function>
      <constant symbol="s_3"/>
     </constantatomformula>
    </formulalist>
    <formulalist>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <function symbol="∪">
        <function symbol="∩">
         <constant symbol="s_1"/>
         <constant symbol="s_2"/>
        </function>
        <constant symbol="s_1"/>
       </function>
       <constant symbol="s_1"/>
      </function>
      <function symbol="∪">
       <function symbol="∩">
        <constant symbol="s_1"/>
        <constant symbol="s_2"/>
       </function>
       <constant symbol="s_1"/>
      </function>
     </constantatomformula>
     <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
    </formulalist>
   </sequent>
   <rule type="defr" symbol="d:r" param="();[()];()">
    <sequent>
     <formulalist>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <function symbol="∪">
         <function symbol="∩">
          <constant symbol="s_1"/>
          <constant symbol="s_2"/>
         </function>
         <constant symbol="s_1"/>
        </function>
        <constant symbol="s_1"/>
       </function>
       <function symbol="∪">
        <function symbol="∩">
         <constant symbol="s_1"/>
         <constant symbol="s_2"/>
        </function>
        <constant symbol="s_1"/>
       </function>
      </constantatomformula>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <function symbol="∪">
         <constant symbol="s_3"/>
         <constant symbol="s_4"/>
        </function>
        <constant symbol="s_3"/>
       </function>
       <constant symbol="s_3"/>
      </constantatomformula>
     </formulalist>
     <formulalist>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <function symbol="∪">
         <function symbol="∩">
          <constant symbol="s_1"/>
          <constant symbol="s_2"/>
         </function>
         <constant symbol="s_1"/>
        </function>
        <constant symbol="s_1"/>
       </function>
       <function symbol="∪">
        <function symbol="∩">
         <constant symbol="s_1"/>
         <constant symbol="s_2"/>
        </function>
        <constant symbol="s_1"/>
       </function>
      </constantatomformula>
      <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
     </formulalist>
    </sequent>
    <rule type="andr" symbol="\and:r">
     <sequent>
      <formulalist>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <function symbol="∪">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
         <constant symbol="s_1"/>
        </function>
        <function symbol="∪">
         <function symbol="∩">
          <constant symbol="s_1"/>
          <constant symbol="s_2"/>
         </function>
         <constant symbol="s_1"/>
        </function>
       </constantatomformula>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <function symbol="∪">
          <constant symbol="s_3"/>
          <constant symbol="s_4"/>
         </function>
         <constant symbol="s_3"/>
        </function>
        <constant symbol="s_3"/>
       </constantatomformula>
      </formulalist>
      <formulalist>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <function symbol="∪">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
         <constant symbol="s_1"/>
        </function>
        <function symbol="∪">
         <function symbol="∩">
          <constant symbol="s_1"/>
          <constant symbol="s_2"/>
         </function>
         <constant symbol="s_1"/>
        </function>
       </constantatomformula>
       <conjunctiveformula type="and">
        <constantatomformula symbol="=">
         <function symbol="∪">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
         <constant symbol="s_1"/>
        </constantatomformula>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
          <constant symbol="s_3"/>
         </function>
         <constant symbol="s_3"/>
        </constantatomformula>
       </conjunctiveformula>
      </formulalist>
     </sequent>
     <rule type="weakr" symbol="w:r">
      <sequent>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <function symbol="∪">
           <function symbol="∩">
            <constant symbol="s_1"/>
            <constant symbol="s_2"/>
           </function>
           <constant symbol="s_1"/>
          </function>
          <constant symbol="s_1"/>
         </function>
         <function symbol="∪">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
        </constantatomformula>
       </formulalist>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <function symbol="∪">
           <function symbol="∩">
            <constant symbol="s_1"/>
            <constant symbol="s_2"/>
           </function>
           <constant symbol="s_1"/>
          </function>
          <constant symbol="s_1"/>
         </function>
         <function symbol="∪">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
        </constantatomformula>
        <constantatomformula symbol="=">
         <function symbol="∪">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
         <constant symbol="s_1"/>
        </constantatomformula>
       </formulalist>
      </sequent>
      <rule type="axiom" symbol="axiom">
       <sequent>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∩">
           <function symbol="∪">
            <function symbol="∩">
             <constant symbol="s_1"/>
             <constant symbol="s_2"/>
            </function>
            <constant symbol="s_1"/>
           </function>
           <constant symbol="s_1"/>
          </function>
          <function symbol="∪">
           <function symbol="∩">
            <constant symbol="s_1"/>
            <constant symbol="s_2"/>
           </function>
           <constant symbol="s_1"/>
          </function>
         </constantatomformula>
        </formulalist>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∩">
           <function symbol="∪">
            <function symbol="∩">
             <constant symbol="s_1"/>
             <constant symbol="s_2"/>
            </function>
            <constant symbol="s_1"/>
           </function>
           <constant symbol="s_1"/>
          </function>
          <function symbol="∪">
           <function symbol="∩">
            <constant symbol="s_1"/>
            <constant symbol="s_2"/>
           </function>
           <constant symbol="s_1"/>
          </function>
         </constantatomformula>
        </formulalist>
       </sequent>
      </rule>
     </rule>
     <rule type="axiom" symbol="axiom">
      <sequent>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
          <constant symbol="s_3"/>
         </function>
         <constant symbol="s_3"/>
        </constantatomformula>
       </formulalist>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
          <constant symbol="s_3"/>
         </function>
         <constant symbol="s_3"/>
        </constantatomformula>
       </formulalist>
      </sequent>
     </rule>
    </rule>
   </rule>
  </rule>
 </proof>
 <proof symbol="\psi_{12}" calculus="LK">
  <rule type="weakl" symbol="w:l">
   <sequent>
    <formulalist>
     <constantatomformula symbol="L1" definition="L1"/>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <function symbol="∪">
        <function symbol="∩">
         <constant symbol="s_1"/>
         <constant symbol="s_2"/>
        </function>
        <constant symbol="s_1"/>
       </function>
       <constant symbol="s_1"/>
      </function>
      <function symbol="∪">
       <function symbol="∩">
        <constant symbol="s_1"/>
        <constant symbol="s_2"/>
       </function>
       <constant symbol="s_1"/>
      </function>
     </constantatomformula>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <constant symbol="s_3"/>
       <function symbol="∩">
        <function symbol="∪">
         <constant symbol="s_3"/>
         <constant symbol="s_4"/>
        </function>
        <constant symbol="s_3"/>
       </function>
      </function>
      <constant symbol="s_3"/>
     </constantatomformula>
    </formulalist>
    <formulalist>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <function symbol="∪">
        <function symbol="∩">
         <constant symbol="s_1"/>
         <constant symbol="s_2"/>
        </function>
        <constant symbol="s_1"/>
       </function>
       <constant symbol="s_1"/>
      </function>
      <function symbol="∪">
       <function symbol="∩">
        <constant symbol="s_1"/>
        <constant symbol="s_2"/>
       </function>
       <constant symbol="s_1"/>
      </function>
     </constantatomformula>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <constant symbol="s_3"/>
       <function symbol="∩">
        <function symbol="∪">
         <constant symbol="s_3"/>
         <constant symbol="s_4"/>
        </function>
        <constant symbol="s_3"/>
       </function>
      </function>
      <constant symbol="s_3"/>
     </constantatomformula>
     <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
    </formulalist>
   </sequent>
   <rule type="defr" symbol="d:r" param="();[()];()">
    <sequent>
     <formulalist>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <function symbol="∪">
         <function symbol="∩">
          <constant symbol="s_1"/>
          <constant symbol="s_2"/>
         </function>
         <constant symbol="s_1"/>
        </function>
        <constant symbol="s_1"/>
       </function>
       <function symbol="∪">
        <function symbol="∩">
         <constant symbol="s_1"/>
         <constant symbol="s_2"/>
        </function>
        <constant symbol="s_1"/>
       </function>
      </constantatomformula>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <constant symbol="s_3"/>
        <function symbol="∩">
         <function symbol="∪">
          <constant symbol="s_3"/>
          <constant symbol="s_4"/>
         </function>
         <constant symbol="s_3"/>
        </function>
       </function>
       <constant symbol="s_3"/>
      </constantatomformula>
     </formulalist>
     <formulalist>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <function symbol="∪">
         <function symbol="∩">
          <constant symbol="s_1"/>
          <constant symbol="s_2"/>
         </function>
         <constant symbol="s_1"/>
        </function>
        <constant symbol="s_1"/>
       </function>
       <function symbol="∪">
        <function symbol="∩">
         <constant symbol="s_1"/>
         <constant symbol="s_2"/>
        </function>
        <constant symbol="s_1"/>
       </function>
      </constantatomformula>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <constant symbol="s_3"/>
        <function symbol="∩">
         <function symbol="∪">
          <constant symbol="s_3"/>
          <constant symbol="s_4"/>
         </function>
         <constant symbol="s_3"/>
        </function>
       </function>
       <constant symbol="s_3"/>
      </constantatomformula>
      <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
     </formulalist>
    </sequent>
    <rule type="andr" symbol="\and:r">
     <sequent>
      <formulalist>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <function symbol="∪">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
         <constant symbol="s_1"/>
        </function>
        <function symbol="∪">
         <function symbol="∩">
          <constant symbol="s_1"/>
          <constant symbol="s_2"/>
         </function>
         <constant symbol="s_1"/>
        </function>
       </constantatomformula>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <constant symbol="s_3"/>
         <function symbol="∩">
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
          <constant symbol="s_3"/>
         </function>
        </function>
        <constant symbol="s_3"/>
       </constantatomformula>
      </formulalist>
      <formulalist>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <function symbol="∪">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
         <constant symbol="s_1"/>
        </function>
        <function symbol="∪">
         <function symbol="∩">
          <constant symbol="s_1"/>
          <constant symbol="s_2"/>
         </function>
         <constant symbol="s_1"/>
        </function>
       </constantatomformula>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <constant symbol="s_3"/>
         <function symbol="∩">
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
          <constant symbol="s_3"/>
         </function>
        </function>
        <constant symbol="s_3"/>
       </constantatomformula>
       <conjunctiveformula type="and">
        <constantatomformula symbol="=">
         <function symbol="∪">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
         <constant symbol="s_1"/>
        </constantatomformula>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
          <constant symbol="s_3"/>
         </function>
         <constant symbol="s_3"/>
        </constantatomformula>
       </conjunctiveformula>
      </formulalist>
     </sequent>
     <rule type="weakr" symbol="w:r">
      <sequent>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <function symbol="∪">
           <function symbol="∩">
            <constant symbol="s_1"/>
            <constant symbol="s_2"/>
           </function>
           <constant symbol="s_1"/>
          </function>
          <constant symbol="s_1"/>
         </function>
         <function symbol="∪">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
        </constantatomformula>
       </formulalist>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <function symbol="∪">
           <function symbol="∩">
            <constant symbol="s_1"/>
            <constant symbol="s_2"/>
           </function>
           <constant symbol="s_1"/>
          </function>
          <constant symbol="s_1"/>
         </function>
         <function symbol="∪">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
        </constantatomformula>
        <constantatomformula symbol="=">
         <function symbol="∪">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
         <constant symbol="s_1"/>
        </constantatomformula>
       </formulalist>
      </sequent>
      <rule type="axiom" symbol="axiom">
       <sequent>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∩">
           <function symbol="∪">
            <function symbol="∩">
             <constant symbol="s_1"/>
             <constant symbol="s_2"/>
            </function>
            <constant symbol="s_1"/>
           </function>
           <constant symbol="s_1"/>
          </function>
          <function symbol="∪">
           <function symbol="∩">
            <constant symbol="s_1"/>
            <constant symbol="s_2"/>
           </function>
           <constant symbol="s_1"/>
          </function>
         </constantatomformula>
        </formulalist>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∩">
           <function symbol="∪">
            <function symbol="∩">
             <constant symbol="s_1"/>
             <constant symbol="s_2"/>
            </function>
            <constant symbol="s_1"/>
           </function>
           <constant symbol="s_1"/>
          </function>
          <function symbol="∪">
           <function symbol="∩">
            <constant symbol="s_1"/>
            <constant symbol="s_2"/>
           </function>
           <constant symbol="s_1"/>
          </function>
         </constantatomformula>
        </formulalist>
       </sequent>
      </rule>
     </rule>
     <rule type="weakr" symbol="w:r">
      <sequent>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <constant symbol="s_3"/>
          <function symbol="∩">
           <function symbol="∪">
            <constant symbol="s_3"/>
            <constant symbol="s_4"/>
           </function>
           <constant symbol="s_3"/>
          </function>
         </function>
         <constant symbol="s_3"/>
        </constantatomformula>
       </formulalist>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <constant symbol="s_3"/>
          <function symbol="∩">
           <function symbol="∪">
            <constant symbol="s_3"/>
            <constant symbol="s_4"/>
           </function>
           <constant symbol="s_3"/>
          </function>
         </function>
         <constant symbol="s_3"/>
        </constantatomformula>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
          <constant symbol="s_3"/>
         </function>
         <constant symbol="s_3"/>
        </constantatomformula>
       </formulalist>
      </sequent>
      <rule type="axiom" symbol="axiom">
       <sequent>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∩">
           <constant symbol="s_3"/>
           <function symbol="∩">
            <function symbol="∪">
             <constant symbol="s_3"/>
             <constant symbol="s_4"/>
            </function>
            <constant symbol="s_3"/>
           </function>
          </function>
          <constant symbol="s_3"/>
         </constantatomformula>
        </formulalist>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∩">
           <constant symbol="s_3"/>
           <function symbol="∩">
            <function symbol="∪">
             <constant symbol="s_3"/>
             <constant symbol="s_4"/>
            </function>
            <constant symbol="s_3"/>
           </function>
          </function>
          <constant symbol="s_3"/>
         </constantatomformula>
        </formulalist>
       </sequent>
      </rule>
     </rule>
    </rule>
   </rule>
  </rule>
 </proof>
 <proof symbol="\psi_{13}" calculus="LK">
  <rule type="weakl" symbol="w:l">
   <sequent>
    <formulalist>
     <constantatomformula symbol="L1" definition="L1"/>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <function symbol="∪">
        <function symbol="∩">
         <constant symbol="s_1"/>
         <constant symbol="s_2"/>
        </function>
        <constant symbol="s_1"/>
       </function>
       <constant symbol="s_1"/>
      </function>
      <function symbol="∪">
       <function symbol="∩">
        <constant symbol="s_1"/>
        <constant symbol="s_2"/>
       </function>
       <constant symbol="s_1"/>
      </function>
     </constantatomformula>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <constant symbol="s_3"/>
       <constant symbol="s_3"/>
      </function>
      <constant symbol="s_3"/>
     </constantatomformula>
    </formulalist>
    <formulalist>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <function symbol="∪">
        <function symbol="∩">
         <constant symbol="s_1"/>
         <constant symbol="s_2"/>
        </function>
        <constant symbol="s_1"/>
       </function>
       <constant symbol="s_1"/>
      </function>
      <function symbol="∪">
       <function symbol="∩">
        <constant symbol="s_1"/>
        <constant symbol="s_2"/>
       </function>
       <constant symbol="s_1"/>
      </function>
     </constantatomformula>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <constant symbol="s_3"/>
       <constant symbol="s_3"/>
      </function>
      <constant symbol="s_3"/>
     </constantatomformula>
     <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
    </formulalist>
   </sequent>
   <rule type="defr" symbol="d:r" param="();[()];()">
    <sequent>
     <formulalist>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <function symbol="∪">
         <function symbol="∩">
          <constant symbol="s_1"/>
          <constant symbol="s_2"/>
         </function>
         <constant symbol="s_1"/>
        </function>
        <constant symbol="s_1"/>
       </function>
       <function symbol="∪">
        <function symbol="∩">
         <constant symbol="s_1"/>
         <constant symbol="s_2"/>
        </function>
        <constant symbol="s_1"/>
       </function>
      </constantatomformula>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <constant symbol="s_3"/>
        <constant symbol="s_3"/>
       </function>
       <constant symbol="s_3"/>
      </constantatomformula>
     </formulalist>
     <formulalist>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <function symbol="∪">
         <function symbol="∩">
          <constant symbol="s_1"/>
          <constant symbol="s_2"/>
         </function>
         <constant symbol="s_1"/>
        </function>
        <constant symbol="s_1"/>
       </function>
       <function symbol="∪">
        <function symbol="∩">
         <constant symbol="s_1"/>
         <constant symbol="s_2"/>
        </function>
        <constant symbol="s_1"/>
       </function>
      </constantatomformula>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <constant symbol="s_3"/>
        <constant symbol="s_3"/>
       </function>
       <constant symbol="s_3"/>
      </constantatomformula>
      <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
     </formulalist>
    </sequent>
    <rule type="andr" symbol="\and:r">
     <sequent>
      <formulalist>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <function symbol="∪">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
         <constant symbol="s_1"/>
        </function>
        <function symbol="∪">
         <function symbol="∩">
          <constant symbol="s_1"/>
          <constant symbol="s_2"/>
         </function>
         <constant symbol="s_1"/>
        </function>
       </constantatomformula>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <constant symbol="s_3"/>
         <constant symbol="s_3"/>
        </function>
        <constant symbol="s_3"/>
       </constantatomformula>
      </formulalist>
      <formulalist>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <function symbol="∪">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
         <constant symbol="s_1"/>
        </function>
        <function symbol="∪">
         <function symbol="∩">
          <constant symbol="s_1"/>
          <constant symbol="s_2"/>
         </function>
         <constant symbol="s_1"/>
        </function>
       </constantatomformula>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <constant symbol="s_3"/>
         <constant symbol="s_3"/>
        </function>
        <constant symbol="s_3"/>
       </constantatomformula>
       <conjunctiveformula type="and">
        <constantatomformula symbol="=">
         <function symbol="∪">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
         <constant symbol="s_1"/>
        </constantatomformula>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
          <constant symbol="s_3"/>
         </function>
         <constant symbol="s_3"/>
        </constantatomformula>
       </conjunctiveformula>
      </formulalist>
     </sequent>
     <rule type="weakr" symbol="w:r">
      <sequent>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <function symbol="∪">
           <function symbol="∩">
            <constant symbol="s_1"/>
            <constant symbol="s_2"/>
           </function>
           <constant symbol="s_1"/>
          </function>
          <constant symbol="s_1"/>
         </function>
         <function symbol="∪">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
        </constantatomformula>
       </formulalist>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <function symbol="∪">
           <function symbol="∩">
            <constant symbol="s_1"/>
            <constant symbol="s_2"/>
           </function>
           <constant symbol="s_1"/>
          </function>
          <constant symbol="s_1"/>
         </function>
         <function symbol="∪">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
        </constantatomformula>
        <constantatomformula symbol="=">
         <function symbol="∪">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
         <constant symbol="s_1"/>
        </constantatomformula>
       </formulalist>
      </sequent>
      <rule type="axiom" symbol="axiom">
       <sequent>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∩">
           <function symbol="∪">
            <function symbol="∩">
             <constant symbol="s_1"/>
             <constant symbol="s_2"/>
            </function>
            <constant symbol="s_1"/>
           </function>
           <constant symbol="s_1"/>
          </function>
          <function symbol="∪">
           <function symbol="∩">
            <constant symbol="s_1"/>
            <constant symbol="s_2"/>
           </function>
           <constant symbol="s_1"/>
          </function>
         </constantatomformula>
        </formulalist>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∩">
           <function symbol="∪">
            <function symbol="∩">
             <constant symbol="s_1"/>
             <constant symbol="s_2"/>
            </function>
            <constant symbol="s_1"/>
           </function>
           <constant symbol="s_1"/>
          </function>
          <function symbol="∪">
           <function symbol="∩">
            <constant symbol="s_1"/>
            <constant symbol="s_2"/>
           </function>
           <constant symbol="s_1"/>
          </function>
         </constantatomformula>
        </formulalist>
       </sequent>
      </rule>
     </rule>
     <rule type="weakr" symbol="w:r">
      <sequent>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <constant symbol="s_3"/>
          <constant symbol="s_3"/>
         </function>
         <constant symbol="s_3"/>
        </constantatomformula>
       </formulalist>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <constant symbol="s_3"/>
          <constant symbol="s_3"/>
         </function>
         <constant symbol="s_3"/>
        </constantatomformula>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
          <constant symbol="s_3"/>
         </function>
         <constant symbol="s_3"/>
        </constantatomformula>
       </formulalist>
      </sequent>
      <rule type="axiom" symbol="axiom">
       <sequent>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∩">
           <constant symbol="s_3"/>
           <constant symbol="s_3"/>
          </function>
          <constant symbol="s_3"/>
         </constantatomformula>
        </formulalist>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∩">
           <constant symbol="s_3"/>
           <constant symbol="s_3"/>
          </function>
          <constant symbol="s_3"/>
         </constantatomformula>
        </formulalist>
       </sequent>
      </rule>
     </rule>
    </rule>
   </rule>
  </rule>
 </proof>
 <proof symbol="\psi_{14}" calculus="LK">
  <rule type="weakl" symbol="w:l">
   <sequent>
    <formulalist>
     <constantatomformula symbol="L1" definition="L1"/>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <function symbol="∪">
        <function symbol="∩">
         <constant symbol="s_1"/>
         <constant symbol="s_2"/>
        </function>
        <constant symbol="s_1"/>
       </function>
       <constant symbol="s_1"/>
      </function>
      <function symbol="∪">
       <function symbol="∩">
        <constant symbol="s_1"/>
        <constant symbol="s_2"/>
       </function>
       <constant symbol="s_1"/>
      </function>
     </constantatomformula>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <constant symbol="s_3"/>
       <function symbol="∪">
        <constant symbol="s_3"/>
        <constant symbol="s_4"/>
       </function>
      </function>
      <constant symbol="s_3"/>
     </constantatomformula>
    </formulalist>
    <formulalist>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <function symbol="∪">
        <function symbol="∩">
         <constant symbol="s_1"/>
         <constant symbol="s_2"/>
        </function>
        <constant symbol="s_1"/>
       </function>
       <constant symbol="s_1"/>
      </function>
      <function symbol="∪">
       <function symbol="∩">
        <constant symbol="s_1"/>
        <constant symbol="s_2"/>
       </function>
       <constant symbol="s_1"/>
      </function>
     </constantatomformula>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <constant symbol="s_3"/>
       <function symbol="∪">
        <constant symbol="s_3"/>
        <constant symbol="s_4"/>
       </function>
      </function>
      <constant symbol="s_3"/>
     </constantatomformula>
     <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
    </formulalist>
   </sequent>
   <rule type="defr" symbol="d:r" param="();[()];()">
    <sequent>
     <formulalist>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <function symbol="∪">
         <function symbol="∩">
          <constant symbol="s_1"/>
          <constant symbol="s_2"/>
         </function>
         <constant symbol="s_1"/>
        </function>
        <constant symbol="s_1"/>
       </function>
       <function symbol="∪">
        <function symbol="∩">
         <constant symbol="s_1"/>
         <constant symbol="s_2"/>
        </function>
        <constant symbol="s_1"/>
       </function>
      </constantatomformula>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <constant symbol="s_3"/>
        <function symbol="∪">
         <constant symbol="s_3"/>
         <constant symbol="s_4"/>
        </function>
       </function>
       <constant symbol="s_3"/>
      </constantatomformula>
     </formulalist>
     <formulalist>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <function symbol="∪">
         <function symbol="∩">
          <constant symbol="s_1"/>
          <constant symbol="s_2"/>
         </function>
         <constant symbol="s_1"/>
        </function>
        <constant symbol="s_1"/>
       </function>
       <function symbol="∪">
        <function symbol="∩">
         <constant symbol="s_1"/>
         <constant symbol="s_2"/>
        </function>
        <constant symbol="s_1"/>
       </function>
      </constantatomformula>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <constant symbol="s_3"/>
        <function symbol="∪">
         <constant symbol="s_3"/>
         <constant symbol="s_4"/>
        </function>
       </function>
       <constant symbol="s_3"/>
      </constantatomformula>
      <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
     </formulalist>
    </sequent>
    <rule type="andr" symbol="\and:r">
     <sequent>
      <formulalist>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <function symbol="∪">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
         <constant symbol="s_1"/>
        </function>
        <function symbol="∪">
         <function symbol="∩">
          <constant symbol="s_1"/>
          <constant symbol="s_2"/>
         </function>
         <constant symbol="s_1"/>
        </function>
       </constantatomformula>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <constant symbol="s_3"/>
         <function symbol="∪">
          <constant symbol="s_3"/>
          <constant symbol="s_4"/>
         </function>
        </function>
        <constant symbol="s_3"/>
       </constantatomformula>
      </formulalist>
      <formulalist>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <function symbol="∪">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
         <constant symbol="s_1"/>
        </function>
        <function symbol="∪">
         <function symbol="∩">
          <constant symbol="s_1"/>
          <constant symbol="s_2"/>
         </function>
         <constant symbol="s_1"/>
        </function>
       </constantatomformula>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <constant symbol="s_3"/>
         <function symbol="∪">
          <constant symbol="s_3"/>
          <constant symbol="s_4"/>
         </function>
        </function>
        <constant symbol="s_3"/>
       </constantatomformula>
       <conjunctiveformula type="and">
        <constantatomformula symbol="=">
         <function symbol="∪">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
         <constant symbol="s_1"/>
        </constantatomformula>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
          <constant symbol="s_3"/>
         </function>
         <constant symbol="s_3"/>
        </constantatomformula>
       </conjunctiveformula>
      </formulalist>
     </sequent>
     <rule type="weakr" symbol="w:r">
      <sequent>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <function symbol="∪">
           <function symbol="∩">
            <constant symbol="s_1"/>
            <constant symbol="s_2"/>
           </function>
           <constant symbol="s_1"/>
          </function>
          <constant symbol="s_1"/>
         </function>
         <function symbol="∪">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
        </constantatomformula>
       </formulalist>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <function symbol="∪">
           <function symbol="∩">
            <constant symbol="s_1"/>
            <constant symbol="s_2"/>
           </function>
           <constant symbol="s_1"/>
          </function>
          <constant symbol="s_1"/>
         </function>
         <function symbol="∪">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
        </constantatomformula>
        <constantatomformula symbol="=">
         <function symbol="∪">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
         <constant symbol="s_1"/>
        </constantatomformula>
       </formulalist>
      </sequent>
      <rule type="axiom" symbol="axiom">
       <sequent>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∩">
           <function symbol="∪">
            <function symbol="∩">
             <constant symbol="s_1"/>
             <constant symbol="s_2"/>
            </function>
            <constant symbol="s_1"/>
           </function>
           <constant symbol="s_1"/>
          </function>
          <function symbol="∪">
           <function symbol="∩">
            <constant symbol="s_1"/>
            <constant symbol="s_2"/>
           </function>
           <constant symbol="s_1"/>
          </function>
         </constantatomformula>
        </formulalist>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∩">
           <function symbol="∪">
            <function symbol="∩">
             <constant symbol="s_1"/>
             <constant symbol="s_2"/>
            </function>
            <constant symbol="s_1"/>
           </function>
           <constant symbol="s_1"/>
          </function>
          <function symbol="∪">
           <function symbol="∩">
            <constant symbol="s_1"/>
            <constant symbol="s_2"/>
           </function>
           <constant symbol="s_1"/>
          </function>
         </constantatomformula>
        </formulalist>
       </sequent>
      </rule>
     </rule>
     <rule type="weakr" symbol="w:r">
      <sequent>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <constant symbol="s_3"/>
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
         </function>
         <constant symbol="s_3"/>
        </constantatomformula>
       </formulalist>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <constant symbol="s_3"/>
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
         </function>
         <constant symbol="s_3"/>
        </constantatomformula>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
          <constant symbol="s_3"/>
         </function>
         <constant symbol="s_3"/>
        </constantatomformula>
       </formulalist>
      </sequent>
      <rule type="axiom" symbol="axiom">
       <sequent>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∩">
           <constant symbol="s_3"/>
           <function symbol="∪">
            <constant symbol="s_3"/>
            <constant symbol="s_4"/>
           </function>
          </function>
          <constant symbol="s_3"/>
         </constantatomformula>
        </formulalist>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∩">
           <constant symbol="s_3"/>
           <function symbol="∪">
            <constant symbol="s_3"/>
            <constant symbol="s_4"/>
           </function>
          </function>
          <constant symbol="s_3"/>
         </constantatomformula>
        </formulalist>
       </sequent>
      </rule>
     </rule>
    </rule>
   </rule>
  </rule>
 </proof>
 <proof symbol="\psi_{15}" calculus="LK">
  <rule type="weakl" symbol="w:l">
   <sequent>
    <formulalist>
     <constantatomformula symbol="L1" definition="L1"/>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <function symbol="∪">
        <function symbol="∩">
         <constant symbol="s_1"/>
         <constant symbol="s_2"/>
        </function>
        <constant symbol="s_1"/>
       </function>
       <constant symbol="s_1"/>
      </function>
      <function symbol="∪">
       <function symbol="∩">
        <constant symbol="s_1"/>
        <constant symbol="s_2"/>
       </function>
       <constant symbol="s_1"/>
      </function>
     </constantatomformula>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <function symbol="∩">
        <function symbol="∪">
         <constant symbol="s_3"/>
         <constant symbol="s_4"/>
        </function>
        <constant symbol="s_3"/>
       </function>
       <constant symbol="s_3"/>
      </function>
      <function symbol="∩">
       <function symbol="∪">
        <constant symbol="s_3"/>
        <constant symbol="s_4"/>
       </function>
       <constant symbol="s_3"/>
      </function>
     </constantatomformula>
    </formulalist>
    <formulalist>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <function symbol="∪">
        <function symbol="∩">
         <constant symbol="s_1"/>
         <constant symbol="s_2"/>
        </function>
        <constant symbol="s_1"/>
       </function>
       <constant symbol="s_1"/>
      </function>
      <function symbol="∪">
       <function symbol="∩">
        <constant symbol="s_1"/>
        <constant symbol="s_2"/>
       </function>
       <constant symbol="s_1"/>
      </function>
     </constantatomformula>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <function symbol="∩">
        <function symbol="∪">
         <constant symbol="s_3"/>
         <constant symbol="s_4"/>
        </function>
        <constant symbol="s_3"/>
       </function>
       <constant symbol="s_3"/>
      </function>
      <function symbol="∩">
       <function symbol="∪">
        <constant symbol="s_3"/>
        <constant symbol="s_4"/>
       </function>
       <constant symbol="s_3"/>
      </function>
     </constantatomformula>
     <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
    </formulalist>
   </sequent>
   <rule type="defr" symbol="d:r" param="();[()];()">
    <sequent>
     <formulalist>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <function symbol="∪">
         <function symbol="∩">
          <constant symbol="s_1"/>
          <constant symbol="s_2"/>
         </function>
         <constant symbol="s_1"/>
        </function>
        <constant symbol="s_1"/>
       </function>
       <function symbol="∪">
        <function symbol="∩">
         <constant symbol="s_1"/>
         <constant symbol="s_2"/>
        </function>
        <constant symbol="s_1"/>
       </function>
      </constantatomformula>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <function symbol="∩">
         <function symbol="∪">
          <constant symbol="s_3"/>
          <constant symbol="s_4"/>
         </function>
         <constant symbol="s_3"/>
        </function>
        <constant symbol="s_3"/>
       </function>
       <function symbol="∩">
        <function symbol="∪">
         <constant symbol="s_3"/>
         <constant symbol="s_4"/>
        </function>
        <constant symbol="s_3"/>
       </function>
      </constantatomformula>
     </formulalist>
     <formulalist>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <function symbol="∪">
         <function symbol="∩">
          <constant symbol="s_1"/>
          <constant symbol="s_2"/>
         </function>
         <constant symbol="s_1"/>
        </function>
        <constant symbol="s_1"/>
       </function>
       <function symbol="∪">
        <function symbol="∩">
         <constant symbol="s_1"/>
         <constant symbol="s_2"/>
        </function>
        <constant symbol="s_1"/>
       </function>
      </constantatomformula>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <function symbol="∩">
         <function symbol="∪">
          <constant symbol="s_3"/>
          <constant symbol="s_4"/>
         </function>
         <constant symbol="s_3"/>
        </function>
        <constant symbol="s_3"/>
       </function>
       <function symbol="∩">
        <function symbol="∪">
         <constant symbol="s_3"/>
         <constant symbol="s_4"/>
        </function>
        <constant symbol="s_3"/>
       </function>
      </constantatomformula>
      <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
     </formulalist>
    </sequent>
    <rule type="andr" symbol="\and:r">
     <sequent>
      <formulalist>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <function symbol="∪">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
         <constant symbol="s_1"/>
        </function>
        <function symbol="∪">
         <function symbol="∩">
          <constant symbol="s_1"/>
          <constant symbol="s_2"/>
         </function>
         <constant symbol="s_1"/>
        </function>
       </constantatomformula>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <function symbol="∩">
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
          <constant symbol="s_3"/>
         </function>
         <constant symbol="s_3"/>
        </function>
        <function symbol="∩">
         <function symbol="∪">
          <constant symbol="s_3"/>
          <constant symbol="s_4"/>
         </function>
         <constant symbol="s_3"/>
        </function>
       </constantatomformula>
      </formulalist>
      <formulalist>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <function symbol="∪">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
         <constant symbol="s_1"/>
        </function>
        <function symbol="∪">
         <function symbol="∩">
          <constant symbol="s_1"/>
          <constant symbol="s_2"/>
         </function>
         <constant symbol="s_1"/>
        </function>
       </constantatomformula>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <function symbol="∩">
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
          <constant symbol="s_3"/>
         </function>
         <constant symbol="s_3"/>
        </function>
        <function symbol="∩">
         <function symbol="∪">
          <constant symbol="s_3"/>
          <constant symbol="s_4"/>
         </function>
         <constant symbol="s_3"/>
        </function>
       </constantatomformula>
       <conjunctiveformula type="and">
        <constantatomformula symbol="=">
         <function symbol="∪">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
         <constant symbol="s_1"/>
        </constantatomformula>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
          <constant symbol="s_3"/>
         </function>
         <constant symbol="s_3"/>
        </constantatomformula>
       </conjunctiveformula>
      </formulalist>
     </sequent>
     <rule type="weakr" symbol="w:r">
      <sequent>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <function symbol="∪">
           <function symbol="∩">
            <constant symbol="s_1"/>
            <constant symbol="s_2"/>
           </function>
           <constant symbol="s_1"/>
          </function>
          <constant symbol="s_1"/>
         </function>
         <function symbol="∪">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
        </constantatomformula>
       </formulalist>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <function symbol="∪">
           <function symbol="∩">
            <constant symbol="s_1"/>
            <constant symbol="s_2"/>
           </function>
           <constant symbol="s_1"/>
          </function>
          <constant symbol="s_1"/>
         </function>
         <function symbol="∪">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
        </constantatomformula>
        <constantatomformula symbol="=">
         <function symbol="∪">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
         <constant symbol="s_1"/>
        </constantatomformula>
       </formulalist>
      </sequent>
      <rule type="axiom" symbol="axiom">
       <sequent>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∩">
           <function symbol="∪">
            <function symbol="∩">
             <constant symbol="s_1"/>
             <constant symbol="s_2"/>
            </function>
            <constant symbol="s_1"/>
           </function>
           <constant symbol="s_1"/>
          </function>
          <function symbol="∪">
           <function symbol="∩">
            <constant symbol="s_1"/>
            <constant symbol="s_2"/>
           </function>
           <constant symbol="s_1"/>
          </function>
         </constantatomformula>
        </formulalist>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∩">
           <function symbol="∪">
            <function symbol="∩">
             <constant symbol="s_1"/>
             <constant symbol="s_2"/>
            </function>
            <constant symbol="s_1"/>
           </function>
           <constant symbol="s_1"/>
          </function>
          <function symbol="∪">
           <function symbol="∩">
            <constant symbol="s_1"/>
            <constant symbol="s_2"/>
           </function>
           <constant symbol="s_1"/>
          </function>
         </constantatomformula>
        </formulalist>
       </sequent>
      </rule>
     </rule>
     <rule type="weakr" symbol="w:r">
      <sequent>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <function symbol="∩">
           <function symbol="∪">
            <constant symbol="s_3"/>
            <constant symbol="s_4"/>
           </function>
           <constant symbol="s_3"/>
          </function>
          <constant symbol="s_3"/>
         </function>
         <function symbol="∩">
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
          <constant symbol="s_3"/>
         </function>
        </constantatomformula>
       </formulalist>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <function symbol="∩">
           <function symbol="∪">
            <constant symbol="s_3"/>
            <constant symbol="s_4"/>
           </function>
           <constant symbol="s_3"/>
          </function>
          <constant symbol="s_3"/>
         </function>
         <function symbol="∩">
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
          <constant symbol="s_3"/>
         </function>
        </constantatomformula>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
          <constant symbol="s_3"/>
         </function>
         <constant symbol="s_3"/>
        </constantatomformula>
       </formulalist>
      </sequent>
      <rule type="axiom" symbol="axiom">
       <sequent>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∩">
           <function symbol="∩">
            <function symbol="∪">
             <constant symbol="s_3"/>
             <constant symbol="s_4"/>
            </function>
            <constant symbol="s_3"/>
           </function>
           <constant symbol="s_3"/>
          </function>
          <function symbol="∩">
           <function symbol="∪">
            <constant symbol="s_3"/>
            <constant symbol="s_4"/>
           </function>
           <constant symbol="s_3"/>
          </function>
         </constantatomformula>
        </formulalist>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∩">
           <function symbol="∩">
            <function symbol="∪">
             <constant symbol="s_3"/>
             <constant symbol="s_4"/>
            </function>
            <constant symbol="s_3"/>
           </function>
           <constant symbol="s_3"/>
          </function>
          <function symbol="∩">
           <function symbol="∪">
            <constant symbol="s_3"/>
            <constant symbol="s_4"/>
           </function>
           <constant symbol="s_3"/>
          </function>
         </constantatomformula>
        </formulalist>
       </sequent>
      </rule>
     </rule>
    </rule>
   </rule>
  </rule>
 </proof>
 <proof symbol="\psi_{16}" calculus="LK">
  <rule type="weakl" symbol="w:l">
   <sequent>
    <formulalist>
     <constantatomformula symbol="L1" definition="L1"/>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <constant symbol="s_1"/>
       <constant symbol="s_1"/>
      </function>
      <constant symbol="s_1"/>
     </constantatomformula>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <function symbol="∪">
        <constant symbol="s_3"/>
        <constant symbol="s_4"/>
       </function>
       <constant symbol="s_3"/>
      </function>
      <constant symbol="s_3"/>
     </constantatomformula>
    </formulalist>
    <formulalist>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <constant symbol="s_1"/>
       <constant symbol="s_1"/>
      </function>
      <constant symbol="s_1"/>
     </constantatomformula>
     <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
    </formulalist>
   </sequent>
   <rule type="defr" symbol="d:r" param="();[()];()">
    <sequent>
     <formulalist>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <constant symbol="s_1"/>
        <constant symbol="s_1"/>
       </function>
       <constant symbol="s_1"/>
      </constantatomformula>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <function symbol="∪">
         <constant symbol="s_3"/>
         <constant symbol="s_4"/>
        </function>
        <constant symbol="s_3"/>
       </function>
       <constant symbol="s_3"/>
      </constantatomformula>
     </formulalist>
     <formulalist>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <constant symbol="s_1"/>
        <constant symbol="s_1"/>
       </function>
       <constant symbol="s_1"/>
      </constantatomformula>
      <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
     </formulalist>
    </sequent>
    <rule type="andr" symbol="\and:r">
     <sequent>
      <formulalist>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <constant symbol="s_1"/>
         <constant symbol="s_1"/>
        </function>
        <constant symbol="s_1"/>
       </constantatomformula>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <function symbol="∪">
          <constant symbol="s_3"/>
          <constant symbol="s_4"/>
         </function>
         <constant symbol="s_3"/>
        </function>
        <constant symbol="s_3"/>
       </constantatomformula>
      </formulalist>
      <formulalist>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <constant symbol="s_1"/>
         <constant symbol="s_1"/>
        </function>
        <constant symbol="s_1"/>
       </constantatomformula>
       <conjunctiveformula type="and">
        <constantatomformula symbol="=">
         <function symbol="∪">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
         <constant symbol="s_1"/>
        </constantatomformula>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
          <constant symbol="s_3"/>
         </function>
         <constant symbol="s_3"/>
        </constantatomformula>
       </conjunctiveformula>
      </formulalist>
     </sequent>
     <rule type="weakr" symbol="w:r">
      <sequent>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <constant symbol="s_1"/>
          <constant symbol="s_1"/>
         </function>
         <constant symbol="s_1"/>
        </constantatomformula>
       </formulalist>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <constant symbol="s_1"/>
          <constant symbol="s_1"/>
         </function>
         <constant symbol="s_1"/>
        </constantatomformula>
        <constantatomformula symbol="=">
         <function symbol="∪">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
         <constant symbol="s_1"/>
        </constantatomformula>
       </formulalist>
      </sequent>
      <rule type="axiom" symbol="axiom">
       <sequent>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_1"/>
          </function>
          <constant symbol="s_1"/>
         </constantatomformula>
        </formulalist>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_1"/>
          </function>
          <constant symbol="s_1"/>
         </constantatomformula>
        </formulalist>
       </sequent>
      </rule>
     </rule>
     <rule type="axiom" symbol="axiom">
      <sequent>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
          <constant symbol="s_3"/>
         </function>
         <constant symbol="s_3"/>
        </constantatomformula>
       </formulalist>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
          <constant symbol="s_3"/>
         </function>
         <constant symbol="s_3"/>
        </constantatomformula>
       </formulalist>
      </sequent>
     </rule>
    </rule>
   </rule>
  </rule>
 </proof>
 <proof symbol="\psi_{17}" calculus="LK">
  <rule type="weakl" symbol="w:l">
   <sequent>
    <formulalist>
     <constantatomformula symbol="L1" definition="L1"/>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <constant symbol="s_1"/>
       <constant symbol="s_1"/>
      </function>
      <constant symbol="s_1"/>
     </constantatomformula>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <constant symbol="s_3"/>
       <function symbol="∩">
        <function symbol="∪">
         <constant symbol="s_3"/>
         <constant symbol="s_4"/>
        </function>
        <constant symbol="s_3"/>
       </function>
      </function>
      <constant symbol="s_3"/>
     </constantatomformula>
    </formulalist>
    <formulalist>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <constant symbol="s_1"/>
       <constant symbol="s_1"/>
      </function>
      <constant symbol="s_1"/>
     </constantatomformula>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <constant symbol="s_3"/>
       <function symbol="∩">
        <function symbol="∪">
         <constant symbol="s_3"/>
         <constant symbol="s_4"/>
        </function>
        <constant symbol="s_3"/>
       </function>
      </function>
      <constant symbol="s_3"/>
     </constantatomformula>
     <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
    </formulalist>
   </sequent>
   <rule type="defr" symbol="d:r" param="();[()];()">
    <sequent>
     <formulalist>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <constant symbol="s_1"/>
        <constant symbol="s_1"/>
       </function>
       <constant symbol="s_1"/>
      </constantatomformula>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <constant symbol="s_3"/>
        <function symbol="∩">
         <function symbol="∪">
          <constant symbol="s_3"/>
          <constant symbol="s_4"/>
         </function>
         <constant symbol="s_3"/>
        </function>
       </function>
       <constant symbol="s_3"/>
      </constantatomformula>
     </formulalist>
     <formulalist>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <constant symbol="s_1"/>
        <constant symbol="s_1"/>
       </function>
       <constant symbol="s_1"/>
      </constantatomformula>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <constant symbol="s_3"/>
        <function symbol="∩">
         <function symbol="∪">
          <constant symbol="s_3"/>
          <constant symbol="s_4"/>
         </function>
         <constant symbol="s_3"/>
        </function>
       </function>
       <constant symbol="s_3"/>
      </constantatomformula>
      <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
     </formulalist>
    </sequent>
    <rule type="andr" symbol="\and:r">
     <sequent>
      <formulalist>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <constant symbol="s_1"/>
         <constant symbol="s_1"/>
        </function>
        <constant symbol="s_1"/>
       </constantatomformula>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <constant symbol="s_3"/>
         <function symbol="∩">
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
          <constant symbol="s_3"/>
         </function>
        </function>
        <constant symbol="s_3"/>
       </constantatomformula>
      </formulalist>
      <formulalist>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <constant symbol="s_1"/>
         <constant symbol="s_1"/>
        </function>
        <constant symbol="s_1"/>
       </constantatomformula>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <constant symbol="s_3"/>
         <function symbol="∩">
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
          <constant symbol="s_3"/>
         </function>
        </function>
        <constant symbol="s_3"/>
       </constantatomformula>
       <conjunctiveformula type="and">
        <constantatomformula symbol="=">
         <function symbol="∪">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
         <constant symbol="s_1"/>
        </constantatomformula>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
          <constant symbol="s_3"/>
         </function>
         <constant symbol="s_3"/>
        </constantatomformula>
       </conjunctiveformula>
      </formulalist>
     </sequent>
     <rule type="weakr" symbol="w:r">
      <sequent>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <constant symbol="s_1"/>
          <constant symbol="s_1"/>
         </function>
         <constant symbol="s_1"/>
        </constantatomformula>
       </formulalist>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <constant symbol="s_1"/>
          <constant symbol="s_1"/>
         </function>
         <constant symbol="s_1"/>
        </constantatomformula>
        <constantatomformula symbol="=">
         <function symbol="∪">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
         <constant symbol="s_1"/>
        </constantatomformula>
       </formulalist>
      </sequent>
      <rule type="axiom" symbol="axiom">
       <sequent>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_1"/>
          </function>
          <constant symbol="s_1"/>
         </constantatomformula>
        </formulalist>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_1"/>
          </function>
          <constant symbol="s_1"/>
         </constantatomformula>
        </formulalist>
       </sequent>
      </rule>
     </rule>
     <rule type="weakr" symbol="w:r">
      <sequent>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <constant symbol="s_3"/>
          <function symbol="∩">
           <function symbol="∪">
            <constant symbol="s_3"/>
            <constant symbol="s_4"/>
           </function>
           <constant symbol="s_3"/>
          </function>
         </function>
         <constant symbol="s_3"/>
        </constantatomformula>
       </formulalist>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <constant symbol="s_3"/>
          <function symbol="∩">
           <function symbol="∪">
            <constant symbol="s_3"/>
            <constant symbol="s_4"/>
           </function>
           <constant symbol="s_3"/>
          </function>
         </function>
         <constant symbol="s_3"/>
        </constantatomformula>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
          <constant symbol="s_3"/>
         </function>
         <constant symbol="s_3"/>
        </constantatomformula>
       </formulalist>
      </sequent>
      <rule type="axiom" symbol="axiom">
       <sequent>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∩">
           <constant symbol="s_3"/>
           <function symbol="∩">
            <function symbol="∪">
             <constant symbol="s_3"/>
             <constant symbol="s_4"/>
            </function>
            <constant symbol="s_3"/>
           </function>
          </function>
          <constant symbol="s_3"/>
         </constantatomformula>
        </formulalist>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∩">
           <constant symbol="s_3"/>
           <function symbol="∩">
            <function symbol="∪">
             <constant symbol="s_3"/>
             <constant symbol="s_4"/>
            </function>
            <constant symbol="s_3"/>
           </function>
          </function>
          <constant symbol="s_3"/>
         </constantatomformula>
        </formulalist>
       </sequent>
      </rule>
     </rule>
    </rule>
   </rule>
  </rule>
 </proof>
 <proof symbol="\psi_{18}" calculus="LK">
  <rule type="weakl" symbol="w:l">
   <sequent>
    <formulalist>
     <constantatomformula symbol="L1" definition="L1"/>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <constant symbol="s_1"/>
       <constant symbol="s_1"/>
      </function>
      <constant symbol="s_1"/>
     </constantatomformula>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <constant symbol="s_3"/>
       <constant symbol="s_3"/>
      </function>
      <constant symbol="s_3"/>
     </constantatomformula>
    </formulalist>
    <formulalist>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <constant symbol="s_1"/>
       <constant symbol="s_1"/>
      </function>
      <constant symbol="s_1"/>
     </constantatomformula>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <constant symbol="s_3"/>
       <constant symbol="s_3"/>
      </function>
      <constant symbol="s_3"/>
     </constantatomformula>
     <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
    </formulalist>
   </sequent>
   <rule type="defr" symbol="d:r" param="();[()];()">
    <sequent>
     <formulalist>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <constant symbol="s_1"/>
        <constant symbol="s_1"/>
       </function>
       <constant symbol="s_1"/>
      </constantatomformula>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <constant symbol="s_3"/>
        <constant symbol="s_3"/>
       </function>
       <constant symbol="s_3"/>
      </constantatomformula>
     </formulalist>
     <formulalist>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <constant symbol="s_1"/>
        <constant symbol="s_1"/>
       </function>
       <constant symbol="s_1"/>
      </constantatomformula>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <constant symbol="s_3"/>
        <constant symbol="s_3"/>
       </function>
       <constant symbol="s_3"/>
      </constantatomformula>
      <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
     </formulalist>
    </sequent>
    <rule type="andr" symbol="\and:r">
     <sequent>
      <formulalist>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <constant symbol="s_1"/>
         <constant symbol="s_1"/>
        </function>
        <constant symbol="s_1"/>
       </constantatomformula>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <constant symbol="s_3"/>
         <constant symbol="s_3"/>
        </function>
        <constant symbol="s_3"/>
       </constantatomformula>
      </formulalist>
      <formulalist>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <constant symbol="s_1"/>
         <constant symbol="s_1"/>
        </function>
        <constant symbol="s_1"/>
       </constantatomformula>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <constant symbol="s_3"/>
         <constant symbol="s_3"/>
        </function>
        <constant symbol="s_3"/>
       </constantatomformula>
       <conjunctiveformula type="and">
        <constantatomformula symbol="=">
         <function symbol="∪">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
         <constant symbol="s_1"/>
        </constantatomformula>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
          <constant symbol="s_3"/>
         </function>
         <constant symbol="s_3"/>
        </constantatomformula>
       </conjunctiveformula>
      </formulalist>
     </sequent>
     <rule type="weakr" symbol="w:r">
      <sequent>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <constant symbol="s_1"/>
          <constant symbol="s_1"/>
         </function>
         <constant symbol="s_1"/>
        </constantatomformula>
       </formulalist>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <constant symbol="s_1"/>
          <constant symbol="s_1"/>
         </function>
         <constant symbol="s_1"/>
        </constantatomformula>
        <constantatomformula symbol="=">
         <function symbol="∪">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
         <constant symbol="s_1"/>
        </constantatomformula>
       </formulalist>
      </sequent>
      <rule type="axiom" symbol="axiom">
       <sequent>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_1"/>
          </function>
          <constant symbol="s_1"/>
         </constantatomformula>
        </formulalist>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_1"/>
          </function>
          <constant symbol="s_1"/>
         </constantatomformula>
        </formulalist>
       </sequent>
      </rule>
     </rule>
     <rule type="weakr" symbol="w:r">
      <sequent>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <constant symbol="s_3"/>
          <constant symbol="s_3"/>
         </function>
         <constant symbol="s_3"/>
        </constantatomformula>
       </formulalist>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <constant symbol="s_3"/>
          <constant symbol="s_3"/>
         </function>
         <constant symbol="s_3"/>
        </constantatomformula>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
          <constant symbol="s_3"/>
         </function>
         <constant symbol="s_3"/>
        </constantatomformula>
       </formulalist>
      </sequent>
      <rule type="axiom" symbol="axiom">
       <sequent>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∩">
           <constant symbol="s_3"/>
           <constant symbol="s_3"/>
          </function>
          <constant symbol="s_3"/>
         </constantatomformula>
        </formulalist>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∩">
           <constant symbol="s_3"/>
           <constant symbol="s_3"/>
          </function>
          <constant symbol="s_3"/>
         </constantatomformula>
        </formulalist>
       </sequent>
      </rule>
     </rule>
    </rule>
   </rule>
  </rule>
 </proof>
 <proof symbol="\psi_{19}" calculus="LK">
  <rule type="weakl" symbol="w:l">
   <sequent>
    <formulalist>
     <constantatomformula symbol="L1" definition="L1"/>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <constant symbol="s_1"/>
       <constant symbol="s_1"/>
      </function>
      <constant symbol="s_1"/>
     </constantatomformula>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <constant symbol="s_3"/>
       <function symbol="∪">
        <constant symbol="s_3"/>
        <constant symbol="s_4"/>
       </function>
      </function>
      <constant symbol="s_3"/>
     </constantatomformula>
    </formulalist>
    <formulalist>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <constant symbol="s_1"/>
       <constant symbol="s_1"/>
      </function>
      <constant symbol="s_1"/>
     </constantatomformula>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <constant symbol="s_3"/>
       <function symbol="∪">
        <constant symbol="s_3"/>
        <constant symbol="s_4"/>
       </function>
      </function>
      <constant symbol="s_3"/>
     </constantatomformula>
     <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
    </formulalist>
   </sequent>
   <rule type="defr" symbol="d:r" param="();[()];()">
    <sequent>
     <formulalist>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <constant symbol="s_1"/>
        <constant symbol="s_1"/>
       </function>
       <constant symbol="s_1"/>
      </constantatomformula>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <constant symbol="s_3"/>
        <function symbol="∪">
         <constant symbol="s_3"/>
         <constant symbol="s_4"/>
        </function>
       </function>
       <constant symbol="s_3"/>
      </constantatomformula>
     </formulalist>
     <formulalist>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <constant symbol="s_1"/>
        <constant symbol="s_1"/>
       </function>
       <constant symbol="s_1"/>
      </constantatomformula>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <constant symbol="s_3"/>
        <function symbol="∪">
         <constant symbol="s_3"/>
         <constant symbol="s_4"/>
        </function>
       </function>
       <constant symbol="s_3"/>
      </constantatomformula>
      <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
     </formulalist>
    </sequent>
    <rule type="andr" symbol="\and:r">
     <sequent>
      <formulalist>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <constant symbol="s_1"/>
         <constant symbol="s_1"/>
        </function>
        <constant symbol="s_1"/>
       </constantatomformula>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <constant symbol="s_3"/>
         <function symbol="∪">
          <constant symbol="s_3"/>
          <constant symbol="s_4"/>
         </function>
        </function>
        <constant symbol="s_3"/>
       </constantatomformula>
      </formulalist>
      <formulalist>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <constant symbol="s_1"/>
         <constant symbol="s_1"/>
        </function>
        <constant symbol="s_1"/>
       </constantatomformula>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <constant symbol="s_3"/>
         <function symbol="∪">
          <constant symbol="s_3"/>
          <constant symbol="s_4"/>
         </function>
        </function>
        <constant symbol="s_3"/>
       </constantatomformula>
       <conjunctiveformula type="and">
        <constantatomformula symbol="=">
         <function symbol="∪">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
         <constant symbol="s_1"/>
        </constantatomformula>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
          <constant symbol="s_3"/>
         </function>
         <constant symbol="s_3"/>
        </constantatomformula>
       </conjunctiveformula>
      </formulalist>
     </sequent>
     <rule type="weakr" symbol="w:r">
      <sequent>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <constant symbol="s_1"/>
          <constant symbol="s_1"/>
         </function>
         <constant symbol="s_1"/>
        </constantatomformula>
       </formulalist>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <constant symbol="s_1"/>
          <constant symbol="s_1"/>
         </function>
         <constant symbol="s_1"/>
        </constantatomformula>
        <constantatomformula symbol="=">
         <function symbol="∪">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
         <constant symbol="s_1"/>
        </constantatomformula>
       </formulalist>
      </sequent>
      <rule type="axiom" symbol="axiom">
       <sequent>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_1"/>
          </function>
          <constant symbol="s_1"/>
         </constantatomformula>
        </formulalist>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_1"/>
          </function>
          <constant symbol="s_1"/>
         </constantatomformula>
        </formulalist>
       </sequent>
      </rule>
     </rule>
     <rule type="weakr" symbol="w:r">
      <sequent>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <constant symbol="s_3"/>
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
         </function>
         <constant symbol="s_3"/>
        </constantatomformula>
       </formulalist>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <constant symbol="s_3"/>
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
         </function>
         <constant symbol="s_3"/>
        </constantatomformula>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
          <constant symbol="s_3"/>
         </function>
         <constant symbol="s_3"/>
        </constantatomformula>
       </formulalist>
      </sequent>
      <rule type="axiom" symbol="axiom">
       <sequent>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∩">
           <constant symbol="s_3"/>
           <function symbol="∪">
            <constant symbol="s_3"/>
            <constant symbol="s_4"/>
           </function>
          </function>
          <constant symbol="s_3"/>
         </constantatomformula>
        </formulalist>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∩">
           <constant symbol="s_3"/>
           <function symbol="∪">
            <constant symbol="s_3"/>
            <constant symbol="s_4"/>
           </function>
          </function>
          <constant symbol="s_3"/>
         </constantatomformula>
        </formulalist>
       </sequent>
      </rule>
     </rule>
    </rule>
   </rule>
  </rule>
 </proof>
 <proof symbol="\psi_{20}" calculus="LK">
  <rule type="weakl" symbol="w:l">
   <sequent>
    <formulalist>
     <constantatomformula symbol="L1" definition="L1"/>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <constant symbol="s_1"/>
       <constant symbol="s_1"/>
      </function>
      <constant symbol="s_1"/>
     </constantatomformula>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <function symbol="∩">
        <function symbol="∪">
         <constant symbol="s_3"/>
         <constant symbol="s_4"/>
        </function>
        <constant symbol="s_3"/>
       </function>
       <constant symbol="s_3"/>
      </function>
      <function symbol="∩">
       <function symbol="∪">
        <constant symbol="s_3"/>
        <constant symbol="s_4"/>
       </function>
       <constant symbol="s_3"/>
      </function>
     </constantatomformula>
    </formulalist>
    <formulalist>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <constant symbol="s_1"/>
       <constant symbol="s_1"/>
      </function>
      <constant symbol="s_1"/>
     </constantatomformula>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <function symbol="∩">
        <function symbol="∪">
         <constant symbol="s_3"/>
         <constant symbol="s_4"/>
        </function>
        <constant symbol="s_3"/>
       </function>
       <constant symbol="s_3"/>
      </function>
      <function symbol="∩">
       <function symbol="∪">
        <constant symbol="s_3"/>
        <constant symbol="s_4"/>
       </function>
       <constant symbol="s_3"/>
      </function>
     </constantatomformula>
     <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
    </formulalist>
   </sequent>
   <rule type="defr" symbol="d:r" param="();[()];()">
    <sequent>
     <formulalist>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <constant symbol="s_1"/>
        <constant symbol="s_1"/>
       </function>
       <constant symbol="s_1"/>
      </constantatomformula>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <function symbol="∩">
         <function symbol="∪">
          <constant symbol="s_3"/>
          <constant symbol="s_4"/>
         </function>
         <constant symbol="s_3"/>
        </function>
        <constant symbol="s_3"/>
       </function>
       <function symbol="∩">
        <function symbol="∪">
         <constant symbol="s_3"/>
         <constant symbol="s_4"/>
        </function>
        <constant symbol="s_3"/>
       </function>
      </constantatomformula>
     </formulalist>
     <formulalist>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <constant symbol="s_1"/>
        <constant symbol="s_1"/>
       </function>
       <constant symbol="s_1"/>
      </constantatomformula>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <function symbol="∩">
         <function symbol="∪">
          <constant symbol="s_3"/>
          <constant symbol="s_4"/>
         </function>
         <constant symbol="s_3"/>
        </function>
        <constant symbol="s_3"/>
       </function>
       <function symbol="∩">
        <function symbol="∪">
         <constant symbol="s_3"/>
         <constant symbol="s_4"/>
        </function>
        <constant symbol="s_3"/>
       </function>
      </constantatomformula>
      <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
     </formulalist>
    </sequent>
    <rule type="andr" symbol="\and:r">
     <sequent>
      <formulalist>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <constant symbol="s_1"/>
         <constant symbol="s_1"/>
        </function>
        <constant symbol="s_1"/>
       </constantatomformula>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <function symbol="∩">
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
          <constant symbol="s_3"/>
         </function>
         <constant symbol="s_3"/>
        </function>
        <function symbol="∩">
         <function symbol="∪">
          <constant symbol="s_3"/>
          <constant symbol="s_4"/>
         </function>
         <constant symbol="s_3"/>
        </function>
       </constantatomformula>
      </formulalist>
      <formulalist>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <constant symbol="s_1"/>
         <constant symbol="s_1"/>
        </function>
        <constant symbol="s_1"/>
       </constantatomformula>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <function symbol="∩">
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
          <constant symbol="s_3"/>
         </function>
         <constant symbol="s_3"/>
        </function>
        <function symbol="∩">
         <function symbol="∪">
          <constant symbol="s_3"/>
          <constant symbol="s_4"/>
         </function>
         <constant symbol="s_3"/>
        </function>
       </constantatomformula>
       <conjunctiveformula type="and">
        <constantatomformula symbol="=">
         <function symbol="∪">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
         <constant symbol="s_1"/>
        </constantatomformula>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
          <constant symbol="s_3"/>
         </function>
         <constant symbol="s_3"/>
        </constantatomformula>
       </conjunctiveformula>
      </formulalist>
     </sequent>
     <rule type="weakr" symbol="w:r">
      <sequent>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <constant symbol="s_1"/>
          <constant symbol="s_1"/>
         </function>
         <constant symbol="s_1"/>
        </constantatomformula>
       </formulalist>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <constant symbol="s_1"/>
          <constant symbol="s_1"/>
         </function>
         <constant symbol="s_1"/>
        </constantatomformula>
        <constantatomformula symbol="=">
         <function symbol="∪">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
         <constant symbol="s_1"/>
        </constantatomformula>
       </formulalist>
      </sequent>
      <rule type="axiom" symbol="axiom">
       <sequent>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_1"/>
          </function>
          <constant symbol="s_1"/>
         </constantatomformula>
        </formulalist>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_1"/>
          </function>
          <constant symbol="s_1"/>
         </constantatomformula>
        </formulalist>
       </sequent>
      </rule>
     </rule>
     <rule type="weakr" symbol="w:r">
      <sequent>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <function symbol="∩">
           <function symbol="∪">
            <constant symbol="s_3"/>
            <constant symbol="s_4"/>
           </function>
           <constant symbol="s_3"/>
          </function>
          <constant symbol="s_3"/>
         </function>
         <function symbol="∩">
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
          <constant symbol="s_3"/>
         </function>
        </constantatomformula>
       </formulalist>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <function symbol="∩">
           <function symbol="∪">
            <constant symbol="s_3"/>
            <constant symbol="s_4"/>
           </function>
           <constant symbol="s_3"/>
          </function>
          <constant symbol="s_3"/>
         </function>
         <function symbol="∩">
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
          <constant symbol="s_3"/>
         </function>
        </constantatomformula>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
          <constant symbol="s_3"/>
         </function>
         <constant symbol="s_3"/>
        </constantatomformula>
       </formulalist>
      </sequent>
      <rule type="axiom" symbol="axiom">
       <sequent>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∩">
           <function symbol="∩">
            <function symbol="∪">
             <constant symbol="s_3"/>
             <constant symbol="s_4"/>
            </function>
            <constant symbol="s_3"/>
           </function>
           <constant symbol="s_3"/>
          </function>
          <function symbol="∩">
           <function symbol="∪">
            <constant symbol="s_3"/>
            <constant symbol="s_4"/>
           </function>
           <constant symbol="s_3"/>
          </function>
         </constantatomformula>
        </formulalist>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∩">
           <function symbol="∩">
            <function symbol="∪">
             <constant symbol="s_3"/>
             <constant symbol="s_4"/>
            </function>
            <constant symbol="s_3"/>
           </function>
           <constant symbol="s_3"/>
          </function>
          <function symbol="∩">
           <function symbol="∪">
            <constant symbol="s_3"/>
            <constant symbol="s_4"/>
           </function>
           <constant symbol="s_3"/>
          </function>
         </constantatomformula>
        </formulalist>
       </sequent>
      </rule>
     </rule>
    </rule>
   </rule>
  </rule>
 </proof>
 <proof symbol="\psi_{21}" calculus="LK">
  <rule type="weakl" symbol="w:l">
   <sequent>
    <formulalist>
     <constantatomformula symbol="L1" definition="L1"/>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <function symbol="∩">
        <constant symbol="s_1"/>
        <constant symbol="s_2"/>
       </function>
       <constant symbol="s_1"/>
      </function>
      <function symbol="∩">
       <constant symbol="s_1"/>
       <constant symbol="s_2"/>
      </function>
     </constantatomformula>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <function symbol="∪">
        <constant symbol="s_3"/>
        <constant symbol="s_4"/>
       </function>
       <constant symbol="s_3"/>
      </function>
      <constant symbol="s_3"/>
     </constantatomformula>
    </formulalist>
    <formulalist>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <function symbol="∩">
        <constant symbol="s_1"/>
        <constant symbol="s_2"/>
       </function>
       <constant symbol="s_1"/>
      </function>
      <function symbol="∩">
       <constant symbol="s_1"/>
       <constant symbol="s_2"/>
      </function>
     </constantatomformula>
     <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
    </formulalist>
   </sequent>
   <rule type="defr" symbol="d:r" param="();[()];()">
    <sequent>
     <formulalist>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <function symbol="∩">
         <constant symbol="s_1"/>
         <constant symbol="s_2"/>
        </function>
        <constant symbol="s_1"/>
       </function>
       <function symbol="∩">
        <constant symbol="s_1"/>
        <constant symbol="s_2"/>
       </function>
      </constantatomformula>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <function symbol="∪">
         <constant symbol="s_3"/>
         <constant symbol="s_4"/>
        </function>
        <constant symbol="s_3"/>
       </function>
       <constant symbol="s_3"/>
      </constantatomformula>
     </formulalist>
     <formulalist>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <function symbol="∩">
         <constant symbol="s_1"/>
         <constant symbol="s_2"/>
        </function>
        <constant symbol="s_1"/>
       </function>
       <function symbol="∩">
        <constant symbol="s_1"/>
        <constant symbol="s_2"/>
       </function>
      </constantatomformula>
      <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
     </formulalist>
    </sequent>
    <rule type="andr" symbol="\and:r">
     <sequent>
      <formulalist>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <function symbol="∩">
          <constant symbol="s_1"/>
          <constant symbol="s_2"/>
         </function>
         <constant symbol="s_1"/>
        </function>
        <function symbol="∩">
         <constant symbol="s_1"/>
         <constant symbol="s_2"/>
        </function>
       </constantatomformula>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <function symbol="∪">
          <constant symbol="s_3"/>
          <constant symbol="s_4"/>
         </function>
         <constant symbol="s_3"/>
        </function>
        <constant symbol="s_3"/>
       </constantatomformula>
      </formulalist>
      <formulalist>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <function symbol="∩">
          <constant symbol="s_1"/>
          <constant symbol="s_2"/>
         </function>
         <constant symbol="s_1"/>
        </function>
        <function symbol="∩">
         <constant symbol="s_1"/>
         <constant symbol="s_2"/>
        </function>
       </constantatomformula>
       <conjunctiveformula type="and">
        <constantatomformula symbol="=">
         <function symbol="∪">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
         <constant symbol="s_1"/>
        </constantatomformula>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
          <constant symbol="s_3"/>
         </function>
         <constant symbol="s_3"/>
        </constantatomformula>
       </conjunctiveformula>
      </formulalist>
     </sequent>
     <rule type="weakr" symbol="w:r">
      <sequent>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
         <function symbol="∩">
          <constant symbol="s_1"/>
          <constant symbol="s_2"/>
         </function>
        </constantatomformula>
       </formulalist>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
         <function symbol="∩">
          <constant symbol="s_1"/>
          <constant symbol="s_2"/>
         </function>
        </constantatomformula>
        <constantatomformula symbol="=">
         <function symbol="∪">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
         <constant symbol="s_1"/>
        </constantatomformula>
       </formulalist>
      </sequent>
      <rule type="axiom" symbol="axiom">
       <sequent>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∩">
           <function symbol="∩">
            <constant symbol="s_1"/>
            <constant symbol="s_2"/>
           </function>
           <constant symbol="s_1"/>
          </function>
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
         </constantatomformula>
        </formulalist>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∩">
           <function symbol="∩">
            <constant symbol="s_1"/>
            <constant symbol="s_2"/>
           </function>
           <constant symbol="s_1"/>
          </function>
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
         </constantatomformula>
        </formulalist>
       </sequent>
      </rule>
     </rule>
     <rule type="axiom" symbol="axiom">
      <sequent>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
          <constant symbol="s_3"/>
         </function>
         <constant symbol="s_3"/>
        </constantatomformula>
       </formulalist>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
          <constant symbol="s_3"/>
         </function>
         <constant symbol="s_3"/>
        </constantatomformula>
       </formulalist>
      </sequent>
     </rule>
    </rule>
   </rule>
  </rule>
 </proof>
 <proof symbol="\psi_{22}" calculus="LK">
  <rule type="weakl" symbol="w:l">
   <sequent>
    <formulalist>
     <constantatomformula symbol="L1" definition="L1"/>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <function symbol="∩">
        <constant symbol="s_1"/>
        <constant symbol="s_2"/>
       </function>
       <constant symbol="s_1"/>
      </function>
      <function symbol="∩">
       <constant symbol="s_1"/>
       <constant symbol="s_2"/>
      </function>
     </constantatomformula>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <constant symbol="s_3"/>
       <function symbol="∩">
        <function symbol="∪">
         <constant symbol="s_3"/>
         <constant symbol="s_4"/>
        </function>
        <constant symbol="s_3"/>
       </function>
      </function>
      <constant symbol="s_3"/>
     </constantatomformula>
    </formulalist>
    <formulalist>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <function symbol="∩">
        <constant symbol="s_1"/>
        <constant symbol="s_2"/>
       </function>
       <constant symbol="s_1"/>
      </function>
      <function symbol="∩">
       <constant symbol="s_1"/>
       <constant symbol="s_2"/>
      </function>
     </constantatomformula>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <constant symbol="s_3"/>
       <function symbol="∩">
        <function symbol="∪">
         <constant symbol="s_3"/>
         <constant symbol="s_4"/>
        </function>
        <constant symbol="s_3"/>
       </function>
      </function>
      <constant symbol="s_3"/>
     </constantatomformula>
     <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
    </formulalist>
   </sequent>
   <rule type="defr" symbol="d:r" param="();[()];()">
    <sequent>
     <formulalist>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <function symbol="∩">
         <constant symbol="s_1"/>
         <constant symbol="s_2"/>
        </function>
        <constant symbol="s_1"/>
       </function>
       <function symbol="∩">
        <constant symbol="s_1"/>
        <constant symbol="s_2"/>
       </function>
      </constantatomformula>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <constant symbol="s_3"/>
        <function symbol="∩">
         <function symbol="∪">
          <constant symbol="s_3"/>
          <constant symbol="s_4"/>
         </function>
         <constant symbol="s_3"/>
        </function>
       </function>
       <constant symbol="s_3"/>
      </constantatomformula>
     </formulalist>
     <formulalist>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <function symbol="∩">
         <constant symbol="s_1"/>
         <constant symbol="s_2"/>
        </function>
        <constant symbol="s_1"/>
       </function>
       <function symbol="∩">
        <constant symbol="s_1"/>
        <constant symbol="s_2"/>
       </function>
      </constantatomformula>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <constant symbol="s_3"/>
        <function symbol="∩">
         <function symbol="∪">
          <constant symbol="s_3"/>
          <constant symbol="s_4"/>
         </function>
         <constant symbol="s_3"/>
        </function>
       </function>
       <constant symbol="s_3"/>
      </constantatomformula>
      <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
     </formulalist>
    </sequent>
    <rule type="andr" symbol="\and:r">
     <sequent>
      <formulalist>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <function symbol="∩">
          <constant symbol="s_1"/>
          <constant symbol="s_2"/>
         </function>
         <constant symbol="s_1"/>
        </function>
        <function symbol="∩">
         <constant symbol="s_1"/>
         <constant symbol="s_2"/>
        </function>
       </constantatomformula>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <constant symbol="s_3"/>
         <function symbol="∩">
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
          <constant symbol="s_3"/>
         </function>
        </function>
        <constant symbol="s_3"/>
       </constantatomformula>
      </formulalist>
      <formulalist>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <function symbol="∩">
          <constant symbol="s_1"/>
          <constant symbol="s_2"/>
         </function>
         <constant symbol="s_1"/>
        </function>
        <function symbol="∩">
         <constant symbol="s_1"/>
         <constant symbol="s_2"/>
        </function>
       </constantatomformula>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <constant symbol="s_3"/>
         <function symbol="∩">
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
          <constant symbol="s_3"/>
         </function>
        </function>
        <constant symbol="s_3"/>
       </constantatomformula>
       <conjunctiveformula type="and">
        <constantatomformula symbol="=">
         <function symbol="∪">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
         <constant symbol="s_1"/>
        </constantatomformula>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
          <constant symbol="s_3"/>
         </function>
         <constant symbol="s_3"/>
        </constantatomformula>
       </conjunctiveformula>
      </formulalist>
     </sequent>
     <rule type="weakr" symbol="w:r">
      <sequent>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
         <function symbol="∩">
          <constant symbol="s_1"/>
          <constant symbol="s_2"/>
         </function>
        </constantatomformula>
       </formulalist>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
         <function symbol="∩">
          <constant symbol="s_1"/>
          <constant symbol="s_2"/>
         </function>
        </constantatomformula>
        <constantatomformula symbol="=">
         <function symbol="∪">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
         <constant symbol="s_1"/>
        </constantatomformula>
       </formulalist>
      </sequent>
      <rule type="axiom" symbol="axiom">
       <sequent>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∩">
           <function symbol="∩">
            <constant symbol="s_1"/>
            <constant symbol="s_2"/>
           </function>
           <constant symbol="s_1"/>
          </function>
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
         </constantatomformula>
        </formulalist>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∩">
           <function symbol="∩">
            <constant symbol="s_1"/>
            <constant symbol="s_2"/>
           </function>
           <constant symbol="s_1"/>
          </function>
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
         </constantatomformula>
        </formulalist>
       </sequent>
      </rule>
     </rule>
     <rule type="weakr" symbol="w:r">
      <sequent>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <constant symbol="s_3"/>
          <function symbol="∩">
           <function symbol="∪">
            <constant symbol="s_3"/>
            <constant symbol="s_4"/>
           </function>
           <constant symbol="s_3"/>
          </function>
         </function>
         <constant symbol="s_3"/>
        </constantatomformula>
       </formulalist>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <constant symbol="s_3"/>
          <function symbol="∩">
           <function symbol="∪">
            <constant symbol="s_3"/>
            <constant symbol="s_4"/>
           </function>
           <constant symbol="s_3"/>
          </function>
         </function>
         <constant symbol="s_3"/>
        </constantatomformula>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
          <constant symbol="s_3"/>
         </function>
         <constant symbol="s_3"/>
        </constantatomformula>
       </formulalist>
      </sequent>
      <rule type="axiom" symbol="axiom">
       <sequent>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∩">
           <constant symbol="s_3"/>
           <function symbol="∩">
            <function symbol="∪">
             <constant symbol="s_3"/>
             <constant symbol="s_4"/>
            </function>
            <constant symbol="s_3"/>
           </function>
          </function>
          <constant symbol="s_3"/>
         </constantatomformula>
        </formulalist>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∩">
           <constant symbol="s_3"/>
           <function symbol="∩">
            <function symbol="∪">
             <constant symbol="s_3"/>
             <constant symbol="s_4"/>
            </function>
            <constant symbol="s_3"/>
           </function>
          </function>
          <constant symbol="s_3"/>
         </constantatomformula>
        </formulalist>
       </sequent>
      </rule>
     </rule>
    </rule>
   </rule>
  </rule>
 </proof>
 <proof symbol="\psi_{23}" calculus="LK">
  <rule type="weakl" symbol="w:l">
   <sequent>
    <formulalist>
     <constantatomformula symbol="L1" definition="L1"/>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <function symbol="∩">
        <constant symbol="s_1"/>
        <constant symbol="s_2"/>
       </function>
       <constant symbol="s_1"/>
      </function>
      <function symbol="∩">
       <constant symbol="s_1"/>
       <constant symbol="s_2"/>
      </function>
     </constantatomformula>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <constant symbol="s_3"/>
       <constant symbol="s_3"/>
      </function>
      <constant symbol="s_3"/>
     </constantatomformula>
    </formulalist>
    <formulalist>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <function symbol="∩">
        <constant symbol="s_1"/>
        <constant symbol="s_2"/>
       </function>
       <constant symbol="s_1"/>
      </function>
      <function symbol="∩">
       <constant symbol="s_1"/>
       <constant symbol="s_2"/>
      </function>
     </constantatomformula>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <constant symbol="s_3"/>
       <constant symbol="s_3"/>
      </function>
      <constant symbol="s_3"/>
     </constantatomformula>
     <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
    </formulalist>
   </sequent>
   <rule type="defr" symbol="d:r" param="();[()];()">
    <sequent>
     <formulalist>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <function symbol="∩">
         <constant symbol="s_1"/>
         <constant symbol="s_2"/>
        </function>
        <constant symbol="s_1"/>
       </function>
       <function symbol="∩">
        <constant symbol="s_1"/>
        <constant symbol="s_2"/>
       </function>
      </constantatomformula>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <constant symbol="s_3"/>
        <constant symbol="s_3"/>
       </function>
       <constant symbol="s_3"/>
      </constantatomformula>
     </formulalist>
     <formulalist>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <function symbol="∩">
         <constant symbol="s_1"/>
         <constant symbol="s_2"/>
        </function>
        <constant symbol="s_1"/>
       </function>
       <function symbol="∩">
        <constant symbol="s_1"/>
        <constant symbol="s_2"/>
       </function>
      </constantatomformula>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <constant symbol="s_3"/>
        <constant symbol="s_3"/>
       </function>
       <constant symbol="s_3"/>
      </constantatomformula>
      <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
     </formulalist>
    </sequent>
    <rule type="andr" symbol="\and:r">
     <sequent>
      <formulalist>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <function symbol="∩">
          <constant symbol="s_1"/>
          <constant symbol="s_2"/>
         </function>
         <constant symbol="s_1"/>
        </function>
        <function symbol="∩">
         <constant symbol="s_1"/>
         <constant symbol="s_2"/>
        </function>
       </constantatomformula>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <constant symbol="s_3"/>
         <constant symbol="s_3"/>
        </function>
        <constant symbol="s_3"/>
       </constantatomformula>
      </formulalist>
      <formulalist>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <function symbol="∩">
          <constant symbol="s_1"/>
          <constant symbol="s_2"/>
         </function>
         <constant symbol="s_1"/>
        </function>
        <function symbol="∩">
         <constant symbol="s_1"/>
         <constant symbol="s_2"/>
        </function>
       </constantatomformula>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <constant symbol="s_3"/>
         <constant symbol="s_3"/>
        </function>
        <constant symbol="s_3"/>
       </constantatomformula>
       <conjunctiveformula type="and">
        <constantatomformula symbol="=">
         <function symbol="∪">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
         <constant symbol="s_1"/>
        </constantatomformula>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
          <constant symbol="s_3"/>
         </function>
         <constant symbol="s_3"/>
        </constantatomformula>
       </conjunctiveformula>
      </formulalist>
     </sequent>
     <rule type="weakr" symbol="w:r">
      <sequent>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
         <function symbol="∩">
          <constant symbol="s_1"/>
          <constant symbol="s_2"/>
         </function>
        </constantatomformula>
       </formulalist>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
         <function symbol="∩">
          <constant symbol="s_1"/>
          <constant symbol="s_2"/>
         </function>
        </constantatomformula>
        <constantatomformula symbol="=">
         <function symbol="∪">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
         <constant symbol="s_1"/>
        </constantatomformula>
       </formulalist>
      </sequent>
      <rule type="axiom" symbol="axiom">
       <sequent>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∩">
           <function symbol="∩">
            <constant symbol="s_1"/>
            <constant symbol="s_2"/>
           </function>
           <constant symbol="s_1"/>
          </function>
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
         </constantatomformula>
        </formulalist>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∩">
           <function symbol="∩">
            <constant symbol="s_1"/>
            <constant symbol="s_2"/>
           </function>
           <constant symbol="s_1"/>
          </function>
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
         </constantatomformula>
        </formulalist>
       </sequent>
      </rule>
     </rule>
     <rule type="weakr" symbol="w:r">
      <sequent>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <constant symbol="s_3"/>
          <constant symbol="s_3"/>
         </function>
         <constant symbol="s_3"/>
        </constantatomformula>
       </formulalist>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <constant symbol="s_3"/>
          <constant symbol="s_3"/>
         </function>
         <constant symbol="s_3"/>
        </constantatomformula>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
          <constant symbol="s_3"/>
         </function>
         <constant symbol="s_3"/>
        </constantatomformula>
       </formulalist>
      </sequent>
      <rule type="axiom" symbol="axiom">
       <sequent>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∩">
           <constant symbol="s_3"/>
           <constant symbol="s_3"/>
          </function>
          <constant symbol="s_3"/>
         </constantatomformula>
        </formulalist>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∩">
           <constant symbol="s_3"/>
           <constant symbol="s_3"/>
          </function>
          <constant symbol="s_3"/>
         </constantatomformula>
        </formulalist>
       </sequent>
      </rule>
     </rule>
    </rule>
   </rule>
  </rule>
 </proof>
 <proof symbol="\psi_{24}" calculus="LK">
  <rule type="weakl" symbol="w:l">
   <sequent>
    <formulalist>
     <constantatomformula symbol="L1" definition="L1"/>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <function symbol="∩">
        <constant symbol="s_1"/>
        <constant symbol="s_2"/>
       </function>
       <constant symbol="s_1"/>
      </function>
      <function symbol="∩">
       <constant symbol="s_1"/>
       <constant symbol="s_2"/>
      </function>
     </constantatomformula>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <constant symbol="s_3"/>
       <function symbol="∪">
        <constant symbol="s_3"/>
        <constant symbol="s_4"/>
       </function>
      </function>
      <constant symbol="s_3"/>
     </constantatomformula>
    </formulalist>
    <formulalist>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <function symbol="∩">
        <constant symbol="s_1"/>
        <constant symbol="s_2"/>
       </function>
       <constant symbol="s_1"/>
      </function>
      <function symbol="∩">
       <constant symbol="s_1"/>
       <constant symbol="s_2"/>
      </function>
     </constantatomformula>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <constant symbol="s_3"/>
       <function symbol="∪">
        <constant symbol="s_3"/>
        <constant symbol="s_4"/>
       </function>
      </function>
      <constant symbol="s_3"/>
     </constantatomformula>
     <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
    </formulalist>
   </sequent>
   <rule type="defr" symbol="d:r" param="();[()];()">
    <sequent>
     <formulalist>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <function symbol="∩">
         <constant symbol="s_1"/>
         <constant symbol="s_2"/>
        </function>
        <constant symbol="s_1"/>
       </function>
       <function symbol="∩">
        <constant symbol="s_1"/>
        <constant symbol="s_2"/>
       </function>
      </constantatomformula>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <constant symbol="s_3"/>
        <function symbol="∪">
         <constant symbol="s_3"/>
         <constant symbol="s_4"/>
        </function>
       </function>
       <constant symbol="s_3"/>
      </constantatomformula>
     </formulalist>
     <formulalist>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <function symbol="∩">
         <constant symbol="s_1"/>
         <constant symbol="s_2"/>
        </function>
        <constant symbol="s_1"/>
       </function>
       <function symbol="∩">
        <constant symbol="s_1"/>
        <constant symbol="s_2"/>
       </function>
      </constantatomformula>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <constant symbol="s_3"/>
        <function symbol="∪">
         <constant symbol="s_3"/>
         <constant symbol="s_4"/>
        </function>
       </function>
       <constant symbol="s_3"/>
      </constantatomformula>
      <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
     </formulalist>
    </sequent>
    <rule type="andr" symbol="\and:r">
     <sequent>
      <formulalist>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <function symbol="∩">
          <constant symbol="s_1"/>
          <constant symbol="s_2"/>
         </function>
         <constant symbol="s_1"/>
        </function>
        <function symbol="∩">
         <constant symbol="s_1"/>
         <constant symbol="s_2"/>
        </function>
       </constantatomformula>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <constant symbol="s_3"/>
         <function symbol="∪">
          <constant symbol="s_3"/>
          <constant symbol="s_4"/>
         </function>
        </function>
        <constant symbol="s_3"/>
       </constantatomformula>
      </formulalist>
      <formulalist>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <function symbol="∩">
          <constant symbol="s_1"/>
          <constant symbol="s_2"/>
         </function>
         <constant symbol="s_1"/>
        </function>
        <function symbol="∩">
         <constant symbol="s_1"/>
         <constant symbol="s_2"/>
        </function>
       </constantatomformula>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <constant symbol="s_3"/>
         <function symbol="∪">
          <constant symbol="s_3"/>
          <constant symbol="s_4"/>
         </function>
        </function>
        <constant symbol="s_3"/>
       </constantatomformula>
       <conjunctiveformula type="and">
        <constantatomformula symbol="=">
         <function symbol="∪">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
         <constant symbol="s_1"/>
        </constantatomformula>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
          <constant symbol="s_3"/>
         </function>
         <constant symbol="s_3"/>
        </constantatomformula>
       </conjunctiveformula>
      </formulalist>
     </sequent>
     <rule type="weakr" symbol="w:r">
      <sequent>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
         <function symbol="∩">
          <constant symbol="s_1"/>
          <constant symbol="s_2"/>
         </function>
        </constantatomformula>
       </formulalist>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
         <function symbol="∩">
          <constant symbol="s_1"/>
          <constant symbol="s_2"/>
         </function>
        </constantatomformula>
        <constantatomformula symbol="=">
         <function symbol="∪">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
         <constant symbol="s_1"/>
        </constantatomformula>
       </formulalist>
      </sequent>
      <rule type="axiom" symbol="axiom">
       <sequent>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∩">
           <function symbol="∩">
            <constant symbol="s_1"/>
            <constant symbol="s_2"/>
           </function>
           <constant symbol="s_1"/>
          </function>
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
         </constantatomformula>
        </formulalist>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∩">
           <function symbol="∩">
            <constant symbol="s_1"/>
            <constant symbol="s_2"/>
           </function>
           <constant symbol="s_1"/>
          </function>
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
         </constantatomformula>
        </formulalist>
       </sequent>
      </rule>
     </rule>
     <rule type="weakr" symbol="w:r">
      <sequent>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <constant symbol="s_3"/>
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
         </function>
         <constant symbol="s_3"/>
        </constantatomformula>
       </formulalist>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <constant symbol="s_3"/>
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
         </function>
         <constant symbol="s_3"/>
        </constantatomformula>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
          <constant symbol="s_3"/>
         </function>
         <constant symbol="s_3"/>
        </constantatomformula>
       </formulalist>
      </sequent>
      <rule type="axiom" symbol="axiom">
       <sequent>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∩">
           <constant symbol="s_3"/>
           <function symbol="∪">
            <constant symbol="s_3"/>
            <constant symbol="s_4"/>
           </function>
          </function>
          <constant symbol="s_3"/>
         </constantatomformula>
        </formulalist>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∩">
           <constant symbol="s_3"/>
           <function symbol="∪">
            <constant symbol="s_3"/>
            <constant symbol="s_4"/>
           </function>
          </function>
          <constant symbol="s_3"/>
         </constantatomformula>
        </formulalist>
       </sequent>
      </rule>
     </rule>
    </rule>
   </rule>
  </rule>
 </proof>
 <proof symbol="\psi_{25}" calculus="LK">
  <rule type="weakl" symbol="w:l">
   <sequent>
    <formulalist>
     <constantatomformula symbol="L1" definition="L1"/>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <function symbol="∩">
        <constant symbol="s_1"/>
        <constant symbol="s_2"/>
       </function>
       <constant symbol="s_1"/>
      </function>
      <function symbol="∩">
       <constant symbol="s_1"/>
       <constant symbol="s_2"/>
      </function>
     </constantatomformula>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <function symbol="∩">
        <function symbol="∪">
         <constant symbol="s_3"/>
         <constant symbol="s_4"/>
        </function>
        <constant symbol="s_3"/>
       </function>
       <constant symbol="s_3"/>
      </function>
      <function symbol="∩">
       <function symbol="∪">
        <constant symbol="s_3"/>
        <constant symbol="s_4"/>
       </function>
       <constant symbol="s_3"/>
      </function>
     </constantatomformula>
    </formulalist>
    <formulalist>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <function symbol="∩">
        <constant symbol="s_1"/>
        <constant symbol="s_2"/>
       </function>
       <constant symbol="s_1"/>
      </function>
      <function symbol="∩">
       <constant symbol="s_1"/>
       <constant symbol="s_2"/>
      </function>
     </constantatomformula>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <function symbol="∩">
        <function symbol="∪">
         <constant symbol="s_3"/>
         <constant symbol="s_4"/>
        </function>
        <constant symbol="s_3"/>
       </function>
       <constant symbol="s_3"/>
      </function>
      <function symbol="∩">
       <function symbol="∪">
        <constant symbol="s_3"/>
        <constant symbol="s_4"/>
       </function>
       <constant symbol="s_3"/>
      </function>
     </constantatomformula>
     <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
    </formulalist>
   </sequent>
   <rule type="defr" symbol="d:r" param="();[()];()">
    <sequent>
     <formulalist>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <function symbol="∩">
         <constant symbol="s_1"/>
         <constant symbol="s_2"/>
        </function>
        <constant symbol="s_1"/>
       </function>
       <function symbol="∩">
        <constant symbol="s_1"/>
        <constant symbol="s_2"/>
       </function>
      </constantatomformula>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <function symbol="∩">
         <function symbol="∪">
          <constant symbol="s_3"/>
          <constant symbol="s_4"/>
         </function>
         <constant symbol="s_3"/>
        </function>
        <constant symbol="s_3"/>
       </function>
       <function symbol="∩">
        <function symbol="∪">
         <constant symbol="s_3"/>
         <constant symbol="s_4"/>
        </function>
        <constant symbol="s_3"/>
       </function>
      </constantatomformula>
     </formulalist>
     <formulalist>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <function symbol="∩">
         <constant symbol="s_1"/>
         <constant symbol="s_2"/>
        </function>
        <constant symbol="s_1"/>
       </function>
       <function symbol="∩">
        <constant symbol="s_1"/>
        <constant symbol="s_2"/>
       </function>
      </constantatomformula>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <function symbol="∩">
         <function symbol="∪">
          <constant symbol="s_3"/>
          <constant symbol="s_4"/>
         </function>
         <constant symbol="s_3"/>
        </function>
        <constant symbol="s_3"/>
       </function>
       <function symbol="∩">
        <function symbol="∪">
         <constant symbol="s_3"/>
         <constant symbol="s_4"/>
        </function>
        <constant symbol="s_3"/>
       </function>
      </constantatomformula>
      <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
     </formulalist>
    </sequent>
    <rule type="andr" symbol="\and:r">
     <sequent>
      <formulalist>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <function symbol="∩">
          <constant symbol="s_1"/>
          <constant symbol="s_2"/>
         </function>
         <constant symbol="s_1"/>
        </function>
        <function symbol="∩">
         <constant symbol="s_1"/>
         <constant symbol="s_2"/>
        </function>
       </constantatomformula>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <function symbol="∩">
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
          <constant symbol="s_3"/>
         </function>
         <constant symbol="s_3"/>
        </function>
        <function symbol="∩">
         <function symbol="∪">
          <constant symbol="s_3"/>
          <constant symbol="s_4"/>
         </function>
         <constant symbol="s_3"/>
        </function>
       </constantatomformula>
      </formulalist>
      <formulalist>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <function symbol="∩">
          <constant symbol="s_1"/>
          <constant symbol="s_2"/>
         </function>
         <constant symbol="s_1"/>
        </function>
        <function symbol="∩">
         <constant symbol="s_1"/>
         <constant symbol="s_2"/>
        </function>
       </constantatomformula>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <function symbol="∩">
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
          <constant symbol="s_3"/>
         </function>
         <constant symbol="s_3"/>
        </function>
        <function symbol="∩">
         <function symbol="∪">
          <constant symbol="s_3"/>
          <constant symbol="s_4"/>
         </function>
         <constant symbol="s_3"/>
        </function>
       </constantatomformula>
       <conjunctiveformula type="and">
        <constantatomformula symbol="=">
         <function symbol="∪">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
         <constant symbol="s_1"/>
        </constantatomformula>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
          <constant symbol="s_3"/>
         </function>
         <constant symbol="s_3"/>
        </constantatomformula>
       </conjunctiveformula>
      </formulalist>
     </sequent>
     <rule type="weakr" symbol="w:r">
      <sequent>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
         <function symbol="∩">
          <constant symbol="s_1"/>
          <constant symbol="s_2"/>
         </function>
        </constantatomformula>
       </formulalist>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
         <function symbol="∩">
          <constant symbol="s_1"/>
          <constant symbol="s_2"/>
         </function>
        </constantatomformula>
        <constantatomformula symbol="=">
         <function symbol="∪">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
         <constant symbol="s_1"/>
        </constantatomformula>
       </formulalist>
      </sequent>
      <rule type="axiom" symbol="axiom">
       <sequent>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∩">
           <function symbol="∩">
            <constant symbol="s_1"/>
            <constant symbol="s_2"/>
           </function>
           <constant symbol="s_1"/>
          </function>
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
         </constantatomformula>
        </formulalist>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∩">
           <function symbol="∩">
            <constant symbol="s_1"/>
            <constant symbol="s_2"/>
           </function>
           <constant symbol="s_1"/>
          </function>
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
         </constantatomformula>
        </formulalist>
       </sequent>
      </rule>
     </rule>
     <rule type="weakr" symbol="w:r">
      <sequent>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <function symbol="∩">
           <function symbol="∪">
            <constant symbol="s_3"/>
            <constant symbol="s_4"/>
           </function>
           <constant symbol="s_3"/>
          </function>
          <constant symbol="s_3"/>
         </function>
         <function symbol="∩">
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
          <constant symbol="s_3"/>
         </function>
        </constantatomformula>
       </formulalist>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <function symbol="∩">
           <function symbol="∪">
            <constant symbol="s_3"/>
            <constant symbol="s_4"/>
           </function>
           <constant symbol="s_3"/>
          </function>
          <constant symbol="s_3"/>
         </function>
         <function symbol="∩">
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
          <constant symbol="s_3"/>
         </function>
        </constantatomformula>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <function symbol="∪">
           <constant symbol="s_3"/>
           <constant symbol="s_4"/>
          </function>
          <constant symbol="s_3"/>
         </function>
         <constant symbol="s_3"/>
        </constantatomformula>
       </formulalist>
      </sequent>
      <rule type="axiom" symbol="axiom">
       <sequent>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∩">
           <function symbol="∩">
            <function symbol="∪">
             <constant symbol="s_3"/>
             <constant symbol="s_4"/>
            </function>
            <constant symbol="s_3"/>
           </function>
           <constant symbol="s_3"/>
          </function>
          <function symbol="∩">
           <function symbol="∪">
            <constant symbol="s_3"/>
            <constant symbol="s_4"/>
           </function>
           <constant symbol="s_3"/>
          </function>
         </constantatomformula>
        </formulalist>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∩">
           <function symbol="∩">
            <function symbol="∪">
             <constant symbol="s_3"/>
             <constant symbol="s_4"/>
            </function>
            <constant symbol="s_3"/>
           </function>
           <constant symbol="s_3"/>
          </function>
          <function symbol="∩">
           <function symbol="∪">
            <constant symbol="s_3"/>
            <constant symbol="s_4"/>
           </function>
           <constant symbol="s_3"/>
          </function>
         </constantatomformula>
        </formulalist>
       </sequent>
      </rule>
     </rule>
    </rule>
   </rule>
  </rule>
 </proof>
 <proof symbol="\psi_{26}" calculus="LK">
  <rule type="weakr" symbol="w:r">
   <sequent>
    <formulalist>
     <constantatomformula symbol="L1" definition="L1"/>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <variable symbol="x_0"/>
       <variable symbol="z_0"/>
      </function>
      <variable symbol="x_0"/>
     </constantatomformula>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <variable symbol="y_0"/>
       <variable symbol="z_0"/>
      </function>
      <variable symbol="y_0"/>
     </constantatomformula>
    </formulalist>
    <formulalist>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <function symbol="∪">
        <variable symbol="x_0"/>
        <variable symbol="y_0"/>
       </function>
       <variable symbol="z_0"/>
      </function>
      <function symbol="∪">
       <variable symbol="x_0"/>
       <variable symbol="y_0"/>
      </function>
     </constantatomformula>
     <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
    </formulalist>
   </sequent>
   <rule type="contrl" symbol="c:l" param="2,1,1">
    <sequent>
     <formulalist>
      <constantatomformula symbol="L1" definition="L1"/>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <variable symbol="x_0"/>
        <variable symbol="z_0"/>
       </function>
       <variable symbol="x_0"/>
      </constantatomformula>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <variable symbol="y_0"/>
        <variable symbol="z_0"/>
       </function>
       <variable symbol="y_0"/>
      </constantatomformula>
     </formulalist>
     <formulalist>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <function symbol="∪">
         <variable symbol="x_0"/>
         <variable symbol="y_0"/>
        </function>
        <variable symbol="z_0"/>
       </function>
       <function symbol="∪">
        <variable symbol="x_0"/>
        <variable symbol="y_0"/>
       </function>
      </constantatomformula>
     </formulalist>
    </sequent>
    <rule type="weakl" symbol="w:l">
     <sequent>
      <formulalist>
       <constantatomformula symbol="L1" definition="L1"/>
       <constantatomformula symbol="L1" definition="L1"/>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <variable symbol="x_0"/>
         <variable symbol="z_0"/>
        </function>
        <variable symbol="x_0"/>
       </constantatomformula>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <variable symbol="y_0"/>
         <variable symbol="z_0"/>
        </function>
        <variable symbol="y_0"/>
       </constantatomformula>
      </formulalist>
      <formulalist>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <function symbol="∪">
          <variable symbol="x_0"/>
          <variable symbol="y_0"/>
         </function>
         <variable symbol="z_0"/>
        </function>
        <function symbol="∪">
         <variable symbol="x_0"/>
         <variable symbol="y_0"/>
        </function>
       </constantatomformula>
      </formulalist>
     </sequent>
     <rule type="defl" symbol="d:l" param="();[()];()">
      <sequent>
       <formulalist>
        <constantatomformula symbol="L1" definition="L1"/>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <variable symbol="x_0"/>
          <variable symbol="z_0"/>
         </function>
         <variable symbol="x_0"/>
        </constantatomformula>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <variable symbol="y_0"/>
          <variable symbol="z_0"/>
         </function>
         <variable symbol="y_0"/>
        </constantatomformula>
       </formulalist>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <function symbol="∪">
           <variable symbol="x_0"/>
           <variable symbol="y_0"/>
          </function>
          <variable symbol="z_0"/>
         </function>
         <function symbol="∪">
          <variable symbol="x_0"/>
          <variable symbol="y_0"/>
         </function>
        </constantatomformula>
       </formulalist>
      </sequent>
      <rule type="contrl" symbol="c:l" param="2,1,1">
       <sequent>
        <formulalist>
         <quantifiedformula type="all">
          <variable symbol="x"/>
          <quantifiedformula type="all">
           <variable symbol="y"/>
           <conjunctiveformula type="and">
            <conjunctiveformula type="impl">
             <constantatomformula symbol="=">
              <function symbol="∩">
               <variable symbol="x"/>
               <variable symbol="y"/>
              </function>
              <variable symbol="x"/>
             </constantatomformula>
             <constantatomformula symbol="=">
              <function symbol="∪">
               <variable symbol="x"/>
               <variable symbol="y"/>
              </function>
              <variable symbol="y"/>
             </constantatomformula>
            </conjunctiveformula>
            <conjunctiveformula type="impl">
             <constantatomformula symbol="=">
              <function symbol="∪">
               <variable symbol="x"/>
               <variable symbol="y"/>
              </function>
              <variable symbol="y"/>
             </constantatomformula>
             <constantatomformula symbol="=">
              <function symbol="∩">
               <variable symbol="x"/>
               <variable symbol="y"/>
              </function>
              <variable symbol="x"/>
             </constantatomformula>
            </conjunctiveformula>
           </conjunctiveformula>
          </quantifiedformula>
         </quantifiedformula>
         <constantatomformula symbol="=">
          <function symbol="∩">
           <variable symbol="x_0"/>
           <variable symbol="z_0"/>
          </function>
          <variable symbol="x_0"/>
         </constantatomformula>
         <constantatomformula symbol="=">
          <function symbol="∩">
           <variable symbol="y_0"/>
           <variable symbol="z_0"/>
          </function>
          <variable symbol="y_0"/>
         </constantatomformula>
        </formulalist>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∩">
           <function symbol="∪">
            <variable symbol="x_0"/>
            <variable symbol="y_0"/>
           </function>
           <variable symbol="z_0"/>
          </function>
          <function symbol="∪">
           <variable symbol="x_0"/>
           <variable symbol="y_0"/>
          </function>
         </constantatomformula>
        </formulalist>
       </sequent>
       <rule type="foralll" symbol="\all:l">
        <sequent>
         <formulalist>
          <quantifiedformula type="all">
           <variable symbol="x"/>
           <quantifiedformula type="all">
            <variable symbol="y"/>
            <conjunctiveformula type="and">
             <conjunctiveformula type="impl">
              <constantatomformula symbol="=">
               <function symbol="∩">
                <variable symbol="x"/>
                <variable symbol="y"/>
               </function>
               <variable symbol="x"/>
              </constantatomformula>
              <constantatomformula symbol="=">
               <function symbol="∪">
                <variable symbol="x"/>
                <variable symbol="y"/>
               </function>
               <variable symbol="y"/>
              </constantatomformula>
             </conjunctiveformula>
             <conjunctiveformula type="impl">
              <constantatomformula symbol="=">
               <function symbol="∪">
                <variable symbol="x"/>
                <variable symbol="y"/>
               </function>
               <variable symbol="y"/>
              </constantatomformula>
              <constantatomformula symbol="=">
               <function symbol="∩">
                <variable symbol="x"/>
                <variable symbol="y"/>
               </function>
               <variable symbol="x"/>
              </constantatomformula>
             </conjunctiveformula>
            </conjunctiveformula>
           </quantifiedformula>
          </quantifiedformula>
          <quantifiedformula type="all">
           <variable symbol="x"/>
           <quantifiedformula type="all">
            <variable symbol="y"/>
            <conjunctiveformula type="and">
             <conjunctiveformula type="impl">
              <constantatomformula symbol="=">
               <function symbol="∩">
                <variable symbol="x"/>
                <variable symbol="y"/>
               </function>
               <variable symbol="x"/>
              </constantatomformula>
              <constantatomformula symbol="=">
               <function symbol="∪">
                <variable symbol="x"/>
                <variable symbol="y"/>
               </function>
               <variable symbol="y"/>
              </constantatomformula>
             </conjunctiveformula>
             <conjunctiveformula type="impl">
              <constantatomformula symbol="=">
               <function symbol="∪">
                <variable symbol="x"/>
                <variable symbol="y"/>
               </function>
               <variable symbol="y"/>
              </constantatomformula>
              <constantatomformula symbol="=">
               <function symbol="∩">
                <variable symbol="x"/>
                <variable symbol="y"/>
               </function>
               <variable symbol="x"/>
              </constantatomformula>
             </conjunctiveformula>
            </conjunctiveformula>
           </quantifiedformula>
          </quantifiedformula>
          <constantatomformula symbol="=">
           <function symbol="∩">
            <variable symbol="x_0"/>
            <variable symbol="z_0"/>
           </function>
           <variable symbol="x_0"/>
          </constantatomformula>
          <constantatomformula symbol="=">
           <function symbol="∩">
            <variable symbol="y_0"/>
            <variable symbol="z_0"/>
           </function>
           <variable symbol="y_0"/>
          </constantatomformula>
         </formulalist>
         <formulalist>
          <constantatomformula symbol="=">
           <function symbol="∩">
            <function symbol="∪">
             <variable symbol="x_0"/>
             <variable symbol="y_0"/>
            </function>
            <variable symbol="z_0"/>
           </function>
           <function symbol="∪">
            <variable symbol="x_0"/>
            <variable symbol="y_0"/>
           </function>
          </constantatomformula>
         </formulalist>
        </sequent>
        <rule type="foralll" symbol="\all:l">
         <sequent>
          <formulalist>
           <quantifiedformula type="all">
            <variable symbol="y"/>
            <conjunctiveformula type="and">
             <conjunctiveformula type="impl">
              <constantatomformula symbol="=">
               <function symbol="∩">
                <variable symbol="x_0"/>
                <variable symbol="y"/>
               </function>
               <variable symbol="x_0"/>
              </constantatomformula>
              <constantatomformula symbol="=">
               <function symbol="∪">
                <variable symbol="x_0"/>
                <variable symbol="y"/>
               </function>
               <variable symbol="y"/>
              </constantatomformula>
             </conjunctiveformula>
             <conjunctiveformula type="impl">
              <constantatomformula symbol="=">
               <function symbol="∪">
                <variable symbol="x_0"/>
                <variable symbol="y"/>
               </function>
               <variable symbol="y"/>
              </constantatomformula>
              <constantatomformula symbol="=">
               <function symbol="∩">
                <variable symbol="x_0"/>
                <variable symbol="y"/>
               </function>
               <variable symbol="x_0"/>
              </constantatomformula>
             </conjunctiveformula>
            </conjunctiveformula>
           </quantifiedformula>
           <quantifiedformula type="all">
            <variable symbol="x"/>
            <quantifiedformula type="all">
             <variable symbol="y"/>
             <conjunctiveformula type="and">
              <conjunctiveformula type="impl">
               <constantatomformula symbol="=">
                <function symbol="∩">
                 <variable symbol="x"/>
                 <variable symbol="y"/>
                </function>
                <variable symbol="x"/>
               </constantatomformula>
               <constantatomformula symbol="=">
                <function symbol="∪">
                 <variable symbol="x"/>
                 <variable symbol="y"/>
                </function>
                <variable symbol="y"/>
               </constantatomformula>
              </conjunctiveformula>
              <conjunctiveformula type="impl">
               <constantatomformula symbol="=">
                <function symbol="∪">
                 <variable symbol="x"/>
                 <variable symbol="y"/>
                </function>
                <variable symbol="y"/>
               </constantatomformula>
               <constantatomformula symbol="=">
                <function symbol="∩">
                 <variable symbol="x"/>
                 <variable symbol="y"/>
                </function>
                <variable symbol="x"/>
               </constantatomformula>
              </conjunctiveformula>
             </conjunctiveformula>
            </quantifiedformula>
           </quantifiedformula>
           <constantatomformula symbol="=">
            <function symbol="∩">
             <variable symbol="x_0"/>
             <variable symbol="z_0"/>
            </function>
            <variable symbol="x_0"/>
           </constantatomformula>
           <constantatomformula symbol="=">
            <function symbol="∩">
             <variable symbol="y_0"/>
             <variable symbol="z_0"/>
            </function>
            <variable symbol="y_0"/>
           </constantatomformula>
          </formulalist>
          <formulalist>
           <constantatomformula symbol="=">
            <function symbol="∩">
             <function symbol="∪">
              <variable symbol="x_0"/>
              <variable symbol="y_0"/>
             </function>
             <variable symbol="z_0"/>
            </function>
            <function symbol="∪">
             <variable symbol="x_0"/>
             <variable symbol="y_0"/>
            </function>
           </constantatomformula>
          </formulalist>
         </sequent>
         <rule type="andl1" symbol="\and:l1">
          <sequent>
           <formulalist>
            <conjunctiveformula type="and">
             <conjunctiveformula type="impl">
              <constantatomformula symbol="=">
               <function symbol="∩">
                <variable symbol="x_0"/>
                <variable symbol="z_0"/>
               </function>
               <variable symbol="x_0"/>
              </constantatomformula>
              <constantatomformula symbol="=">
               <function symbol="∪">
                <variable symbol="x_0"/>
                <variable symbol="z_0"/>
               </function>
               <variable symbol="z_0"/>
              </constantatomformula>
             </conjunctiveformula>
             <conjunctiveformula type="impl">
              <constantatomformula symbol="=">
               <function symbol="∪">
                <variable symbol="x_0"/>
                <variable symbol="z_0"/>
               </function>
               <variable symbol="z_0"/>
              </constantatomformula>
              <constantatomformula symbol="=">
               <function symbol="∩">
                <variable symbol="x_0"/>
                <variable symbol="z_0"/>
               </function>
               <variable symbol="x_0"/>
              </constantatomformula>
             </conjunctiveformula>
            </conjunctiveformula>
            <quantifiedformula type="all">
             <variable symbol="x"/>
             <quantifiedformula type="all">
              <variable symbol="y"/>
              <conjunctiveformula type="and">
               <conjunctiveformula type="impl">
                <constantatomformula symbol="=">
                 <function symbol="∩">
                  <variable symbol="x"/>
                  <variable symbol="y"/>
                 </function>
                 <variable symbol="x"/>
                </constantatomformula>
                <constantatomformula symbol="=">
                 <function symbol="∪">
                  <variable symbol="x"/>
                  <variable symbol="y"/>
                 </function>
                 <variable symbol="y"/>
                </constantatomformula>
               </conjunctiveformula>
               <conjunctiveformula type="impl">
                <constantatomformula symbol="=">
                 <function symbol="∪">
                  <variable symbol="x"/>
                  <variable symbol="y"/>
                 </function>
                 <variable symbol="y"/>
                </constantatomformula>
                <constantatomformula symbol="=">
                 <function symbol="∩">
                  <variable symbol="x"/>
                  <variable symbol="y"/>
                 </function>
                 <variable symbol="x"/>
                </constantatomformula>
               </conjunctiveformula>
              </conjunctiveformula>
             </quantifiedformula>
            </quantifiedformula>
            <constantatomformula symbol="=">
             <function symbol="∩">
              <variable symbol="x_0"/>
              <variable symbol="z_0"/>
             </function>
             <variable symbol="x_0"/>
            </constantatomformula>
            <constantatomformula symbol="=">
             <function symbol="∩">
              <variable symbol="y_0"/>
              <variable symbol="z_0"/>
             </function>
             <variable symbol="y_0"/>
            </constantatomformula>
           </formulalist>
           <formulalist>
            <constantatomformula symbol="=">
             <function symbol="∩">
              <function symbol="∪">
               <variable symbol="x_0"/>
               <variable symbol="y_0"/>
              </function>
              <variable symbol="z_0"/>
             </function>
             <function symbol="∪">
              <variable symbol="x_0"/>
              <variable symbol="y_0"/>
             </function>
            </constantatomformula>
           </formulalist>
          </sequent>
          <rule type="perml" symbol="\pi:l" param="(2 3)">
           <sequent>
            <formulalist>
             <conjunctiveformula type="impl">
              <constantatomformula symbol="=">
               <function symbol="∩">
                <variable symbol="x_0"/>
                <variable symbol="z_0"/>
               </function>
               <variable symbol="x_0"/>
              </constantatomformula>
              <constantatomformula symbol="=">
               <function symbol="∪">
                <variable symbol="x_0"/>
                <variable symbol="z_0"/>
               </function>
               <variable symbol="z_0"/>
              </constantatomformula>
             </conjunctiveformula>
             <quantifiedformula type="all">
              <variable symbol="x"/>
              <quantifiedformula type="all">
               <variable symbol="y"/>
               <conjunctiveformula type="and">
                <conjunctiveformula type="impl">
                 <constantatomformula symbol="=">
                  <function symbol="∩">
                   <variable symbol="x"/>
                   <variable symbol="y"/>
                  </function>
                  <variable symbol="x"/>
                 </constantatomformula>
                 <constantatomformula symbol="=">
                  <function symbol="∪">
                   <variable symbol="x"/>
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                           <function symbol="∩">
                            <variable symbol="x"/>
                            <variable symbol="y"/>
                           </function>
                           <variable symbol="x"/>
                          </constantatomformula>
                          <constantatomformula symbol="=">
                           <function symbol="∪">
                            <variable symbol="x"/>
                            <variable symbol="y"/>
                           </function>
                           <variable symbol="y"/>
                          </constantatomformula>
                         </conjunctiveformula>
                         <conjunctiveformula type="impl">
                          <constantatomformula symbol="=">
                           <function symbol="∪">
                            <variable symbol="x"/>
                            <variable symbol="y"/>
                           </function>
                           <variable symbol="y"/>
                          </constantatomformula>
                          <constantatomformula symbol="=">
                           <function symbol="∩">
                            <variable symbol="x"/>
                            <variable symbol="y"/>
                           </function>
                           <variable symbol="x"/>
                          </constantatomformula>
                         </conjunctiveformula>
                        </conjunctiveformula>
                       </quantifiedformula>
                      </quantifiedformula>
                      <constantatomformula symbol="=">
                       <function symbol="∪">
                        <variable symbol="x_0"/>
                        <variable symbol="z_0"/>
                       </function>
                       <variable symbol="z_0"/>
                      </constantatomformula>
                     </formulalist>
                     <formulalist>
                      <constantatomformula symbol="=">
                       <function symbol="∩">
                        <function symbol="∪">
                         <variable symbol="x_0"/>
                         <variable symbol="y_0"/>
                        </function>
                        <variable symbol="z_0"/>
                       </function>
                       <function symbol="∪">
                        <variable symbol="x_0"/>
                        <variable symbol="y_0"/>
                       </function>
                      </constantatomformula>
                     </formulalist>
                    </sequent>
                    <rule type="foralll" symbol="\all:l">
                     <sequent>
                      <formulalist>
                       <quantifiedformula type="all">
                        <variable symbol="x"/>
                        <quantifiedformula type="all">
                         <variable symbol="y"/>
                         <conjunctiveformula type="and">
                          <conjunctiveformula type="impl">
                           <constantatomformula symbol="=">
                            <function symbol="∩">
                             <variable symbol="x"/>
                             <variable symbol="y"/>
                            </function>
                            <variable symbol="x"/>
                           </constantatomformula>
                           <constantatomformula symbol="=">
                            <function symbol="∪">
                             <variable symbol="x"/>
                             <variable symbol="y"/>
                            </function>
                            <variable symbol="y"/>
                           </constantatomformula>
                          </conjunctiveformula>
                          <conjunctiveformula type="impl">
                           <constantatomformula symbol="=">
                            <function symbol="∪">
                             <variable symbol="x"/>
                             <variable symbol="y"/>
                            </function>
                            <variable symbol="y"/>
                           </constantatomformula>
                           <constantatomformula symbol="=">
                            <function symbol="∩">
                             <variable symbol="x"/>
                             <variable symbol="y"/>
                            </function>
                            <variable symbol="x"/>
                           </constantatomformula>
                          </conjunctiveformula>
                         </conjunctiveformula>
                        </quantifiedformula>
                       </quantifiedformula>
                       <constantatomformula symbol="=">
                        <function symbol="∪">
                         <variable symbol="y_0"/>
                         <variable symbol="z_0"/>
                        </function>
                        <variable symbol="z_0"/>
                       </constantatomformula>
                       <constantatomformula symbol="=">
                        <function symbol="∪">
                         <variable symbol="x_0"/>
                         <variable symbol="z_0"/>
                        </function>
                        <variable symbol="z_0"/>
                       </constantatomformula>
                      </formulalist>
                      <formulalist>
                       <constantatomformula symbol="=">
                        <function symbol="∩">
                         <function symbol="∪">
                          <variable symbol="x_0"/>
                          <variable symbol="y_0"/>
                         </function>
                         <variable symbol="z_0"/>
                        </function>
                        <function symbol="∪">
                         <variable symbol="x_0"/>
                         <variable symbol="y_0"/>
                        </function>
                       </constantatomformula>
                      </formulalist>
                     </sequent>
                     <rule type="foralll" symbol="\all:l">
                      <sequent>
                       <formulalist>
                        <quantifiedformula type="all">
                         <variable symbol="y"/>
                         <conjunctiveformula type="and">
                          <conjunctiveformula type="impl">
                           <constantatomformula symbol="=">
                            <function symbol="∩">
                             <function symbol="∪">
                              <variable symbol="x_0"/>
                              <variable symbol="y_0"/>
                             </function>
                             <variable symbol="y"/>
                            </function>
                            <function symbol="∪">
                             <variable symbol="x_0"/>
                             <variable symbol="y_0"/>
                            </function>
                           </constantatomformula>
                           <constantatomformula symbol="=">
                            <function symbol="∪">
                             <function symbol="∪">
                              <variable symbol="x_0"/>
                              <variable symbol="y_0"/>
                             </function>
                             <variable symbol="y"/>
                            </function>
                            <variable symbol="y"/>
                           </constantatomformula>
                          </conjunctiveformula>
                          <conjunctiveformula type="impl">
                           <constantatomformula symbol="=">
                            <function symbol="∪">
                             <function symbol="∪">
                              <variable symbol="x_0"/>
                              <variable symbol="y_0"/>
                             </function>
                             <variable symbol="y"/>
                            </function>
                            <variable symbol="y"/>
                           </constantatomformula>
                           <constantatomformula symbol="=">
                            <function symbol="∩">
                             <function symbol="∪">
                              <variable symbol="x_0"/>
                              <variable symbol="y_0"/>
                             </function>
                             <variable symbol="y"/>
                            </function>
                            <function symbol="∪">
                             <variable symbol="x_0"/>
                             <variable symbol="y_0"/>
                            </function>
                           </constantatomformula>
                          </conjunctiveformula>
                         </conjunctiveformula>
                        </quantifiedformula>
                        <constantatomformula symbol="=">
                         <function symbol="∪">
                          <variable symbol="y_0"/>
                          <variable symbol="z_0"/>
                         </function>
                         <variable symbol="z_0"/>
                        </constantatomformula>
                        <constantatomformula symbol="=">
                         <function symbol="∪">
                          <variable symbol="x_0"/>
                          <variable symbol="z_0"/>
                         </function>
                         <variable symbol="z_0"/>
                        </constantatomformula>
                       </formulalist>
                       <formulalist>
                        <constantatomformula symbol="=">
                         <function symbol="∩">
                          <function symbol="∪">
                           <variable symbol="x_0"/>
                           <variable symbol="y_0"/>
                          </function>
                          <variable symbol="z_0"/>
                         </function>
                         <function symbol="∪">
                          <variable symbol="x_0"/>
                          <variable symbol="y_0"/>
                         </function>
                        </constantatomformula>
                       </formulalist>
                      </sequent>
                      <rule type="andl2" symbol="\and:l2">
                       <sequent>
                        <formulalist>
                         <conjunctiveformula type="and">
                          <conjunctiveformula type="impl">
                           <constantatomformula symbol="=">
                            <function symbol="∩">
                             <function symbol="∪">
                              <variable symbol="x_0"/>
                              <variable symbol="y_0"/>
                             </function>
                             <variable symbol="z_0"/>
                            </function>
                            <function symbol="∪">
                             <variable symbol="x_0"/>
                             <variable symbol="y_0"/>
                            </function>
                           </constantatomformula>
                           <constantatomformula symbol="=">
                            <function symbol="∪">
                             <function symbol="∪">
                              <variable symbol="x_0"/>
                              <variable symbol="y_0"/>
                             </function>
                             <variable symbol="z_0"/>
                            </function>
                            <variable symbol="z_0"/>
                           </constantatomformula>
                          </conjunctiveformula>
                          <conjunctiveformula type="impl">
                           <constantatomformula symbol="=">
                            <function symbol="∪">
                             <function symbol="∪">
                              <variable symbol="x_0"/>
                              <variable symbol="y_0"/>
                             </function>
                             <variable symbol="z_0"/>
                            </function>
                            <variable symbol="z_0"/>
                           </constantatomformula>
                           <constantatomformula symbol="=">
                            <function symbol="∩">
                             <function symbol="∪">
                              <variable symbol="x_0"/>
                              <variable symbol="y_0"/>
                             </function>
                             <variable symbol="z_0"/>
                            </function>
                            <function symbol="∪">
                             <variable symbol="x_0"/>
                             <variable symbol="y_0"/>
                            </function>
                           </constantatomformula>
                          </conjunctiveformula>
                         </conjunctiveformula>
                         <constantatomformula symbol="=">
                          <function symbol="∪">
                           <variable symbol="y_0"/>
                           <variable symbol="z_0"/>
                          </function>
                          <variable symbol="z_0"/>
                         </constantatomformula>
                         <constantatomformula symbol="=">
                          <function symbol="∪">
                           <variable symbol="x_0"/>
                           <variable symbol="z_0"/>
                          </function>
                          <variable symbol="z_0"/>
                         </constantatomformula>
                        </formulalist>
                        <formulalist>
                         <constantatomformula symbol="=">
                          <function symbol="∩">
                           <function symbol="∪">
                            <variable symbol="x_0"/>
                            <variable symbol="y_0"/>
                           </function>
                           <variable symbol="z_0"/>
                          </function>
                          <function symbol="∪">
                           <variable symbol="x_0"/>
                           <variable symbol="y_0"/>
                          </function>
                         </constantatomformula>
                        </formulalist>
                       </sequent>
                       <rule type="impll" symbol="\impl:l">
                        <sequent>
                         <formulalist>
                          <conjunctiveformula type="impl">
                           <constantatomformula symbol="=">
                            <function symbol="∪">
                             <function symbol="∪">
                              <variable symbol="x_0"/>
                              <variable symbol="y_0"/>
                             </function>
                             <variable symbol="z_0"/>
                            </function>
                            <variable symbol="z_0"/>
                           </constantatomformula>
                           <constantatomformula symbol="=">
                            <function symbol="∩">
                             <function symbol="∪">
                              <variable symbol="x_0"/>
                              <variable symbol="y_0"/>
                             </function>
                             <variable symbol="z_0"/>
                            </function>
                            <function symbol="∪">
                             <variable symbol="x_0"/>
                             <variable symbol="y_0"/>
                            </function>
                           </constantatomformula>
                          </conjunctiveformula>
                          <constantatomformula symbol="=">
                           <function symbol="∪">
                            <variable symbol="y_0"/>
                            <variable symbol="z_0"/>
                           </function>
                           <variable symbol="z_0"/>
                          </constantatomformula>
                          <constantatomformula symbol="=">
                           <function symbol="∪">
                            <variable symbol="x_0"/>
                            <variable symbol="z_0"/>
                           </function>
                           <variable symbol="z_0"/>
                          </constantatomformula>
                         </formulalist>
                         <formulalist>
                          <constantatomformula symbol="=">
                           <function symbol="∩">
                            <function symbol="∪">
                             <variable symbol="x_0"/>
                             <variable symbol="y_0"/>
                            </function>
                            <variable symbol="z_0"/>
                           </function>
                           <function symbol="∪">
                            <variable symbol="x_0"/>
                            <variable symbol="y_0"/>
                           </function>
                          </constantatomformula>
                         </formulalist>
                        </sequent>
                        <rule type="eqr2" symbol="=:r2" param="(1)">
                         <sequent>
                          <formulalist>
                           <constantatomformula symbol="=">
                            <function symbol="∪">
                             <variable symbol="y_0"/>
                             <variable symbol="z_0"/>
                            </function>
                            <variable symbol="z_0"/>
                           </constantatomformula>
                           <constantatomformula symbol="=">
                            <function symbol="∪">
                             <variable symbol="x_0"/>
                             <variable symbol="z_0"/>
                            </function>
                            <variable symbol="z_0"/>
                           </constantatomformula>
                          </formulalist>
                          <formulalist>
                           <constantatomformula symbol="=">
                            <function symbol="∪">
                             <function symbol="∪">
                              <variable symbol="x_0"/>
                              <variable symbol="y_0"/>
                             </function>
                             <variable symbol="z_0"/>
                            </function>
                            <variable symbol="z_0"/>
                           </constantatomformula>
                          </formulalist>
                         </sequent>
                         <rule type="axiom" symbol="axiom">
                          <sequent>
                           <formulalist/>
                           <formulalist>
                            <constantatomformula symbol="=">
                             <function symbol="∪">
                              <function symbol="∪">
                               <variable symbol="x_0"/>
                               <variable symbol="y_0"/>
                              </function>
                              <variable symbol="z_0"/>
                             </function>
                             <function symbol="∪">
                              <variable symbol="x_0"/>
                              <function symbol="∪">
                               <variable symbol="y_0"/>
                               <variable symbol="z_0"/>
                              </function>
                             </function>
                            </constantatomformula>
                           </formulalist>
                          </sequent>
                         </rule>
                         <rule type="eqr2" symbol="=:r2" param="(1 2)">
                          <sequent>
                           <formulalist>
                            <constantatomformula symbol="=">
                             <function symbol="∪">
                              <variable symbol="y_0"/>
                              <variable symbol="z_0"/>
                             </function>
                             <variable symbol="z_0"/>
                            </constantatomformula>
                            <constantatomformula symbol="=">
                             <function symbol="∪">
                              <variable symbol="x_0"/>
                              <variable symbol="z_0"/>
                             </function>
                             <variable symbol="z_0"/>
                            </constantatomformula>
                           </formulalist>
                           <formulalist>
                            <constantatomformula symbol="=">
                             <function symbol="∪">
                              <variable symbol="x_0"/>
                              <function symbol="∪">
                               <variable symbol="y_0"/>
                               <variable symbol="z_0"/>
                              </function>
                             </function>
                             <variable symbol="z_0"/>
                            </constantatomformula>
                           </formulalist>
                          </sequent>
                          <rule type="axiom" symbol="axiom">
                           <sequent>
                            <formulalist>
                             <constantatomformula symbol="=">
                              <function symbol="∪">
                               <variable symbol="y_0"/>
                               <variable symbol="z_0"/>
                              </function>
                              <variable symbol="z_0"/>
                             </constantatomformula>
                            </formulalist>
                            <formulalist>
                             <constantatomformula symbol="=">
                              <function symbol="∪">
                               <variable symbol="y_0"/>
                               <variable symbol="z_0"/>
                              </function>
                              <variable symbol="z_0"/>
                             </constantatomformula>
                            </formulalist>
                           </sequent>
                          </rule>
                          <rule type="contrr" symbol="c:r" param="2">
                           <sequent>
                            <formulalist>
                             <constantatomformula symbol="=">
                              <function symbol="∪">
                               <variable symbol="x_0"/>
                               <variable symbol="z_0"/>
                              </function>
                              <variable symbol="z_0"/>
                             </constantatomformula>
                            </formulalist>
                            <formulalist>
                             <constantatomformula symbol="=">
                              <function symbol="∪">
                               <variable symbol="x_0"/>
                               <variable symbol="z_0"/>
                              </function>
                              <variable symbol="z_0"/>
                             </constantatomformula>
                            </formulalist>
                           </sequent>
                           <rule type="contrl" symbol="c:l" param="2">
                            <sequent>
                             <formulalist>
                              <constantatomformula symbol="=">
                               <function symbol="∪">
                                <variable symbol="x_0"/>
                                <variable symbol="z_0"/>
                               </function>
                               <variable symbol="z_0"/>
                              </constantatomformula>
                             </formulalist>
                             <formulalist>
                              <constantatomformula symbol="=">
                               <function symbol="∪">
                                <variable symbol="x_0"/>
                                <variable symbol="z_0"/>
                               </function>
                               <variable symbol="z_0"/>
                              </constantatomformula>
                              <constantatomformula symbol="=">
                               <function symbol="∪">
                                <variable symbol="x_0"/>
                                <variable symbol="z_0"/>
                               </function>
                               <variable symbol="z_0"/>
                              </constantatomformula>
                             </formulalist>
                            </sequent>
                            <rule type="eqr2" symbol="=:r2" param="(1)">
                             <sequent>
                              <formulalist>
                               <constantatomformula symbol="=">
                                <function symbol="∪">
                                 <variable symbol="x_0"/>
                                 <variable symbol="z_0"/>
                                </function>
                                <variable symbol="z_0"/>
                               </constantatomformula>
                               <constantatomformula symbol="=">
                                <function symbol="∪">
                                 <variable symbol="x_0"/>
                                 <variable symbol="z_0"/>
                                </function>
                                <variable symbol="z_0"/>
                               </constantatomformula>
                              </formulalist>
                              <formulalist>
                               <constantatomformula symbol="=">
                                <function symbol="∪">
                                 <variable symbol="x_0"/>
                                 <variable symbol="z_0"/>
                                </function>
                                <variable symbol="z_0"/>
                               </constantatomformula>
                               <constantatomformula symbol="=">
                                <function symbol="∪">
                                 <variable symbol="x_0"/>
                                 <variable symbol="z_0"/>
                                </function>
                                <variable symbol="z_0"/>
                               </constantatomformula>
                              </formulalist>
                             </sequent>
                             <rule type="axiom" symbol="axiom">
                              <sequent>
                               <formulalist>
                                <constantatomformula symbol="=">
                                 <function symbol="∪">
                                  <variable symbol="x_0"/>
                                  <variable symbol="z_0"/>
                                 </function>
                                 <variable symbol="z_0"/>
                                </constantatomformula>
                               </formulalist>
                               <formulalist>
                                <constantatomformula symbol="=">
                                 <function symbol="∪">
                                  <variable symbol="x_0"/>
                                  <variable symbol="z_0"/>
                                 </function>
                                 <variable symbol="z_0"/>
                                </constantatomformula>
                               </formulalist>
                              </sequent>
                             </rule>
                             <rule type="weakr" symbol="w:r">
                              <sequent>
                               <formulalist>
                                <constantatomformula symbol="=">
                                 <function symbol="∪">
                                  <variable symbol="x_0"/>
                                  <variable symbol="z_0"/>
                                 </function>
                                 <variable symbol="z_0"/>
                                </constantatomformula>
                               </formulalist>
                               <formulalist>
                                <constantatomformula symbol="=">
                                 <function symbol="∪">
                                  <variable symbol="x_0"/>
                                  <variable symbol="z_0"/>
                                 </function>
                                 <variable symbol="z_0"/>
                                </constantatomformula>
                                <constantatomformula symbol="=">
                                 <variable symbol="z_0"/>
                                 <variable symbol="z_0"/>
                                </constantatomformula>
                               </formulalist>
                              </sequent>
                              <rule type="axiom" symbol="axiom">
                               <sequent>
                                <formulalist>
                                 <constantatomformula symbol="=">
                                  <function symbol="∪">
                                   <variable symbol="x_0"/>
                                   <variable symbol="z_0"/>
                                  </function>
                                  <variable symbol="z_0"/>
                                 </constantatomformula>
                                </formulalist>
                                <formulalist>
                                 <constantatomformula symbol="=">
                                  <function symbol="∪">
                                   <variable symbol="x_0"/>
                                   <variable symbol="z_0"/>
                                  </function>
                                  <variable symbol="z_0"/>
                                 </constantatomformula>
                                </formulalist>
                               </sequent>
                              </rule>
                             </rule>
                            </rule>
                           </rule>
                          </rule>
                         </rule>
                        </rule>
                        <rule type="axiom" symbol="axiom">
                         <sequent>
                          <formulalist>
                           <constantatomformula symbol="=">
                            <function symbol="∩">
                             <function symbol="∪">
                              <variable symbol="x_0"/>
                              <variable symbol="y_0"/>
                             </function>
                             <variable symbol="z_0"/>
                            </function>
                            <function symbol="∪">
                             <variable symbol="x_0"/>
                             <variable symbol="y_0"/>
                            </function>
                           </constantatomformula>
                          </formulalist>
                          <formulalist>
                           <constantatomformula symbol="=">
                            <function symbol="∩">
                             <function symbol="∪">
                              <variable symbol="x_0"/>
                              <variable symbol="y_0"/>
                             </function>
                             <variable symbol="z_0"/>
                            </function>
                            <function symbol="∪">
                             <variable symbol="x_0"/>
                             <variable symbol="y_0"/>
                            </function>
                           </constantatomformula>
                          </formulalist>
                         </sequent>
                        </rule>
                       </rule>
                      </rule>
                     </rule>
                    </rule>
                   </rule>
                  </rule>
                 </rule>
                </rule>
               </rule>
              </rule>
             </rule>
            </rule>
           </rule>
          </rule>
         </rule>
        </rule>
       </rule>
      </rule>
     </rule>
    </rule>
   </rule>
  </rule>
 </proof>
 <proof symbol="\psi_{27}" calculus="LK">
  <rule type="weakr" symbol="w:r">
   <sequent>
    <formulalist>
     <constantatomformula symbol="L1" definition="L1"/>
    </formulalist>
    <formulalist>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <variable symbol="y_0"/>
       <function symbol="∪">
        <variable symbol="y_0"/>
        <variable symbol="x_0"/>
       </function>
      </function>
      <variable symbol="y_0"/>
     </constantatomformula>
     <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
    </formulalist>
   </sequent>
   <rule type="contrl" symbol="c:l" param="2">
    <sequent>
     <formulalist>
      <constantatomformula symbol="L1" definition="L1"/>
     </formulalist>
     <formulalist>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <variable symbol="y_0"/>
        <function symbol="∪">
         <variable symbol="y_0"/>
         <variable symbol="x_0"/>
        </function>
       </function>
       <variable symbol="y_0"/>
      </constantatomformula>
     </formulalist>
    </sequent>
    <rule type="perml" symbol="\pi:l" param="(1 2)">
     <sequent>
      <formulalist>
       <constantatomformula symbol="L1" definition="L1"/>
       <constantatomformula symbol="L1" definition="L1"/>
      </formulalist>
      <formulalist>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <variable symbol="y_0"/>
         <function symbol="∪">
          <variable symbol="y_0"/>
          <variable symbol="x_0"/>
         </function>
        </function>
        <variable symbol="y_0"/>
       </constantatomformula>
      </formulalist>
     </sequent>
     <rule type="weakl" symbol="w:l">
      <sequent>
       <formulalist>
        <constantatomformula symbol="L1" definition="L1"/>
        <constantatomformula symbol="L1" definition="L1"/>
       </formulalist>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <variable symbol="y_0"/>
          <function symbol="∪">
           <variable symbol="y_0"/>
           <variable symbol="x_0"/>
          </function>
         </function>
         <variable symbol="y_0"/>
        </constantatomformula>
       </formulalist>
      </sequent>
      <rule type="contrl" symbol="c:l" param="2">
       <sequent>
        <formulalist>
         <constantatomformula symbol="L1" definition="L1"/>
        </formulalist>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∩">
           <variable symbol="y_0"/>
           <function symbol="∪">
            <variable symbol="y_0"/>
            <variable symbol="x_0"/>
           </function>
          </function>
          <variable symbol="y_0"/>
         </constantatomformula>
        </formulalist>
       </sequent>
       <rule type="weakl" symbol="w:l">
        <sequent>
         <formulalist>
          <constantatomformula symbol="L1" definition="L1"/>
          <constantatomformula symbol="L1" definition="L1"/>
         </formulalist>
         <formulalist>
          <constantatomformula symbol="=">
           <function symbol="∩">
            <variable symbol="y_0"/>
            <function symbol="∪">
             <variable symbol="y_0"/>
             <variable symbol="x_0"/>
            </function>
           </function>
           <variable symbol="y_0"/>
          </constantatomformula>
         </formulalist>
        </sequent>
        <rule type="defl" symbol="d:l" param="();[()];()">
         <sequent>
          <formulalist>
           <constantatomformula symbol="L1" definition="L1"/>
          </formulalist>
          <formulalist>
           <constantatomformula symbol="=">
            <function symbol="∩">
             <variable symbol="y_0"/>
             <function symbol="∪">
              <variable symbol="y_0"/>
              <variable symbol="x_0"/>
             </function>
            </function>
            <variable symbol="y_0"/>
           </constantatomformula>
          </formulalist>
         </sequent>
         <rule type="foralll" symbol="\all:l">
          <sequent>
           <formulalist>
            <quantifiedformula type="all">
             <variable symbol="x"/>
             <quantifiedformula type="all">
              <variable symbol="y"/>
              <conjunctiveformula type="and">
               <conjunctiveformula type="impl">
                <constantatomformula symbol="=">
                 <function symbol="∩">
                  <variable symbol="x"/>
                  <variable symbol="y"/>
                 </function>
                 <variable symbol="x"/>
                </constantatomformula>
                <constantatomformula symbol="=">
                 <function symbol="∪">
                  <variable symbol="x"/>
                  <variable symbol="y"/>
                 </function>
                 <variable symbol="y"/>
                </constantatomformula>
               </conjunctiveformula>
               <conjunctiveformula type="impl">
                <constantatomformula symbol="=">
                 <function symbol="∪">
                  <variable symbol="x"/>
                  <variable symbol="y"/>
                 </function>
                 <variable symbol="y"/>
                </constantatomformula>
                <constantatomformula symbol="=">
                 <function symbol="∩">
                  <variable symbol="x"/>
                  <variable symbol="y"/>
                 </function>
                 <variable symbol="x"/>
                </constantatomformula>
               </conjunctiveformula>
              </conjunctiveformula>
             </quantifiedformula>
            </quantifiedformula>
           </formulalist>
           <formulalist>
            <constantatomformula symbol="=">
             <function symbol="∩">
              <variable symbol="y_0"/>
              <function symbol="∪">
               <variable symbol="y_0"/>
               <variable symbol="x_0"/>
              </function>
             </function>
             <variable symbol="y_0"/>
            </constantatomformula>
           </formulalist>
          </sequent>
          <rule type="foralll" symbol="\all:l">
           <sequent>
            <formulalist>
             <quantifiedformula type="all">
              <variable symbol="y"/>
              <conjunctiveformula type="and">
               <conjunctiveformula type="impl">
                <constantatomformula symbol="=">
                 <function symbol="∩">
                  <variable symbol="y_0"/>
                  <variable symbol="y"/>
                 </function>
                 <variable symbol="y_0"/>
                </constantatomformula>
                <constantatomformula symbol="=">
                 <function symbol="∪">
                  <variable symbol="y_0"/>
                  <variable symbol="y"/>
                 </function>
                 <variable symbol="y"/>
                </constantatomformula>
               </conjunctiveformula>
               <conjunctiveformula type="impl">
                <constantatomformula symbol="=">
                 <function symbol="∪">
                  <variable symbol="y_0"/>
                  <variable symbol="y"/>
                 </function>
                 <variable symbol="y"/>
                </constantatomformula>
                <constantatomformula symbol="=">
                 <function symbol="∩">
                  <variable symbol="y_0"/>
                  <variable symbol="y"/>
                 </function>
                 <variable symbol="y_0"/>
                </constantatomformula>
               </conjunctiveformula>
              </conjunctiveformula>
             </quantifiedformula>
            </formulalist>
            <formulalist>
             <constantatomformula symbol="=">
              <function symbol="∩">
               <variable symbol="y_0"/>
               <function symbol="∪">
                <variable symbol="y_0"/>
                <variable symbol="x_0"/>
               </function>
              </function>
              <variable symbol="y_0"/>
             </constantatomformula>
            </formulalist>
           </sequent>
           <rule type="andl2" symbol="\and:l2">
            <sequent>
             <formulalist>
              <conjunctiveformula type="and">
               <conjunctiveformula type="impl">
                <constantatomformula symbol="=">
                 <function symbol="∩">
                  <variable symbol="y_0"/>
                  <function symbol="∪">
                   <variable symbol="y_0"/>
                   <variable symbol="x_0"/>
                  </function>
                 </function>
                 <variable symbol="y_0"/>
                </constantatomformula>
                <constantatomformula symbol="=">
                 <function symbol="∪">
                  <variable symbol="y_0"/>
                  <function symbol="∪">
                   <variable symbol="y_0"/>
                   <variable symbol="x_0"/>
                  </function>
                 </function>
                 <function symbol="∪">
                  <variable symbol="y_0"/>
                  <variable symbol="x_0"/>
                 </function>
                </constantatomformula>
               </conjunctiveformula>
               <conjunctiveformula type="impl">
                <constantatomformula symbol="=">
                 <function symbol="∪">
                  <variable symbol="y_0"/>
                  <function symbol="∪">
                   <variable symbol="y_0"/>
                   <variable symbol="x_0"/>
                  </function>
                 </function>
                 <function symbol="∪">
                  <variable symbol="y_0"/>
                  <variable symbol="x_0"/>
                 </function>
                </constantatomformula>
                <constantatomformula symbol="=">
                 <function symbol="∩">
                  <variable symbol="y_0"/>
                  <function symbol="∪">
                   <variable symbol="y_0"/>
                   <variable symbol="x_0"/>
                  </function>
                 </function>
                 <variable symbol="y_0"/>
                </constantatomformula>
               </conjunctiveformula>
              </conjunctiveformula>
             </formulalist>
             <formulalist>
              <constantatomformula symbol="=">
               <function symbol="∩">
                <variable symbol="y_0"/>
                <function symbol="∪">
                 <variable symbol="y_0"/>
                 <variable symbol="x_0"/>
                </function>
               </function>
               <variable symbol="y_0"/>
              </constantatomformula>
             </formulalist>
            </sequent>
            <rule type="impll" symbol="\impl:l">
             <sequent>
              <formulalist>
               <conjunctiveformula type="impl">
                <constantatomformula symbol="=">
                 <function symbol="∪">
                  <variable symbol="y_0"/>
                  <function symbol="∪">
                   <variable symbol="y_0"/>
                   <variable symbol="x_0"/>
                  </function>
                 </function>
                 <function symbol="∪">
                  <variable symbol="y_0"/>
                  <variable symbol="x_0"/>
                 </function>
                </constantatomformula>
                <constantatomformula symbol="=">
                 <function symbol="∩">
                  <variable symbol="y_0"/>
                  <function symbol="∪">
                   <variable symbol="y_0"/>
                   <variable symbol="x_0"/>
                  </function>
                 </function>
                 <variable symbol="y_0"/>
                </constantatomformula>
               </conjunctiveformula>
              </formulalist>
              <formulalist>
               <constantatomformula symbol="=">
                <function symbol="∩">
                 <variable symbol="y_0"/>
                 <function symbol="∪">
                  <variable symbol="y_0"/>
                  <variable symbol="x_0"/>
                 </function>
                </function>
                <variable symbol="y_0"/>
               </constantatomformula>
              </formulalist>
             </sequent>
             <rule type="eqr1" symbol="=:r1" param="(1)">
              <sequent>
               <formulalist/>
               <formulalist>
                <constantatomformula symbol="=">
                 <function symbol="∪">
                  <variable symbol="y_0"/>
                  <function symbol="∪">
                   <variable symbol="y_0"/>
                   <variable symbol="x_0"/>
                  </function>
                 </function>
                 <function symbol="∪">
                  <variable symbol="y_0"/>
                  <variable symbol="x_0"/>
                 </function>
                </constantatomformula>
               </formulalist>
              </sequent>
              <rule type="axiom" symbol="axiom">
               <sequent>
                <formulalist/>
                <formulalist>
                 <constantatomformula symbol="=">
                  <function symbol="∪">
                   <function symbol="∪">
                    <variable symbol="y_0"/>
                    <variable symbol="y_0"/>
                   </function>
                   <variable symbol="x_0"/>
                  </function>
                  <function symbol="∪">
                   <variable symbol="y_0"/>
                   <function symbol="∪">
                    <variable symbol="y_0"/>
                    <variable symbol="x_0"/>
                   </function>
                  </function>
                 </constantatomformula>
                </formulalist>
               </sequent>
              </rule>
              <rule type="eqr2" symbol="=:r2" param="(1 1)">
               <sequent>
                <formulalist/>
                <formulalist>
                 <constantatomformula symbol="=">
                  <function symbol="∪">
                   <function symbol="∪">
                    <variable symbol="y_0"/>
                    <variable symbol="y_0"/>
                   </function>
                   <variable symbol="x_0"/>
                  </function>
                  <function symbol="∪">
                   <variable symbol="y_0"/>
                   <variable symbol="x_0"/>
                  </function>
                 </constantatomformula>
                </formulalist>
               </sequent>
               <rule type="axiom" symbol="axiom">
                <sequent>
                 <formulalist/>
                 <formulalist>
                  <constantatomformula symbol="=">
                   <function symbol="∪">
                    <variable symbol="y_0"/>
                    <variable symbol="y_0"/>
                   </function>
                   <variable symbol="y_0"/>
                  </constantatomformula>
                 </formulalist>
                </sequent>
               </rule>
               <rule type="axiom" symbol="axiom">
                <sequent>
                 <formulalist/>
                 <formulalist>
                  <constantatomformula symbol="=">
                   <function symbol="∪">
                    <variable symbol="y_0"/>
                    <variable symbol="x_0"/>
                   </function>
                   <function symbol="∪">
                    <variable symbol="y_0"/>
                    <variable symbol="x_0"/>
                   </function>
                  </constantatomformula>
                 </formulalist>
                </sequent>
               </rule>
              </rule>
             </rule>
             <rule type="axiom" symbol="axiom">
              <sequent>
               <formulalist>
                <constantatomformula symbol="=">
                 <function symbol="∩">
                  <variable symbol="y_0"/>
                  <function symbol="∪">
                   <variable symbol="y_0"/>
                   <variable symbol="x_0"/>
                  </function>
                 </function>
                 <variable symbol="y_0"/>
                </constantatomformula>
               </formulalist>
               <formulalist>
                <constantatomformula symbol="=">
                 <function symbol="∩">
                  <variable symbol="y_0"/>
                  <function symbol="∪">
                   <variable symbol="y_0"/>
                   <variable symbol="x_0"/>
                  </function>
                 </function>
                 <variable symbol="y_0"/>
                </constantatomformula>
               </formulalist>
              </sequent>
             </rule>
            </rule>
           </rule>
          </rule>
         </rule>
        </rule>
       </rule>
      </rule>
     </rule>
    </rule>
   </rule>
  </rule>
 </proof>
 <proof symbol="\psi_{28}" calculus="LK">
  <rule type="weakr" symbol="w:r">
   <sequent>
    <formulalist>
     <constantatomformula symbol="L1" definition="L1"/>
    </formulalist>
    <formulalist>
     <constantatomformula symbol="=">
      <function symbol="∪">
       <variable symbol="x_0"/>
       <variable symbol="y_0"/>
      </function>
      <function symbol="∪">
       <variable symbol="y_0"/>
       <variable symbol="x_0"/>
      </function>
     </constantatomformula>
     <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
    </formulalist>
   </sequent>
   <rule type="contrl" symbol="c:l" param="2">
    <sequent>
     <formulalist>
      <constantatomformula symbol="L1" definition="L1"/>
     </formulalist>
     <formulalist>
      <constantatomformula symbol="=">
       <function symbol="∪">
        <variable symbol="x_0"/>
        <variable symbol="y_0"/>
       </function>
       <function symbol="∪">
        <variable symbol="y_0"/>
        <variable symbol="x_0"/>
       </function>
      </constantatomformula>
     </formulalist>
    </sequent>
    <rule type="perml" symbol="\pi:l" param="(1 2)">
     <sequent>
      <formulalist>
       <constantatomformula symbol="L1" definition="L1"/>
       <constantatomformula symbol="L1" definition="L1"/>
      </formulalist>
      <formulalist>
       <constantatomformula symbol="=">
        <function symbol="∪">
         <variable symbol="x_0"/>
         <variable symbol="y_0"/>
        </function>
        <function symbol="∪">
         <variable symbol="y_0"/>
         <variable symbol="x_0"/>
        </function>
       </constantatomformula>
      </formulalist>
     </sequent>
     <rule type="weakl" symbol="w:l">
      <sequent>
       <formulalist>
        <constantatomformula symbol="L1" definition="L1"/>
        <constantatomformula symbol="L1" definition="L1"/>
       </formulalist>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∪">
          <variable symbol="x_0"/>
          <variable symbol="y_0"/>
         </function>
         <function symbol="∪">
          <variable symbol="y_0"/>
          <variable symbol="x_0"/>
         </function>
        </constantatomformula>
       </formulalist>
      </sequent>
      <rule type="contrl" symbol="c:l" param="2">
       <sequent>
        <formulalist>
         <constantatomformula symbol="L1" definition="L1"/>
        </formulalist>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∪">
           <variable symbol="x_0"/>
           <variable symbol="y_0"/>
          </function>
          <function symbol="∪">
           <variable symbol="y_0"/>
           <variable symbol="x_0"/>
          </function>
         </constantatomformula>
        </formulalist>
       </sequent>
       <rule type="weakl" symbol="w:l">
        <sequent>
         <formulalist>
          <constantatomformula symbol="L1" definition="L1"/>
          <constantatomformula symbol="L1" definition="L1"/>
         </formulalist>
         <formulalist>
          <constantatomformula symbol="=">
           <function symbol="∪">
            <variable symbol="x_0"/>
            <variable symbol="y_0"/>
           </function>
           <function symbol="∪">
            <variable symbol="y_0"/>
            <variable symbol="x_0"/>
           </function>
          </constantatomformula>
         </formulalist>
        </sequent>
        <rule type="weakl" symbol="w:l">
         <sequent>
          <formulalist>
           <constantatomformula symbol="L1" definition="L1"/>
          </formulalist>
          <formulalist>
           <constantatomformula symbol="=">
            <function symbol="∪">
             <variable symbol="x_0"/>
             <variable symbol="y_0"/>
            </function>
            <function symbol="∪">
             <variable symbol="y_0"/>
             <variable symbol="x_0"/>
            </function>
           </constantatomformula>
          </formulalist>
         </sequent>
         <rule type="axiom" symbol="axiom">
          <sequent>
           <formulalist/>
           <formulalist>
            <constantatomformula symbol="=">
             <function symbol="∪">
              <variable symbol="x_0"/>
              <variable symbol="y_0"/>
             </function>
             <function symbol="∪">
              <variable symbol="y_0"/>
              <variable symbol="x_0"/>
             </function>
            </constantatomformula>
           </formulalist>
          </sequent>
         </rule>
        </rule>
       </rule>
      </rule>
     </rule>
    </rule>
   </rule>
  </rule>
 </proof>
 <proof symbol="\psi_{29}" calculus="LK">
  <rule type="weakr" symbol="w:r">
   <sequent>
    <formulalist>
     <constantatomformula symbol="L1" definition="L1"/>
    </formulalist>
    <formulalist>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <variable symbol="x_0"/>
       <function symbol="∪">
        <variable symbol="x_0"/>
        <variable symbol="y_0"/>
       </function>
      </function>
      <variable symbol="x_0"/>
     </constantatomformula>
     <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
    </formulalist>
   </sequent>
   <rule type="contrl" symbol="c:l" param="2">
    <sequent>
     <formulalist>
      <constantatomformula symbol="L1" definition="L1"/>
     </formulalist>
     <formulalist>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <variable symbol="x_0"/>
        <function symbol="∪">
         <variable symbol="x_0"/>
         <variable symbol="y_0"/>
        </function>
       </function>
       <variable symbol="x_0"/>
      </constantatomformula>
     </formulalist>
    </sequent>
    <rule type="perml" symbol="\pi:l" param="(1 2)">
     <sequent>
      <formulalist>
       <constantatomformula symbol="L1" definition="L1"/>
       <constantatomformula symbol="L1" definition="L1"/>
      </formulalist>
      <formulalist>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <variable symbol="x_0"/>
         <function symbol="∪">
          <variable symbol="x_0"/>
          <variable symbol="y_0"/>
         </function>
        </function>
        <variable symbol="x_0"/>
       </constantatomformula>
      </formulalist>
     </sequent>
     <rule type="weakl" symbol="w:l">
      <sequent>
       <formulalist>
        <constantatomformula symbol="L1" definition="L1"/>
        <constantatomformula symbol="L1" definition="L1"/>
       </formulalist>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <variable symbol="x_0"/>
          <function symbol="∪">
           <variable symbol="x_0"/>
           <variable symbol="y_0"/>
          </function>
         </function>
         <variable symbol="x_0"/>
        </constantatomformula>
       </formulalist>
      </sequent>
      <rule type="contrl" symbol="c:l" param="2">
       <sequent>
        <formulalist>
         <constantatomformula symbol="L1" definition="L1"/>
        </formulalist>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∩">
           <variable symbol="x_0"/>
           <function symbol="∪">
            <variable symbol="x_0"/>
            <variable symbol="y_0"/>
           </function>
          </function>
          <variable symbol="x_0"/>
         </constantatomformula>
        </formulalist>
       </sequent>
       <rule type="perml" symbol="\pi:l" param="(1 2)">
        <sequent>
         <formulalist>
          <constantatomformula symbol="L1" definition="L1"/>
          <constantatomformula symbol="L1" definition="L1"/>
         </formulalist>
         <formulalist>
          <constantatomformula symbol="=">
           <function symbol="∩">
            <variable symbol="x_0"/>
            <function symbol="∪">
             <variable symbol="x_0"/>
             <variable symbol="y_0"/>
            </function>
           </function>
           <variable symbol="x_0"/>
          </constantatomformula>
         </formulalist>
        </sequent>
        <rule type="weakl" symbol="w:l">
         <sequent>
          <formulalist>
           <constantatomformula symbol="L1" definition="L1"/>
           <constantatomformula symbol="L1" definition="L1"/>
          </formulalist>
          <formulalist>
           <constantatomformula symbol="=">
            <function symbol="∩">
             <variable symbol="x_0"/>
             <function symbol="∪">
              <variable symbol="x_0"/>
              <variable symbol="y_0"/>
             </function>
            </function>
            <variable symbol="x_0"/>
           </constantatomformula>
          </formulalist>
         </sequent>
         <rule type="defl" symbol="d:l" param="();[()];()">
          <sequent>
           <formulalist>
            <constantatomformula symbol="L1" definition="L1"/>
           </formulalist>
           <formulalist>
            <constantatomformula symbol="=">
             <function symbol="∩">
              <variable symbol="x_0"/>
              <function symbol="∪">
               <variable symbol="x_0"/>
               <variable symbol="y_0"/>
              </function>
             </function>
             <variable symbol="x_0"/>
            </constantatomformula>
           </formulalist>
          </sequent>
          <rule type="foralll" symbol="\all:l">
           <sequent>
            <formulalist>
             <quantifiedformula type="all">
              <variable symbol="x"/>
              <quantifiedformula type="all">
               <variable symbol="y"/>
               <conjunctiveformula type="and">
                <conjunctiveformula type="impl">
                 <constantatomformula symbol="=">
                  <function symbol="∩">
                   <variable symbol="x"/>
                   <variable symbol="y"/>
                  </function>
                  <variable symbol="x"/>
                 </constantatomformula>
                 <constantatomformula symbol="=">
                  <function symbol="∪">
                   <variable symbol="x"/>
                   <variable symbol="y"/>
                  </function>
                  <variable symbol="y"/>
                 </constantatomformula>
                </conjunctiveformula>
                <conjunctiveformula type="impl">
                 <constantatomformula symbol="=">
                  <function symbol="∪">
                   <variable symbol="x"/>
                   <variable symbol="y"/>
                  </function>
                  <variable symbol="y"/>
                 </constantatomformula>
                 <constantatomformula symbol="=">
                  <function symbol="∩">
                   <variable symbol="x"/>
                   <variable symbol="y"/>
                  </function>
                  <variable symbol="x"/>
                 </constantatomformula>
                </conjunctiveformula>
               </conjunctiveformula>
              </quantifiedformula>
             </quantifiedformula>
            </formulalist>
            <formulalist>
             <constantatomformula symbol="=">
              <function symbol="∩">
               <variable symbol="x_0"/>
               <function symbol="∪">
                <variable symbol="x_0"/>
                <variable symbol="y_0"/>
               </function>
              </function>
              <variable symbol="x_0"/>
             </constantatomformula>
            </formulalist>
           </sequent>
           <rule type="foralll" symbol="\all:l">
            <sequent>
             <formulalist>
              <quantifiedformula type="all">
               <variable symbol="y"/>
               <conjunctiveformula type="and">
                <conjunctiveformula type="impl">
                 <constantatomformula symbol="=">
                  <function symbol="∩">
                   <variable symbol="x_0"/>
                   <variable symbol="y"/>
                  </function>
                  <variable symbol="x_0"/>
                 </constantatomformula>
                 <constantatomformula symbol="=">
                  <function symbol="∪">
                   <variable symbol="x_0"/>
                   <variable symbol="y"/>
                  </function>
                  <variable symbol="y"/>
                 </constantatomformula>
                </conjunctiveformula>
                <conjunctiveformula type="impl">
                 <constantatomformula symbol="=">
                  <function symbol="∪">
                   <variable symbol="x_0"/>
                   <variable symbol="y"/>
                  </function>
                  <variable symbol="y"/>
                 </constantatomformula>
                 <constantatomformula symbol="=">
                  <function symbol="∩">
                   <variable symbol="x_0"/>
                   <variable symbol="y"/>
                  </function>
                  <variable symbol="x_0"/>
                 </constantatomformula>
                </conjunctiveformula>
               </conjunctiveformula>
              </quantifiedformula>
             </formulalist>
             <formulalist>
              <constantatomformula symbol="=">
               <function symbol="∩">
                <variable symbol="x_0"/>
                <function symbol="∪">
                 <variable symbol="x_0"/>
                 <variable symbol="y_0"/>
                </function>
               </function>
               <variable symbol="x_0"/>
              </constantatomformula>
             </formulalist>
            </sequent>
            <rule type="andl2" symbol="\and:l2">
             <sequent>
              <formulalist>
               <conjunctiveformula type="and">
                <conjunctiveformula type="impl">
                 <constantatomformula symbol="=">
                  <function symbol="∩">
                   <variable symbol="x_0"/>
                   <function symbol="∪">
                    <variable symbol="x_0"/>
                    <variable symbol="y_0"/>
                   </function>
                  </function>
                  <variable symbol="x_0"/>
                 </constantatomformula>
                 <constantatomformula symbol="=">
                  <function symbol="∪">
                   <variable symbol="x_0"/>
                   <function symbol="∪">
                    <variable symbol="x_0"/>
                    <variable symbol="y_0"/>
                   </function>
                  </function>
                  <function symbol="∪">
                   <variable symbol="x_0"/>
                   <variable symbol="y_0"/>
                  </function>
                 </constantatomformula>
                </conjunctiveformula>
                <conjunctiveformula type="impl">
                 <constantatomformula symbol="=">
                  <function symbol="∪">
                   <variable symbol="x_0"/>
                   <function symbol="∪">
                    <variable symbol="x_0"/>
                    <variable symbol="y_0"/>
                   </function>
                  </function>
                  <function symbol="∪">
                   <variable symbol="x_0"/>
                   <variable symbol="y_0"/>
                  </function>
                 </constantatomformula>
                 <constantatomformula symbol="=">
                  <function symbol="∩">
                   <variable symbol="x_0"/>
                   <function symbol="∪">
                    <variable symbol="x_0"/>
                    <variable symbol="y_0"/>
                   </function>
                  </function>
                  <variable symbol="x_0"/>
                 </constantatomformula>
                </conjunctiveformula>
               </conjunctiveformula>
              </formulalist>
              <formulalist>
               <constantatomformula symbol="=">
                <function symbol="∩">
                 <variable symbol="x_0"/>
                 <function symbol="∪">
                  <variable symbol="x_0"/>
                  <variable symbol="y_0"/>
                 </function>
                </function>
                <variable symbol="x_0"/>
               </constantatomformula>
              </formulalist>
             </sequent>
             <rule type="impll" symbol="\impl:l">
              <sequent>
               <formulalist>
                <conjunctiveformula type="impl">
                 <constantatomformula symbol="=">
                  <function symbol="∪">
                   <variable symbol="x_0"/>
                   <function symbol="∪">
                    <variable symbol="x_0"/>
                    <variable symbol="y_0"/>
                   </function>
                  </function>
                  <function symbol="∪">
                   <variable symbol="x_0"/>
                   <variable symbol="y_0"/>
                  </function>
                 </constantatomformula>
                 <constantatomformula symbol="=">
                  <function symbol="∩">
                   <variable symbol="x_0"/>
                   <function symbol="∪">
                    <variable symbol="x_0"/>
                    <variable symbol="y_0"/>
                   </function>
                  </function>
                  <variable symbol="x_0"/>
                 </constantatomformula>
                </conjunctiveformula>
               </formulalist>
               <formulalist>
                <constantatomformula symbol="=">
                 <function symbol="∩">
                  <variable symbol="x_0"/>
                  <function symbol="∪">
                   <variable symbol="x_0"/>
                   <variable symbol="y_0"/>
                  </function>
                 </function>
                 <variable symbol="x_0"/>
                </constantatomformula>
               </formulalist>
              </sequent>
              <rule type="eqr1" symbol="=:r1" param="(1)">
               <sequent>
                <formulalist/>
                <formulalist>
                 <constantatomformula symbol="=">
                  <function symbol="∪">
                   <variable symbol="x_0"/>
                   <function symbol="∪">
                    <variable symbol="x_0"/>
                    <variable symbol="y_0"/>
                   </function>
                  </function>
                  <function symbol="∪">
                   <variable symbol="x_0"/>
                   <variable symbol="y_0"/>
                  </function>
                 </constantatomformula>
                </formulalist>
               </sequent>
               <rule type="axiom" symbol="axiom">
                <sequent>
                 <formulalist/>
                 <formulalist>
                  <constantatomformula symbol="=">
                   <function symbol="∪">
                    <function symbol="∪">
                     <variable symbol="x_0"/>
                     <variable symbol="x_0"/>
                    </function>
                    <variable symbol="y_0"/>
                   </function>
                   <function symbol="∪">
                    <variable symbol="x_0"/>
                    <function symbol="∪">
                     <variable symbol="x_0"/>
                     <variable symbol="y_0"/>
                    </function>
                   </function>
                  </constantatomformula>
                 </formulalist>
                </sequent>
               </rule>
               <rule type="eqr2" symbol="=:r2" param="(1 1)">
                <sequent>
                 <formulalist/>
                 <formulalist>
                  <constantatomformula symbol="=">
                   <function symbol="∪">
                    <function symbol="∪">
                     <variable symbol="x_0"/>
                     <variable symbol="x_0"/>
                    </function>
                    <variable symbol="y_0"/>
                   </function>
                   <function symbol="∪">
                    <variable symbol="x_0"/>
                    <variable symbol="y_0"/>
                   </function>
                  </constantatomformula>
                 </formulalist>
                </sequent>
                <rule type="axiom" symbol="axiom">
                 <sequent>
                  <formulalist/>
                  <formulalist>
                   <constantatomformula symbol="=">
                    <function symbol="∪">
                     <variable symbol="x_0"/>
                     <variable symbol="x_0"/>
                    </function>
                    <variable symbol="x_0"/>
                   </constantatomformula>
                  </formulalist>
                 </sequent>
                </rule>
                <rule type="axiom" symbol="axiom">
                 <sequent>
                  <formulalist/>
                  <formulalist>
                   <constantatomformula symbol="=">
                    <function symbol="∪">
                     <variable symbol="x_0"/>
                     <variable symbol="y_0"/>
                    </function>
                    <function symbol="∪">
                     <variable symbol="x_0"/>
                     <variable symbol="y_0"/>
                    </function>
                   </constantatomformula>
                  </formulalist>
                 </sequent>
                </rule>
               </rule>
              </rule>
              <rule type="axiom" symbol="axiom">
               <sequent>
                <formulalist>
                 <constantatomformula symbol="=">
                  <function symbol="∩">
                   <variable symbol="x_0"/>
                   <function symbol="∪">
                    <variable symbol="x_0"/>
                    <variable symbol="y_0"/>
                   </function>
                  </function>
                  <variable symbol="x_0"/>
                 </constantatomformula>
                </formulalist>
                <formulalist>
                 <constantatomformula symbol="=">
                  <function symbol="∩">
                   <variable symbol="x_0"/>
                   <function symbol="∪">
                    <variable symbol="x_0"/>
                    <variable symbol="y_0"/>
                   </function>
                  </function>
                  <variable symbol="x_0"/>
                 </constantatomformula>
                </formulalist>
               </sequent>
              </rule>
             </rule>
            </rule>
           </rule>
          </rule>
         </rule>
        </rule>
       </rule>
      </rule>
     </rule>
    </rule>
   </rule>
  </rule>
 </proof>
 <proof symbol="\psi_{30}" calculus="LK">
  <rule type="weakr" symbol="w:r">
   <sequent>
    <formulalist>
     <constantatomformula symbol="L1" definition="L1"/>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <variable symbol="z_0"/>
       <variable symbol="y_0"/>
      </function>
      <variable symbol="z_0"/>
     </constantatomformula>
    </formulalist>
    <formulalist>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <variable symbol="z_0"/>
       <variable symbol="y_0"/>
      </function>
      <variable symbol="z_0"/>
     </constantatomformula>
     <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
    </formulalist>
   </sequent>
   <rule type="weakl" symbol="w:l">
    <sequent>
     <formulalist>
      <constantatomformula symbol="L1" definition="L1"/>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <variable symbol="z_0"/>
        <variable symbol="y_0"/>
       </function>
       <variable symbol="z_0"/>
      </constantatomformula>
     </formulalist>
     <formulalist>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <variable symbol="z_0"/>
        <variable symbol="y_0"/>
       </function>
       <variable symbol="z_0"/>
      </constantatomformula>
     </formulalist>
    </sequent>
    <rule type="axiom" symbol="axiom">
     <sequent>
      <formulalist>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <variable symbol="z_0"/>
         <variable symbol="y_0"/>
        </function>
        <variable symbol="z_0"/>
       </constantatomformula>
      </formulalist>
      <formulalist>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <variable symbol="z_0"/>
         <variable symbol="y_0"/>
        </function>
        <variable symbol="z_0"/>
       </constantatomformula>
      </formulalist>
     </sequent>
    </rule>
   </rule>
  </rule>
 </proof>
 <proof symbol="\psi_{31}" calculus="LK">
  <rule type="weakr" symbol="w:r">
   <sequent>
    <formulalist>
     <constantatomformula symbol="L1" definition="L1"/>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <variable symbol="z_0"/>
       <variable symbol="x_0"/>
      </function>
      <variable symbol="z_0"/>
     </constantatomformula>
    </formulalist>
    <formulalist>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <variable symbol="z_0"/>
       <variable symbol="x_0"/>
      </function>
      <variable symbol="z_0"/>
     </constantatomformula>
     <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
    </formulalist>
   </sequent>
   <rule type="weakl" symbol="w:l">
    <sequent>
     <formulalist>
      <constantatomformula symbol="L1" definition="L1"/>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <variable symbol="z_0"/>
        <variable symbol="x_0"/>
       </function>
       <variable symbol="z_0"/>
      </constantatomformula>
     </formulalist>
     <formulalist>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <variable symbol="z_0"/>
        <variable symbol="x_0"/>
       </function>
       <variable symbol="z_0"/>
      </constantatomformula>
     </formulalist>
    </sequent>
    <rule type="axiom" symbol="axiom">
     <sequent>
      <formulalist>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <variable symbol="z_0"/>
         <variable symbol="x_0"/>
        </function>
        <variable symbol="z_0"/>
       </constantatomformula>
      </formulalist>
      <formulalist>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <variable symbol="z_0"/>
         <variable symbol="x_0"/>
        </function>
        <variable symbol="z_0"/>
       </constantatomformula>
      </formulalist>
     </sequent>
    </rule>
   </rule>
  </rule>
 </proof>
 <proof symbol="\psi_{32}" calculus="LK">
  <rule type="weakr" symbol="w:r">
   <sequent>
    <formulalist>
     <constantatomformula symbol="L1" definition="L1"/>
    </formulalist>
    <formulalist>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <function symbol="∩">
        <variable symbol="z_0"/>
        <variable symbol="x_0"/>
       </function>
       <variable symbol="y_0"/>
      </function>
      <function symbol="∩">
       <variable symbol="z_0"/>
       <function symbol="∩">
        <variable symbol="x_0"/>
        <variable symbol="y_0"/>
       </function>
      </function>
     </constantatomformula>
     <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
    </formulalist>
   </sequent>
   <rule type="weakl" symbol="w:l">
    <sequent>
     <formulalist>
      <constantatomformula symbol="L1" definition="L1"/>
     </formulalist>
     <formulalist>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <function symbol="∩">
         <variable symbol="z_0"/>
         <variable symbol="x_0"/>
        </function>
        <variable symbol="y_0"/>
       </function>
       <function symbol="∩">
        <variable symbol="z_0"/>
        <function symbol="∩">
         <variable symbol="x_0"/>
         <variable symbol="y_0"/>
        </function>
       </function>
      </constantatomformula>
     </formulalist>
    </sequent>
    <rule type="axiom" symbol="axiom">
     <sequent>
      <formulalist/>
      <formulalist>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <function symbol="∩">
          <variable symbol="z_0"/>
          <variable symbol="x_0"/>
         </function>
         <variable symbol="y_0"/>
        </function>
        <function symbol="∩">
         <variable symbol="z_0"/>
         <function symbol="∩">
          <variable symbol="x_0"/>
          <variable symbol="y_0"/>
         </function>
        </function>
       </constantatomformula>
      </formulalist>
     </sequent>
    </rule>
   </rule>
  </rule>
 </proof>
 <proof symbol="\psi_{33}" calculus="LK">
  <rule type="weakr" symbol="w:r">
   <sequent>
    <formulalist>
     <constantatomformula symbol="L1" definition="L1"/>
    </formulalist>
    <formulalist>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <variable symbol="x_0"/>
       <variable symbol="y_0"/>
      </function>
      <function symbol="∩">
       <variable symbol="x_0"/>
       <variable symbol="y_0"/>
      </function>
     </constantatomformula>
     <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
    </formulalist>
   </sequent>
   <rule type="weakl" symbol="w:l">
    <sequent>
     <formulalist>
      <constantatomformula symbol="L1" definition="L1"/>
     </formulalist>
     <formulalist>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <variable symbol="x_0"/>
        <variable symbol="y_0"/>
       </function>
       <function symbol="∩">
        <variable symbol="x_0"/>
        <variable symbol="y_0"/>
       </function>
      </constantatomformula>
     </formulalist>
    </sequent>
    <rule type="axiom" symbol="axiom">
     <sequent>
      <formulalist/>
      <formulalist>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <variable symbol="x_0"/>
         <variable symbol="y_0"/>
        </function>
        <function symbol="∩">
         <variable symbol="x_0"/>
         <variable symbol="y_0"/>
        </function>
       </constantatomformula>
      </formulalist>
     </sequent>
    </rule>
   </rule>
  </rule>
 </proof>
 <proof symbol="\psi_{34}" calculus="LK">
  <rule type="weakr" symbol="w:r">
   <sequent>
    <formulalist>
     <constantatomformula symbol="L1" definition="L1"/>
    </formulalist>
    <formulalist>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <variable symbol="y_0"/>
       <variable symbol="y_0"/>
      </function>
      <variable symbol="y_0"/>
     </constantatomformula>
     <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
    </formulalist>
   </sequent>
   <rule type="weakl" symbol="w:l">
    <sequent>
     <formulalist>
      <constantatomformula symbol="L1" definition="L1"/>
     </formulalist>
     <formulalist>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <variable symbol="y_0"/>
        <variable symbol="y_0"/>
       </function>
       <variable symbol="y_0"/>
      </constantatomformula>
     </formulalist>
    </sequent>
    <rule type="axiom" symbol="axiom">
     <sequent>
      <formulalist/>
      <formulalist>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <variable symbol="y_0"/>
         <variable symbol="y_0"/>
        </function>
        <variable symbol="y_0"/>
       </constantatomformula>
      </formulalist>
     </sequent>
    </rule>
   </rule>
  </rule>
 </proof>
 <proof symbol="\psi_{35}" calculus="LK">
  <rule type="weakr" symbol="w:r">
   <sequent>
    <formulalist>
     <constantatomformula symbol="L1" definition="L1"/>
    </formulalist>
    <formulalist>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <function symbol="∩">
        <variable symbol="x_0"/>
        <variable symbol="y_0"/>
       </function>
       <variable symbol="y_0"/>
      </function>
      <function symbol="∩">
       <variable symbol="x_0"/>
       <function symbol="∩">
        <variable symbol="y_0"/>
        <variable symbol="y_0"/>
       </function>
      </function>
     </constantatomformula>
     <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
    </formulalist>
   </sequent>
   <rule type="weakl" symbol="w:l">
    <sequent>
     <formulalist>
      <constantatomformula symbol="L1" definition="L1"/>
     </formulalist>
     <formulalist>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <function symbol="∩">
         <variable symbol="x_0"/>
         <variable symbol="y_0"/>
        </function>
        <variable symbol="y_0"/>
       </function>
       <function symbol="∩">
        <variable symbol="x_0"/>
        <function symbol="∩">
         <variable symbol="y_0"/>
         <variable symbol="y_0"/>
        </function>
       </function>
      </constantatomformula>
     </formulalist>
    </sequent>
    <rule type="axiom" symbol="axiom">
     <sequent>
      <formulalist/>
      <formulalist>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <function symbol="∩">
          <variable symbol="x_0"/>
          <variable symbol="y_0"/>
         </function>
         <variable symbol="y_0"/>
        </function>
        <function symbol="∩">
         <variable symbol="x_0"/>
         <function symbol="∩">
          <variable symbol="y_0"/>
          <variable symbol="y_0"/>
         </function>
        </function>
       </constantatomformula>
      </formulalist>
     </sequent>
    </rule>
   </rule>
  </rule>
 </proof>
 <proof symbol="\psi_{36}" calculus="LK">
  <rule type="weakr" symbol="w:r">
   <sequent>
    <formulalist>
     <constantatomformula symbol="L1" definition="L1"/>
    </formulalist>
    <formulalist>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <variable symbol="x_0"/>
       <variable symbol="y_0"/>
      </function>
      <function symbol="∩">
       <variable symbol="x_0"/>
       <variable symbol="y_0"/>
      </function>
     </constantatomformula>
     <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
    </formulalist>
   </sequent>
   <rule type="weakl" symbol="w:l">
    <sequent>
     <formulalist>
      <constantatomformula symbol="L1" definition="L1"/>
     </formulalist>
     <formulalist>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <variable symbol="x_0"/>
        <variable symbol="y_0"/>
       </function>
       <function symbol="∩">
        <variable symbol="x_0"/>
        <variable symbol="y_0"/>
       </function>
      </constantatomformula>
     </formulalist>
    </sequent>
    <rule type="axiom" symbol="axiom">
     <sequent>
      <formulalist/>
      <formulalist>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <variable symbol="x_0"/>
         <variable symbol="y_0"/>
        </function>
        <function symbol="∩">
         <variable symbol="x_0"/>
         <variable symbol="y_0"/>
        </function>
       </constantatomformula>
      </formulalist>
     </sequent>
    </rule>
   </rule>
  </rule>
 </proof>
 <proof symbol="\psi_{37}" calculus="LK">
  <rule type="weakr" symbol="w:r">
   <sequent>
    <formulalist>
     <constantatomformula symbol="L1" definition="L1"/>
    </formulalist>
    <formulalist>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <variable symbol="x_0"/>
       <variable symbol="x_0"/>
      </function>
      <variable symbol="x_0"/>
     </constantatomformula>
     <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
    </formulalist>
   </sequent>
   <rule type="weakl" symbol="w:l">
    <sequent>
     <formulalist>
      <constantatomformula symbol="L1" definition="L1"/>
     </formulalist>
     <formulalist>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <variable symbol="x_0"/>
        <variable symbol="x_0"/>
       </function>
       <variable symbol="x_0"/>
      </constantatomformula>
     </formulalist>
    </sequent>
    <rule type="axiom" symbol="axiom">
     <sequent>
      <formulalist/>
      <formulalist>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <variable symbol="x_0"/>
         <variable symbol="x_0"/>
        </function>
        <variable symbol="x_0"/>
       </constantatomformula>
      </formulalist>
     </sequent>
    </rule>
   </rule>
  </rule>
 </proof>
 <proof symbol="\psi_{38}" calculus="LK">
  <rule type="weakr" symbol="w:r">
   <sequent>
    <formulalist>
     <constantatomformula symbol="L1" definition="L1"/>
    </formulalist>
    <formulalist>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <function symbol="∩">
        <variable symbol="x_0"/>
        <variable symbol="x_0"/>
       </function>
       <variable symbol="y_0"/>
      </function>
      <function symbol="∩">
       <variable symbol="x_0"/>
       <function symbol="∩">
        <variable symbol="x_0"/>
        <variable symbol="y_0"/>
       </function>
      </function>
     </constantatomformula>
     <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
    </formulalist>
   </sequent>
   <rule type="weakl" symbol="w:l">
    <sequent>
     <formulalist>
      <constantatomformula symbol="L1" definition="L1"/>
     </formulalist>
     <formulalist>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <function symbol="∩">
         <variable symbol="x_0"/>
         <variable symbol="x_0"/>
        </function>
        <variable symbol="y_0"/>
       </function>
       <function symbol="∩">
        <variable symbol="x_0"/>
        <function symbol="∩">
         <variable symbol="x_0"/>
         <variable symbol="y_0"/>
        </function>
       </function>
      </constantatomformula>
     </formulalist>
    </sequent>
    <rule type="axiom" symbol="axiom">
     <sequent>
      <formulalist/>
      <formulalist>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <function symbol="∩">
          <variable symbol="x_0"/>
          <variable symbol="x_0"/>
         </function>
         <variable symbol="y_0"/>
        </function>
        <function symbol="∩">
         <variable symbol="x_0"/>
         <function symbol="∩">
          <variable symbol="x_0"/>
          <variable symbol="y_0"/>
         </function>
        </function>
       </constantatomformula>
      </formulalist>
     </sequent>
    </rule>
   </rule>
  </rule>
 </proof>
 <proof symbol="\psi_{39}" calculus="LK">
  <rule type="weakr" symbol="w:r">
   <sequent>
    <formulalist>
     <constantatomformula symbol="L1" definition="L1"/>
    </formulalist>
    <formulalist>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <function symbol="∩">
        <variable symbol="x_0"/>
        <variable symbol="y_0"/>
       </function>
       <variable symbol="x_0"/>
      </function>
      <function symbol="∩">
       <variable symbol="x_0"/>
       <function symbol="∩">
        <variable symbol="x_0"/>
        <variable symbol="y_0"/>
       </function>
      </function>
     </constantatomformula>
     <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
    </formulalist>
   </sequent>
   <rule type="weakl" symbol="w:l">
    <sequent>
     <formulalist>
      <constantatomformula symbol="L1" definition="L1"/>
     </formulalist>
     <formulalist>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <function symbol="∩">
         <variable symbol="x_0"/>
         <variable symbol="y_0"/>
        </function>
        <variable symbol="x_0"/>
       </function>
       <function symbol="∩">
        <variable symbol="x_0"/>
        <function symbol="∩">
         <variable symbol="x_0"/>
         <variable symbol="y_0"/>
        </function>
       </function>
      </constantatomformula>
     </formulalist>
    </sequent>
    <rule type="axiom" symbol="axiom">
     <sequent>
      <formulalist/>
      <formulalist>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <function symbol="∩">
          <variable symbol="x_0"/>
          <variable symbol="y_0"/>
         </function>
         <variable symbol="x_0"/>
        </function>
        <function symbol="∩">
         <variable symbol="x_0"/>
         <function symbol="∩">
          <variable symbol="x_0"/>
          <variable symbol="y_0"/>
         </function>
        </function>
       </constantatomformula>
      </formulalist>
     </sequent>
    </rule>
   </rule>
  </rule>
 </proof>
 <proof symbol="\psi_{40}" calculus="LK">
  <rule type="weakr" symbol="w:r">
   <sequent>
    <formulalist>
     <constantatomformula symbol="L1" definition="L1"/>
    </formulalist>
    <formulalist>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <function symbol="∩">
        <variable symbol="x_0"/>
        <variable symbol="y_0"/>
       </function>
       <variable symbol="z_0"/>
      </function>
      <function symbol="∩">
       <variable symbol="x_0"/>
       <function symbol="∩">
        <variable symbol="y_0"/>
        <variable symbol="z_0"/>
       </function>
      </function>
     </constantatomformula>
     <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
    </formulalist>
   </sequent>
   <rule type="weakl" symbol="w:l">
    <sequent>
     <formulalist>
      <constantatomformula symbol="L1" definition="L1"/>
     </formulalist>
     <formulalist>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <function symbol="∩">
         <variable symbol="x_0"/>
         <variable symbol="y_0"/>
        </function>
        <variable symbol="z_0"/>
       </function>
       <function symbol="∩">
        <variable symbol="x_0"/>
        <function symbol="∩">
         <variable symbol="y_0"/>
         <variable symbol="z_0"/>
        </function>
       </function>
      </constantatomformula>
     </formulalist>
    </sequent>
    <rule type="axiom" symbol="axiom">
     <sequent>
      <formulalist/>
      <formulalist>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <function symbol="∩">
          <variable symbol="x_0"/>
          <variable symbol="y_0"/>
         </function>
         <variable symbol="z_0"/>
        </function>
        <function symbol="∩">
         <variable symbol="x_0"/>
         <function symbol="∩">
          <variable symbol="y_0"/>
          <variable symbol="z_0"/>
         </function>
        </function>
       </constantatomformula>
      </formulalist>
     </sequent>
    </rule>
   </rule>
  </rule>
 </proof>
 <proof symbol="\psi_{41}" calculus="LK">
  <rule type="weakr" symbol="w:r">
   <sequent>
    <formulalist>
     <constantatomformula symbol="L1" definition="L1"/>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <variable symbol="y_0"/>
       <variable symbol="z_0"/>
      </function>
      <variable symbol="y_0"/>
     </constantatomformula>
    </formulalist>
    <formulalist>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <variable symbol="y_0"/>
       <variable symbol="z_0"/>
      </function>
      <variable symbol="y_0"/>
     </constantatomformula>
     <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
    </formulalist>
   </sequent>
   <rule type="weakl" symbol="w:l">
    <sequent>
     <formulalist>
      <constantatomformula symbol="L1" definition="L1"/>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <variable symbol="y_0"/>
        <variable symbol="z_0"/>
       </function>
       <variable symbol="y_0"/>
      </constantatomformula>
     </formulalist>
     <formulalist>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <variable symbol="y_0"/>
        <variable symbol="z_0"/>
       </function>
       <variable symbol="y_0"/>
      </constantatomformula>
     </formulalist>
    </sequent>
    <rule type="axiom" symbol="axiom">
     <sequent>
      <formulalist>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <variable symbol="y_0"/>
         <variable symbol="z_0"/>
        </function>
        <variable symbol="y_0"/>
       </constantatomformula>
      </formulalist>
      <formulalist>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <variable symbol="y_0"/>
         <variable symbol="z_0"/>
        </function>
        <variable symbol="y_0"/>
       </constantatomformula>
      </formulalist>
     </sequent>
    </rule>
   </rule>
  </rule>
 </proof>
 <proof symbol="\psi_{42}" calculus="LK">
  <rule type="weakr" symbol="w:r">
   <sequent>
    <formulalist>
     <constantatomformula symbol="L1" definition="L1"/>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <variable symbol="x_0"/>
       <variable symbol="y_0"/>
      </function>
      <variable symbol="x_0"/>
     </constantatomformula>
    </formulalist>
    <formulalist>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <variable symbol="x_0"/>
       <variable symbol="y_0"/>
      </function>
      <variable symbol="x_0"/>
     </constantatomformula>
     <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
    </formulalist>
   </sequent>
   <rule type="weakl" symbol="w:l">
    <sequent>
     <formulalist>
      <constantatomformula symbol="L1" definition="L1"/>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <variable symbol="x_0"/>
        <variable symbol="y_0"/>
       </function>
       <variable symbol="x_0"/>
      </constantatomformula>
     </formulalist>
     <formulalist>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <variable symbol="x_0"/>
        <variable symbol="y_0"/>
       </function>
       <variable symbol="x_0"/>
      </constantatomformula>
     </formulalist>
    </sequent>
    <rule type="axiom" symbol="axiom">
     <sequent>
      <formulalist>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <variable symbol="x_0"/>
         <variable symbol="y_0"/>
        </function>
        <variable symbol="x_0"/>
       </constantatomformula>
      </formulalist>
      <formulalist>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <variable symbol="x_0"/>
         <variable symbol="y_0"/>
        </function>
        <variable symbol="x_0"/>
       </constantatomformula>
      </formulalist>
     </sequent>
    </rule>
   </rule>
  </rule>
 </proof>
 <proof symbol="\psi_{43}" calculus="LK">
  <rule type="weakr" symbol="w:r">
   <sequent>
    <formulalist>
     <constantatomformula symbol="L1" definition="L1"/>
     <constantatomformula symbol="=">
      <variable symbol="x_0"/>
      <variable symbol="y_0"/>
     </constantatomformula>
    </formulalist>
    <formulalist>
     <constantatomformula symbol="=">
      <variable symbol="x_0"/>
      <variable symbol="y_0"/>
     </constantatomformula>
     <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
    </formulalist>
   </sequent>
   <rule type="weakl" symbol="w:l">
    <sequent>
     <formulalist>
      <constantatomformula symbol="L1" definition="L1"/>
      <constantatomformula symbol="=">
       <variable symbol="x_0"/>
       <variable symbol="y_0"/>
      </constantatomformula>
     </formulalist>
     <formulalist>
      <constantatomformula symbol="=">
       <variable symbol="x_0"/>
       <variable symbol="y_0"/>
      </constantatomformula>
     </formulalist>
    </sequent>
    <rule type="axiom" symbol="axiom">
     <sequent>
      <formulalist>
       <constantatomformula symbol="=">
        <variable symbol="x_0"/>
        <variable symbol="y_0"/>
       </constantatomformula>
      </formulalist>
      <formulalist>
       <constantatomformula symbol="=">
        <variable symbol="x_0"/>
        <variable symbol="y_0"/>
       </constantatomformula>
      </formulalist>
     </sequent>
    </rule>
   </rule>
  </rule>
 </proof>
 <proof symbol="\psi_{44}" calculus="LK">
  <rule type="weakr" symbol="w:r">
   <sequent>
    <formulalist>
     <constantatomformula symbol="L1" definition="L1"/>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <variable symbol="x_0"/>
       <variable symbol="y_0"/>
      </function>
      <variable symbol="x_0"/>
     </constantatomformula>
    </formulalist>
    <formulalist>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <variable symbol="x_0"/>
       <variable symbol="y_0"/>
      </function>
      <variable symbol="x_0"/>
     </constantatomformula>
     <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
    </formulalist>
   </sequent>
   <rule type="weakl" symbol="w:l">
    <sequent>
     <formulalist>
      <constantatomformula symbol="L1" definition="L1"/>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <variable symbol="x_0"/>
        <variable symbol="y_0"/>
       </function>
       <variable symbol="x_0"/>
      </constantatomformula>
     </formulalist>
     <formulalist>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <variable symbol="x_0"/>
        <variable symbol="y_0"/>
       </function>
       <variable symbol="x_0"/>
      </constantatomformula>
     </formulalist>
    </sequent>
    <rule type="axiom" symbol="axiom">
     <sequent>
      <formulalist>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <variable symbol="x_0"/>
         <variable symbol="y_0"/>
        </function>
        <variable symbol="x_0"/>
       </constantatomformula>
      </formulalist>
      <formulalist>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <variable symbol="x_0"/>
         <variable symbol="y_0"/>
        </function>
        <variable symbol="x_0"/>
       </constantatomformula>
      </formulalist>
     </sequent>
    </rule>
   </rule>
  </rule>
 </proof>
 <proof symbol="\psi_{45}" calculus="LK">
  <rule type="weakr" symbol="w:r">
   <sequent>
    <formulalist>
     <constantatomformula symbol="L1" definition="L1"/>
    </formulalist>
    <formulalist>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <variable symbol="x_0"/>
       <variable symbol="y_0"/>
      </function>
      <function symbol="∩">
       <variable symbol="y_0"/>
       <variable symbol="x_0"/>
      </function>
     </constantatomformula>
     <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
    </formulalist>
   </sequent>
   <rule type="weakl" symbol="w:l">
    <sequent>
     <formulalist>
      <constantatomformula symbol="L1" definition="L1"/>
     </formulalist>
     <formulalist>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <variable symbol="x_0"/>
        <variable symbol="y_0"/>
       </function>
       <function symbol="∩">
        <variable symbol="y_0"/>
        <variable symbol="x_0"/>
       </function>
      </constantatomformula>
     </formulalist>
    </sequent>
    <rule type="axiom" symbol="axiom">
     <sequent>
      <formulalist/>
      <formulalist>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <variable symbol="x_0"/>
         <variable symbol="y_0"/>
        </function>
        <function symbol="∩">
         <variable symbol="y_0"/>
         <variable symbol="x_0"/>
        </function>
       </constantatomformula>
      </formulalist>
     </sequent>
    </rule>
   </rule>
  </rule>
 </proof>
 <proof symbol="\psi_{46}" calculus="LK">
  <rule type="weakr" symbol="w:r">
   <sequent>
    <formulalist>
     <constantatomformula symbol="L1" definition="L1"/>
    </formulalist>
    <formulalist>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <variable symbol="x_0"/>
       <variable symbol="x_0"/>
      </function>
      <variable symbol="x_0"/>
     </constantatomformula>
     <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
    </formulalist>
   </sequent>
   <rule type="weakl" symbol="w:l">
    <sequent>
     <formulalist>
      <constantatomformula symbol="L1" definition="L1"/>
     </formulalist>
     <formulalist>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <variable symbol="x_0"/>
        <variable symbol="x_0"/>
       </function>
       <variable symbol="x_0"/>
      </constantatomformula>
     </formulalist>
    </sequent>
    <rule type="axiom" symbol="axiom">
     <sequent>
      <formulalist/>
      <formulalist>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <variable symbol="x_0"/>
         <variable symbol="x_0"/>
        </function>
        <variable symbol="x_0"/>
       </constantatomformula>
      </formulalist>
     </sequent>
    </rule>
   </rule>
  </rule>
 </proof>
 <proof symbol="axiom set" calculus="LK">
  <rule type="axset">
   <sequent>
    <formulalist/>
    <formulalist>
     <constantatomformula symbol="=">
      <function symbol="∪">
       <variable symbol="x"/>
       <variable symbol="y"/>
      </function>
      <function symbol="∪">
       <variable symbol="y"/>
       <variable symbol="x"/>
      </function>
     </constantatomformula>
    </formulalist>
   </sequent>
   <rule type="axset">
    <sequent>
     <formulalist/>
     <formulalist>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <variable symbol="x"/>
        <variable symbol="y"/>
       </function>
       <function symbol="∩">
        <variable symbol="y"/>
        <variable symbol="x"/>
       </function>
      </constantatomformula>
     </formulalist>
    </sequent>
    <rule type="axset">
     <sequent>
      <formulalist/>
      <formulalist>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <variable symbol="x"/>
         <variable symbol="x"/>
        </function>
        <variable symbol="x"/>
       </constantatomformula>
      </formulalist>
     </sequent>
     <rule type="axset">
      <sequent>
       <formulalist/>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∪">
          <function symbol="∪">
           <variable symbol="x"/>
           <variable symbol="y"/>
          </function>
          <variable symbol="z"/>
         </function>
         <function symbol="∪">
          <variable symbol="x"/>
          <function symbol="∪">
           <variable symbol="y"/>
           <variable symbol="z"/>
          </function>
         </function>
        </constantatomformula>
       </formulalist>
      </sequent>
      <rule type="axset">
       <sequent>
        <formulalist/>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∩">
           <function symbol="∩">
            <variable symbol="x"/>
            <variable symbol="y"/>
           </function>
           <variable symbol="z"/>
          </function>
          <function symbol="∩">
           <variable symbol="x"/>
           <function symbol="∩">
            <variable symbol="y"/>
            <variable symbol="z"/>
           </function>
          </function>
         </constantatomformula>
        </formulalist>
       </sequent>
       <rule type="axset">
        <sequent>
         <formulalist/>
         <formulalist>
          <constantatomformula symbol="=">
           <function symbol="∪">
            <variable symbol="x"/>
            <variable symbol="x"/>
           </function>
           <variable symbol="x"/>
          </constantatomformula>
         </formulalist>
        </sequent>
        <rule type="axset">
         <sequent>
          <formulalist/>
          <formulalist>
           <constantatomformula symbol="=">
            <variable symbol="x"/>
            <variable symbol="x"/>
           </constantatomformula>
          </formulalist>
         </sequent>
        </rule>
       </rule>
      </rule>
     </rule>
    </rule>
   </rule>
  </rule>
 </proof>
 <proof symbol="refutation" calculus="RES">
  <rule type="res" symbol="r">
   <sequent>
    <formulalist/>
    <formulalist/>
   </sequent>
   <rule type="init" symbol="">
    <sequent>
     <formulalist/>
     <formulalist>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <constant symbol="s_1"/>
        <constant symbol="s_2"/>
       </function>
       <function symbol="∩">
        <constant symbol="s_1"/>
        <constant symbol="s_2"/>
       </function>
      </constantatomformula>
     </formulalist>
    </sequent>
   </rule>
   <rule type="paral1" symbol="p:l_1" param="(1)">
    <sequent>
     <formulalist>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <constant symbol="s_1"/>
        <constant symbol="s_2"/>
       </function>
       <function symbol="∩">
        <constant symbol="s_1"/>
        <constant symbol="s_2"/>
       </function>
      </constantatomformula>
     </formulalist>
     <formulalist/>
    </sequent>
    <rule type="parar1" symbol="p:r_1" param="(1)">
     <sequent>
      <formulalist/>
      <formulalist>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <constant symbol="s_1"/>
         <function symbol="∩">
          <constant symbol="s_1"/>
          <constant symbol="s_2"/>
         </function>
        </function>
        <function symbol="∩">
         <constant symbol="s_1"/>
         <constant symbol="s_2"/>
        </function>
       </constantatomformula>
      </formulalist>
     </sequent>
     <rule type="parar1" symbol="p:r_1" param="(1 1)">
      <sequent>
       <formulalist/>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <constant symbol="s_1"/>
          <constant symbol="s_2"/>
         </function>
         <function symbol="∩">
          <constant symbol="s_1"/>
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
         </function>
        </constantatomformula>
       </formulalist>
      </sequent>
      <rule type="init" symbol="">
       <sequent>
        <formulalist/>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_1"/>
          </function>
          <constant symbol="s_1"/>
         </constantatomformula>
        </formulalist>
       </sequent>
      </rule>
      <rule type="init" symbol="">
       <sequent>
        <formulalist/>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∩">
           <function symbol="∩">
            <constant symbol="s_1"/>
            <constant symbol="s_1"/>
           </function>
           <constant symbol="s_2"/>
          </function>
          <function symbol="∩">
           <constant symbol="s_1"/>
           <function symbol="∩">
            <constant symbol="s_1"/>
            <constant symbol="s_2"/>
           </function>
          </function>
         </constantatomformula>
        </formulalist>
       </sequent>
      </rule>
     </rule>
     <rule type="init" symbol="">
      <sequent>
       <formulalist/>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <constant symbol="s_1"/>
          <constant symbol="s_2"/>
         </function>
         <function symbol="∩">
          <constant symbol="s_1"/>
          <constant symbol="s_2"/>
         </function>
        </constantatomformula>
       </formulalist>
      </sequent>
     </rule>
    </rule>
    <rule type="paral1" symbol="p:l_1" param="(1)">
     <sequent>
      <formulalist>
       <constantatomformula symbol="=">
        <function symbol="∩">
         <constant symbol="s_1"/>
         <function symbol="∩">
          <constant symbol="s_1"/>
          <constant symbol="s_2"/>
         </function>
        </function>
        <function symbol="∩">
         <constant symbol="s_1"/>
         <constant symbol="s_2"/>
        </function>
       </constantatomformula>
      </formulalist>
      <formulalist/>
     </sequent>
     <rule type="init" symbol="">
      <sequent>
       <formulalist/>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
         <function symbol="∩">
          <constant symbol="s_1"/>
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
         </function>
        </constantatomformula>
       </formulalist>
      </sequent>
     </rule>
     <rule type="res" symbol="r">
      <sequent>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="∩">
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
          <constant symbol="s_1"/>
         </function>
         <function symbol="∩">
          <constant symbol="s_1"/>
          <constant symbol="s_2"/>
         </function>
        </constantatomformula>
       </formulalist>
       <formulalist/>
      </sequent>
      <rule type="parar1" symbol="p:r_1" param="(1)">
       <sequent>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∩">
           <function symbol="∩">
            <constant symbol="s_1"/>
            <constant symbol="s_2"/>
           </function>
           <constant symbol="s_1"/>
          </function>
          <function symbol="∩">
           <constant symbol="s_1"/>
           <constant symbol="s_2"/>
          </function>
         </constantatomformula>
        </formulalist>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∪">
           <constant symbol="s_1"/>
           <function symbol="∩">
            <constant symbol="s_1"/>
            <constant symbol="s_2"/>
           </function>
          </function>
          <constant symbol="s_1"/>
         </constantatomformula>
        </formulalist>
       </sequent>
       <rule type="parar1" symbol="p:r_1" param="(1)">
        <sequent>
         <formulalist>
          <constantatomformula symbol="=">
           <function symbol="∩">
            <function symbol="∩">
             <constant symbol="s_1"/>
             <constant symbol="s_2"/>
            </function>
            <constant symbol="s_1"/>
           </function>
           <function symbol="∩">
            <constant symbol="s_1"/>
            <constant symbol="s_2"/>
           </function>
          </constantatomformula>
         </formulalist>
         <formulalist>
          <constantatomformula symbol="=">
           <constant symbol="s_1"/>
           <function symbol="∪">
            <constant symbol="s_1"/>
            <function symbol="∩">
             <constant symbol="s_1"/>
             <constant symbol="s_2"/>
            </function>
           </function>
          </constantatomformula>
         </formulalist>
        </sequent>
        <rule type="init" symbol="">
         <sequent>
          <formulalist/>
          <formulalist>
           <constantatomformula symbol="=">
            <function symbol="∩">
             <constant symbol="s_1"/>
             <function symbol="∪">
              <constant symbol="s_1"/>
              <function symbol="∩">
               <constant symbol="s_1"/>
               <constant symbol="s_2"/>
              </function>
             </function>
            </function>
            <constant symbol="s_1"/>
           </constantatomformula>
          </formulalist>
         </sequent>
        </rule>
        <rule type="parar1" symbol="p:r_1" param="(1)">
         <sequent>
          <formulalist>
           <constantatomformula symbol="=">
            <function symbol="∩">
             <function symbol="∩">
              <constant symbol="s_1"/>
              <constant symbol="s_2"/>
             </function>
             <constant symbol="s_1"/>
            </function>
            <function symbol="∩">
             <constant symbol="s_1"/>
             <constant symbol="s_2"/>
            </function>
           </constantatomformula>
          </formulalist>
          <formulalist>
           <constantatomformula symbol="=">
            <function symbol="∩">
             <constant symbol="s_1"/>
             <function symbol="∪">
              <constant symbol="s_1"/>
              <function symbol="∩">
               <constant symbol="s_1"/>
               <constant symbol="s_2"/>
              </function>
             </function>
            </function>
            <function symbol="∪">
             <constant symbol="s_1"/>
             <function symbol="∩">
              <constant symbol="s_1"/>
              <constant symbol="s_2"/>
             </function>
            </function>
           </constantatomformula>
          </formulalist>
         </sequent>
         <rule type="init" symbol="">
          <sequent>
           <formulalist/>
           <formulalist>
            <constantatomformula symbol="=">
             <function symbol="∩">
              <function symbol="∪">
               <constant symbol="s_1"/>
               <function symbol="∩">
                <constant symbol="s_1"/>
                <constant symbol="s_2"/>
               </function>
              </function>
              <constant symbol="s_1"/>
             </function>
             <function symbol="∩">
              <constant symbol="s_1"/>
              <function symbol="∪">
               <constant symbol="s_1"/>
               <function symbol="∩">
                <constant symbol="s_1"/>
                <constant symbol="s_2"/>
               </function>
              </function>
             </function>
            </constantatomformula>
           </formulalist>
          </sequent>
         </rule>
         <rule type="res" symbol="r">
          <sequent>
           <formulalist>
            <constantatomformula symbol="=">
             <function symbol="∩">
              <function symbol="∩">
               <constant symbol="s_1"/>
               <constant symbol="s_2"/>
              </function>
              <constant symbol="s_1"/>
             </function>
             <function symbol="∩">
              <constant symbol="s_1"/>
              <constant symbol="s_2"/>
             </function>
            </constantatomformula>
           </formulalist>
           <formulalist>
            <constantatomformula symbol="=">
             <function symbol="∩">
              <function symbol="∪">
               <constant symbol="s_1"/>
               <function symbol="∩">
                <constant symbol="s_1"/>
                <constant symbol="s_2"/>
               </function>
              </function>
              <constant symbol="s_1"/>
             </function>
             <function symbol="∪">
              <constant symbol="s_1"/>
              <function symbol="∩">
               <constant symbol="s_1"/>
               <constant symbol="s_2"/>
              </function>
             </function>
            </constantatomformula>
           </formulalist>
          </sequent>
          <rule type="init" symbol="">
           <sequent>
            <formulalist/>
            <formulalist>
             <constantatomformula symbol="=">
              <function symbol="∩">
               <constant symbol="s_1"/>
               <constant symbol="s_1"/>
              </function>
              <constant symbol="s_1"/>
             </constantatomformula>
            </formulalist>
           </sequent>
          </rule>
          <rule type="init" symbol="">
           <sequent>
            <formulalist>
             <constantatomformula symbol="=">
              <function symbol="∩">
               <constant symbol="s_1"/>
               <constant symbol="s_1"/>
              </function>
              <constant symbol="s_1"/>
             </constantatomformula>
             <constantatomformula symbol="=">
              <function symbol="∩">
               <function symbol="∩">
                <constant symbol="s_1"/>
                <constant symbol="s_2"/>
               </function>
               <constant symbol="s_1"/>
              </function>
              <function symbol="∩">
               <constant symbol="s_1"/>
               <constant symbol="s_2"/>
              </function>
             </constantatomformula>
            </formulalist>
            <formulalist>
             <constantatomformula symbol="=">
              <function symbol="∩">
               <function symbol="∪">
                <constant symbol="s_1"/>
                <function symbol="∩">
                 <constant symbol="s_1"/>
                 <constant symbol="s_2"/>
                </function>
               </function>
               <constant symbol="s_1"/>
              </function>
              <function symbol="∪">
               <constant symbol="s_1"/>
               <function symbol="∩">
                <constant symbol="s_1"/>
                <constant symbol="s_2"/>
               </function>
              </function>
             </constantatomformula>
            </formulalist>
           </sequent>
          </rule>
         </rule>
        </rule>
       </rule>
       <rule type="init" symbol="">
        <sequent>
         <formulalist/>
         <formulalist>
          <constantatomformula symbol="=">
           <constant symbol="s_1"/>
           <constant symbol="s_1"/>
          </constantatomformula>
         </formulalist>
        </sequent>
       </rule>
      </rule>
      <rule type="res" symbol="r">
       <sequent>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="∪">
           <constant symbol="s_1"/>
           <function symbol="∩">
            <constant symbol="s_1"/>
            <constant symbol="s_2"/>
           </function>
          </function>
          <constant symbol="s_1"/>
         </constantatomformula>
        </formulalist>
        <formulalist/>
       </sequent>
       <rule type="init" symbol="">
        <sequent>
         <formulalist/>
         <formulalist>
          <constantatomformula symbol="=">
           <constant symbol="s_3"/>
           <constant symbol="s_3"/>
          </constantatomformula>
         </formulalist>
        </sequent>
       </rule>
       <rule type="perml" symbol="\pi:l" param="(1 2)">
        <sequent>
         <formulalist>
          <constantatomformula symbol="=">
           <constant symbol="s_3"/>
           <constant symbol="s_3"/>
          </constantatomformula>
          <constantatomformula symbol="=">
           <function symbol="∪">
            <constant symbol="s_1"/>
            <function symbol="∩">
             <constant symbol="s_1"/>
             <constant symbol="s_2"/>
            </function>
           </function>
           <constant symbol="s_1"/>
          </constantatomformula>
         </formulalist>
         <formulalist/>
        </sequent>
        <rule type="perml" symbol="\pi:l" param="(1 2)">
         <sequent>
          <formulalist>
           <constantatomformula symbol="=">
            <function symbol="∪">
             <constant symbol="s_1"/>
             <function symbol="∩">
              <constant symbol="s_1"/>
              <constant symbol="s_2"/>
             </function>
            </function>
            <constant symbol="s_1"/>
           </constantatomformula>
           <constantatomformula symbol="=">
            <constant symbol="s_3"/>
            <constant symbol="s_3"/>
           </constantatomformula>
          </formulalist>
          <formulalist/>
         </sequent>
         <rule type="paral1" symbol="p:l_1" param="(1)">
          <sequent>
           <formulalist>
            <constantatomformula symbol="=">
             <constant symbol="s_3"/>
             <constant symbol="s_3"/>
            </constantatomformula>
            <constantatomformula symbol="=">
             <function symbol="∪">
              <constant symbol="s_1"/>
              <function symbol="∩">
               <constant symbol="s_1"/>
               <constant symbol="s_2"/>
              </function>
             </function>
             <constant symbol="s_1"/>
            </constantatomformula>
           </formulalist>
           <formulalist/>
          </sequent>
          <rule type="init" symbol="">
           <sequent>
            <formulalist/>
            <formulalist>
             <constantatomformula symbol="=">
              <function symbol="∩">
               <constant symbol="s_3"/>
               <function symbol="∪">
                <constant symbol="s_3"/>
                <constant symbol="s_4"/>
               </function>
              </function>
              <constant symbol="s_3"/>
             </constantatomformula>
            </formulalist>
           </sequent>
          </rule>
          <rule type="perml" symbol="\pi:l" param="(1 2)">
           <sequent>
            <formulalist>
             <constantatomformula symbol="=">
              <function symbol="∩">
               <constant symbol="s_3"/>
               <function symbol="∪">
                <constant symbol="s_3"/>
                <constant symbol="s_4"/>
               </function>
              </function>
              <constant symbol="s_3"/>
             </constantatomformula>
             <constantatomformula symbol="=">
              <function symbol="∪">
               <constant symbol="s_1"/>
               <function symbol="∩">
                <constant symbol="s_1"/>
                <constant symbol="s_2"/>
               </function>
              </function>
              <constant symbol="s_1"/>
             </constantatomformula>
            </formulalist>
            <formulalist/>
           </sequent>
           <rule type="perml" symbol="\pi:l" param="(1 2)">
            <sequent>
             <formulalist>
              <constantatomformula symbol="=">
               <function symbol="∪">
                <constant symbol="s_1"/>
                <function symbol="∩">
                 <constant symbol="s_1"/>
                 <constant symbol="s_2"/>
                </function>
               </function>
               <constant symbol="s_1"/>
              </constantatomformula>
              <constantatomformula symbol="=">
               <function symbol="∩">
                <constant symbol="s_3"/>
                <function symbol="∪">
                 <constant symbol="s_3"/>
                 <constant symbol="s_4"/>
                </function>
               </function>
               <constant symbol="s_3"/>
              </constantatomformula>
             </formulalist>
             <formulalist/>
            </sequent>
            <rule type="paral1" symbol="p:l_1" param="(1)">
             <sequent>
              <formulalist>
               <constantatomformula symbol="=">
                <function symbol="∩">
                 <constant symbol="s_3"/>
                 <function symbol="∪">
                  <constant symbol="s_3"/>
                  <constant symbol="s_4"/>
                 </function>
                </function>
                <constant symbol="s_3"/>
               </constantatomformula>
               <constantatomformula symbol="=">
                <function symbol="∪">
                 <constant symbol="s_1"/>
                 <function symbol="∩">
                  <constant symbol="s_1"/>
                  <constant symbol="s_2"/>
                 </function>
                </function>
                <constant symbol="s_1"/>
               </constantatomformula>
              </formulalist>
              <formulalist/>
             </sequent>
             <rule type="init" symbol="">
              <sequent>
               <formulalist/>
               <formulalist>
                <constantatomformula symbol="=">
                 <function symbol="∩">
                  <function symbol="∪">
                   <constant symbol="s_3"/>
                   <constant symbol="s_4"/>
                  </function>
                  <constant symbol="s_3"/>
                 </function>
                 <function symbol="∩">
                  <constant symbol="s_3"/>
                  <function symbol="∪">
                   <constant symbol="s_3"/>
                   <constant symbol="s_4"/>
                  </function>
                 </function>
                </constantatomformula>
               </formulalist>
              </sequent>
             </rule>
             <rule type="perml" symbol="\pi:l" param="(1 2)">
              <sequent>
               <formulalist>
                <constantatomformula symbol="=">
                 <function symbol="∩">
                  <function symbol="∪">
                   <constant symbol="s_3"/>
                   <constant symbol="s_4"/>
                  </function>
                  <constant symbol="s_3"/>
                 </function>
                 <constant symbol="s_3"/>
                </constantatomformula>
                <constantatomformula symbol="=">
                 <function symbol="∪">
                  <constant symbol="s_1"/>
                  <function symbol="∩">
                   <constant symbol="s_1"/>
                   <constant symbol="s_2"/>
                  </function>
                 </function>
                 <constant symbol="s_1"/>
                </constantatomformula>
               </formulalist>
               <formulalist/>
              </sequent>
              <rule type="paral1" symbol="p:l_1" param="(1)">
               <sequent>
                <formulalist>
                 <constantatomformula symbol="=">
                  <function symbol="∪">
                   <constant symbol="s_1"/>
                   <function symbol="∩">
                    <constant symbol="s_1"/>
                    <constant symbol="s_2"/>
                   </function>
                  </function>
                  <constant symbol="s_1"/>
                 </constantatomformula>
                 <constantatomformula symbol="=">
                  <function symbol="∩">
                   <function symbol="∪">
                    <constant symbol="s_3"/>
                    <constant symbol="s_4"/>
                   </function>
                   <constant symbol="s_3"/>
                  </function>
                  <constant symbol="s_3"/>
                 </constantatomformula>
                </formulalist>
                <formulalist/>
               </sequent>
               <rule type="init" symbol="">
                <sequent>
                 <formulalist/>
                 <formulalist>
                  <constantatomformula symbol="=">
                   <function symbol="∪">
                    <function symbol="∩">
                     <constant symbol="s_1"/>
                     <constant symbol="s_2"/>
                    </function>
                    <constant symbol="s_1"/>
                   </function>
                   <function symbol="∪">
                    <constant symbol="s_1"/>
                    <function symbol="∩">
                     <constant symbol="s_1"/>
                     <constant symbol="s_2"/>
                    </function>
                   </function>
                  </constantatomformula>
                 </formulalist>
                </sequent>
               </rule>
               <rule type="init" symbol="">
                <sequent>
                 <formulalist>
                  <constantatomformula symbol="=">
                   <function symbol="∪">
                    <function symbol="∩">
                     <constant symbol="s_1"/>
                     <constant symbol="s_2"/>
                    </function>
                    <constant symbol="s_1"/>
                   </function>
                   <constant symbol="s_1"/>
                  </constantatomformula>
                  <constantatomformula symbol="=">
                   <function symbol="∩">
                    <function symbol="∪">
                     <constant symbol="s_3"/>
                     <constant symbol="s_4"/>
                    </function>
                    <constant symbol="s_3"/>
                   </function>
                   <constant symbol="s_3"/>
                  </constantatomformula>
                 </formulalist>
                 <formulalist/>
                </sequent>
               </rule>
              </rule>
             </rule>
            </rule>
           </rule>
          </rule>
         </rule>
        </rule>
       </rule>
      </rule>
     </rule>
    </rule>
   </rule>
  </rule>
 </proof>
 <proof symbol="acnf" calculus="LK">
  <rule type="contrr" symbol="c:r" param="2">
   <sequent>
    <formulalist>
     <constantatomformula symbol="L1" definition="L1"/>
    </formulalist>
    <formulalist>
     <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
    </formulalist>
   </sequent>
   <rule type="permr" symbol="\pi:r" param="">
    <sequent>
     <formulalist>
      <constantatomformula symbol="L1" definition="L1"/>
     </formulalist>
     <formulalist>
      <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
      <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
     </formulalist>
    </sequent>
    <rule type="contrl" symbol="c:l" param="2">
     <sequent>
      <formulalist>
       <constantatomformula symbol="L1" definition="L1"/>
      </formulalist>
      <formulalist>
       <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
       <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
      </formulalist>
     </sequent>
     <rule type="perml" symbol="\pi:l" param="">
      <sequent>
       <formulalist>
        <constantatomformula symbol="L1" definition="L1"/>
        <constantatomformula symbol="L1" definition="L1"/>
       </formulalist>
       <formulalist>
        <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
        <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
       </formulalist>
      </sequent>
      <rule type="cut" symbol="cut">
       <sequent>
        <formulalist>
         <constantatomformula symbol="L1" definition="L1"/>
         <constantatomformula symbol="L1" definition="L1"/>
        </formulalist>
        <formulalist>
         <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
         <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
        </formulalist>
       </sequent>
       <rule type="permr" symbol="\pi:r" param="(1 2)">
        <sequent>
         <formulalist>
          <constantatomformula symbol="L1" definition="L1"/>
         </formulalist>
         <formulalist>
          <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
          <constantatomformula symbol="=">
           <function symbol="∩">
            <constant symbol="s_1"/>
            <constant symbol="s_2"/>
           </function>
           <function symbol="∩">
            <constant symbol="s_1"/>
            <constant symbol="s_2"/>
           </function>
          </constantatomformula>
         </formulalist>
        </sequent>
        <rule type="weakr" symbol="w:r">
         <sequent>
          <formulalist>
           <constantatomformula symbol="L1" definition="L1"/>
          </formulalist>
          <formulalist>
           <constantatomformula symbol="=">
            <function symbol="∩">
             <constant symbol="s_1"/>
             <constant symbol="s_2"/>
            </function>
            <function symbol="∩">
             <constant symbol="s_1"/>
             <constant symbol="s_2"/>
            </function>
           </constantatomformula>
           <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
          </formulalist>
         </sequent>
         <rule type="perml" symbol="\pi:l" param="">
          <sequent>
           <formulalist>
            <constantatomformula symbol="L1" definition="L1"/>
           </formulalist>
           <formulalist>
            <constantatomformula symbol="=">
             <function symbol="∩">
              <constant symbol="s_1"/>
              <constant symbol="s_2"/>
             </function>
             <function symbol="∩">
              <constant symbol="s_1"/>
              <constant symbol="s_2"/>
             </function>
            </constantatomformula>
           </formulalist>
          </sequent>
          <rule type="weakl" symbol="w:l">
           <sequent>
            <formulalist>
             <constantatomformula symbol="L1" definition="L1"/>
            </formulalist>
            <formulalist>
             <constantatomformula symbol="=">
              <function symbol="∩">
               <constant symbol="s_1"/>
               <constant symbol="s_2"/>
              </function>
              <function symbol="∩">
               <constant symbol="s_1"/>
               <constant symbol="s_2"/>
              </function>
             </constantatomformula>
            </formulalist>
           </sequent>
           <rule type="axiom" symbol="axiom">
            <sequent>
             <formulalist/>
             <formulalist>
              <constantatomformula symbol="=">
               <function symbol="∩">
                <constant symbol="s_1"/>
                <constant symbol="s_2"/>
               </function>
               <function symbol="∩">
                <constant symbol="s_1"/>
                <constant symbol="s_2"/>
               </function>
              </constantatomformula>
             </formulalist>
            </sequent>
           </rule>
          </rule>
         </rule>
        </rule>
       </rule>
       <rule type="contrr" symbol="c:r" param="2">
        <sequent>
         <formulalist>
          <constantatomformula symbol="=">
           <function symbol="∩">
            <constant symbol="s_1"/>
            <constant symbol="s_2"/>
           </function>
           <function symbol="∩">
            <constant symbol="s_1"/>
            <constant symbol="s_2"/>
           </function>
          </constantatomformula>
          <constantatomformula symbol="L1" definition="L1"/>
         </formulalist>
         <formulalist>
          <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
         </formulalist>
        </sequent>
        <rule type="permr" symbol="\pi:r" param="">
         <sequent>
          <formulalist>
           <constantatomformula symbol="=">
            <function symbol="∩">
             <constant symbol="s_1"/>
             <constant symbol="s_2"/>
            </function>
            <function symbol="∩">
             <constant symbol="s_1"/>
             <constant symbol="s_2"/>
            </function>
           </constantatomformula>
           <constantatomformula symbol="L1" definition="L1"/>
          </formulalist>
          <formulalist>
           <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
           <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
          </formulalist>
         </sequent>
         <rule type="contrl" symbol="c:l" param="1,2">
          <sequent>
           <formulalist>
            <constantatomformula symbol="=">
             <function symbol="∩">
              <constant symbol="s_1"/>
              <constant symbol="s_2"/>
             </function>
             <function symbol="∩">
              <constant symbol="s_1"/>
              <constant symbol="s_2"/>
             </function>
            </constantatomformula>
            <constantatomformula symbol="L1" definition="L1"/>
           </formulalist>
           <formulalist>
            <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
            <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
           </formulalist>
          </sequent>
          <rule type="perml" symbol="\pi:l" param="">
           <sequent>
            <formulalist>
             <constantatomformula symbol="=">
              <function symbol="∩">
               <constant symbol="s_1"/>
               <constant symbol="s_2"/>
              </function>
              <function symbol="∩">
               <constant symbol="s_1"/>
               <constant symbol="s_2"/>
              </function>
             </constantatomformula>
             <constantatomformula symbol="L1" definition="L1"/>
             <constantatomformula symbol="L1" definition="L1"/>
            </formulalist>
            <formulalist>
             <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
             <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
            </formulalist>
           </sequent>
           <rule type="eql1" symbol="=:l1" param="(1)">
            <sequent>
             <formulalist>
              <constantatomformula symbol="=">
               <function symbol="∩">
                <constant symbol="s_1"/>
                <constant symbol="s_2"/>
               </function>
               <function symbol="∩">
                <constant symbol="s_1"/>
                <constant symbol="s_2"/>
               </function>
              </constantatomformula>
              <constantatomformula symbol="L1" definition="L1"/>
              <constantatomformula symbol="L1" definition="L1"/>
             </formulalist>
             <formulalist>
              <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
              <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
             </formulalist>
            </sequent>
            <rule type="contrr" symbol="c:r" param="2,1">
             <sequent>
              <formulalist>
               <constantatomformula symbol="L1" definition="L1"/>
              </formulalist>
              <formulalist>
               <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
               <constantatomformula symbol="=">
                <function symbol="∩">
                 <constant symbol="s_1"/>
                 <function symbol="∩">
                  <constant symbol="s_1"/>
                  <constant symbol="s_2"/>
                 </function>
                </function>
                <function symbol="∩">
                 <constant symbol="s_1"/>
                 <constant symbol="s_2"/>
                </function>
               </constantatomformula>
              </formulalist>
             </sequent>
             <rule type="permr" symbol="\pi:r" param="">
              <sequent>
               <formulalist>
                <constantatomformula symbol="L1" definition="L1"/>
               </formulalist>
               <formulalist>
                <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                <constantatomformula symbol="=">
                 <function symbol="∩">
                  <constant symbol="s_1"/>
                  <function symbol="∩">
                   <constant symbol="s_1"/>
                   <constant symbol="s_2"/>
                  </function>
                 </function>
                 <function symbol="∩">
                  <constant symbol="s_1"/>
                  <constant symbol="s_2"/>
                 </function>
                </constantatomformula>
               </formulalist>
              </sequent>
              <rule type="contrl" symbol="c:l" param="2">
               <sequent>
                <formulalist>
                 <constantatomformula symbol="L1" definition="L1"/>
                </formulalist>
                <formulalist>
                 <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                 <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                 <constantatomformula symbol="=">
                  <function symbol="∩">
                   <constant symbol="s_1"/>
                   <function symbol="∩">
                    <constant symbol="s_1"/>
                    <constant symbol="s_2"/>
                   </function>
                  </function>
                  <function symbol="∩">
                   <constant symbol="s_1"/>
                   <constant symbol="s_2"/>
                  </function>
                 </constantatomformula>
                </formulalist>
               </sequent>
               <rule type="perml" symbol="\pi:l" param="">
                <sequent>
                 <formulalist>
                  <constantatomformula symbol="L1" definition="L1"/>
                  <constantatomformula symbol="L1" definition="L1"/>
                 </formulalist>
                 <formulalist>
                  <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                  <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                  <constantatomformula symbol="=">
                   <function symbol="∩">
                    <constant symbol="s_1"/>
                    <function symbol="∩">
                     <constant symbol="s_1"/>
                     <constant symbol="s_2"/>
                    </function>
                   </function>
                   <function symbol="∩">
                    <constant symbol="s_1"/>
                    <constant symbol="s_2"/>
                   </function>
                  </constantatomformula>
                 </formulalist>
                </sequent>
                <rule type="eqr1" symbol="=:r1" param="(1)">
                 <sequent>
                  <formulalist>
                   <constantatomformula symbol="L1" definition="L1"/>
                   <constantatomformula symbol="L1" definition="L1"/>
                  </formulalist>
                  <formulalist>
                   <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                   <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                   <constantatomformula symbol="=">
                    <function symbol="∩">
                     <constant symbol="s_1"/>
                     <function symbol="∩">
                      <constant symbol="s_1"/>
                      <constant symbol="s_2"/>
                     </function>
                    </function>
                    <function symbol="∩">
                     <constant symbol="s_1"/>
                     <constant symbol="s_2"/>
                    </function>
                   </constantatomformula>
                  </formulalist>
                 </sequent>
                 <rule type="contrr" symbol="c:r" param="2,1">
                  <sequent>
                   <formulalist>
                    <constantatomformula symbol="L1" definition="L1"/>
                   </formulalist>
                   <formulalist>
                    <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                    <constantatomformula symbol="=">
                     <function symbol="∩">
                      <constant symbol="s_1"/>
                      <constant symbol="s_2"/>
                     </function>
                     <function symbol="∩">
                      <constant symbol="s_1"/>
                      <function symbol="∩">
                       <constant symbol="s_1"/>
                       <constant symbol="s_2"/>
                      </function>
                     </function>
                    </constantatomformula>
                   </formulalist>
                  </sequent>
                  <rule type="permr" symbol="\pi:r" param="">
                   <sequent>
                    <formulalist>
                     <constantatomformula symbol="L1" definition="L1"/>
                    </formulalist>
                    <formulalist>
                     <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                     <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                     <constantatomformula symbol="=">
                      <function symbol="∩">
                       <constant symbol="s_1"/>
                       <constant symbol="s_2"/>
                      </function>
                      <function symbol="∩">
                       <constant symbol="s_1"/>
                       <function symbol="∩">
                        <constant symbol="s_1"/>
                        <constant symbol="s_2"/>
                       </function>
                      </function>
                     </constantatomformula>
                    </formulalist>
                   </sequent>
                   <rule type="contrl" symbol="c:l" param="2">
                    <sequent>
                     <formulalist>
                      <constantatomformula symbol="L1" definition="L1"/>
                     </formulalist>
                     <formulalist>
                      <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                      <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                      <constantatomformula symbol="=">
                       <function symbol="∩">
                        <constant symbol="s_1"/>
                        <constant symbol="s_2"/>
                       </function>
                       <function symbol="∩">
                        <constant symbol="s_1"/>
                        <function symbol="∩">
                         <constant symbol="s_1"/>
                         <constant symbol="s_2"/>
                        </function>
                       </function>
                      </constantatomformula>
                     </formulalist>
                    </sequent>
                    <rule type="perml" symbol="\pi:l" param="">
                     <sequent>
                      <formulalist>
                       <constantatomformula symbol="L1" definition="L1"/>
                       <constantatomformula symbol="L1" definition="L1"/>
                      </formulalist>
                      <formulalist>
                       <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                       <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                       <constantatomformula symbol="=">
                        <function symbol="∩">
                         <constant symbol="s_1"/>
                         <constant symbol="s_2"/>
                        </function>
                        <function symbol="∩">
                         <constant symbol="s_1"/>
                         <function symbol="∩">
                          <constant symbol="s_1"/>
                          <constant symbol="s_2"/>
                         </function>
                        </function>
                       </constantatomformula>
                      </formulalist>
                     </sequent>
                     <rule type="eqr1" symbol="=:r1" param="(1 1)">
                      <sequent>
                       <formulalist>
                        <constantatomformula symbol="L1" definition="L1"/>
                        <constantatomformula symbol="L1" definition="L1"/>
                       </formulalist>
                       <formulalist>
                        <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                        <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                        <constantatomformula symbol="=">
                         <function symbol="∩">
                          <constant symbol="s_1"/>
                          <constant symbol="s_2"/>
                         </function>
                         <function symbol="∩">
                          <constant symbol="s_1"/>
                          <function symbol="∩">
                           <constant symbol="s_1"/>
                           <constant symbol="s_2"/>
                          </function>
                         </function>
                        </constantatomformula>
                       </formulalist>
                      </sequent>
                      <rule type="permr" symbol="\pi:r" param="(1 2)">
                       <sequent>
                        <formulalist>
                         <constantatomformula symbol="L1" definition="L1"/>
                        </formulalist>
                        <formulalist>
                         <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                         <constantatomformula symbol="=">
                          <function symbol="∩">
                           <constant symbol="s_1"/>
                           <constant symbol="s_1"/>
                          </function>
                          <constant symbol="s_1"/>
                         </constantatomformula>
                        </formulalist>
                       </sequent>
                       <rule type="weakr" symbol="w:r">
                        <sequent>
                         <formulalist>
                          <constantatomformula symbol="L1" definition="L1"/>
                         </formulalist>
                         <formulalist>
                          <constantatomformula symbol="=">
                           <function symbol="∩">
                            <constant symbol="s_1"/>
                            <constant symbol="s_1"/>
                           </function>
                           <constant symbol="s_1"/>
                          </constantatomformula>
                          <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                         </formulalist>
                        </sequent>
                        <rule type="perml" symbol="\pi:l" param="">
                         <sequent>
                          <formulalist>
                           <constantatomformula symbol="L1" definition="L1"/>
                          </formulalist>
                          <formulalist>
                           <constantatomformula symbol="=">
                            <function symbol="∩">
                             <constant symbol="s_1"/>
                             <constant symbol="s_1"/>
                            </function>
                            <constant symbol="s_1"/>
                           </constantatomformula>
                          </formulalist>
                         </sequent>
                         <rule type="weakl" symbol="w:l">
                          <sequent>
                           <formulalist>
                            <constantatomformula symbol="L1" definition="L1"/>
                           </formulalist>
                           <formulalist>
                            <constantatomformula symbol="=">
                             <function symbol="∩">
                              <constant symbol="s_1"/>
                              <constant symbol="s_1"/>
                             </function>
                             <constant symbol="s_1"/>
                            </constantatomformula>
                           </formulalist>
                          </sequent>
                          <rule type="axiom" symbol="axiom">
                           <sequent>
                            <formulalist/>
                            <formulalist>
                             <constantatomformula symbol="=">
                              <function symbol="∩">
                               <constant symbol="s_1"/>
                               <constant symbol="s_1"/>
                              </function>
                              <constant symbol="s_1"/>
                             </constantatomformula>
                            </formulalist>
                           </sequent>
                          </rule>
                         </rule>
                        </rule>
                       </rule>
                      </rule>
                      <rule type="permr" symbol="\pi:r" param="(1 2)">
                       <sequent>
                        <formulalist>
                         <constantatomformula symbol="L1" definition="L1"/>
                        </formulalist>
                        <formulalist>
                         <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                         <constantatomformula symbol="=">
                          <function symbol="∩">
                           <function symbol="∩">
                            <constant symbol="s_1"/>
                            <constant symbol="s_1"/>
                           </function>
                           <constant symbol="s_2"/>
                          </function>
                          <function symbol="∩">
                           <constant symbol="s_1"/>
                           <function symbol="∩">
                            <constant symbol="s_1"/>
                            <constant symbol="s_2"/>
                           </function>
                          </function>
                         </constantatomformula>
                        </formulalist>
                       </sequent>
                       <rule type="weakr" symbol="w:r">
                        <sequent>
                         <formulalist>
                          <constantatomformula symbol="L1" definition="L1"/>
                         </formulalist>
                         <formulalist>
                          <constantatomformula symbol="=">
                           <function symbol="∩">
                            <function symbol="∩">
                             <constant symbol="s_1"/>
                             <constant symbol="s_1"/>
                            </function>
                            <constant symbol="s_2"/>
                           </function>
                           <function symbol="∩">
                            <constant symbol="s_1"/>
                            <function symbol="∩">
                             <constant symbol="s_1"/>
                             <constant symbol="s_2"/>
                            </function>
                           </function>
                          </constantatomformula>
                          <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                         </formulalist>
                        </sequent>
                        <rule type="perml" symbol="\pi:l" param="">
                         <sequent>
                          <formulalist>
                           <constantatomformula symbol="L1" definition="L1"/>
                          </formulalist>
                          <formulalist>
                           <constantatomformula symbol="=">
                            <function symbol="∩">
                             <function symbol="∩">
                              <constant symbol="s_1"/>
                              <constant symbol="s_1"/>
                             </function>
                             <constant symbol="s_2"/>
                            </function>
                            <function symbol="∩">
                             <constant symbol="s_1"/>
                             <function symbol="∩">
                              <constant symbol="s_1"/>
                              <constant symbol="s_2"/>
                             </function>
                            </function>
                           </constantatomformula>
                          </formulalist>
                         </sequent>
                         <rule type="weakl" symbol="w:l">
                          <sequent>
                           <formulalist>
                            <constantatomformula symbol="L1" definition="L1"/>
                           </formulalist>
                           <formulalist>
                            <constantatomformula symbol="=">
                             <function symbol="∩">
                              <function symbol="∩">
                               <constant symbol="s_1"/>
                               <constant symbol="s_1"/>
                              </function>
                              <constant symbol="s_2"/>
                             </function>
                             <function symbol="∩">
                              <constant symbol="s_1"/>
                              <function symbol="∩">
                               <constant symbol="s_1"/>
                               <constant symbol="s_2"/>
                              </function>
                             </function>
                            </constantatomformula>
                           </formulalist>
                          </sequent>
                          <rule type="axiom" symbol="axiom">
                           <sequent>
                            <formulalist/>
                            <formulalist>
                             <constantatomformula symbol="=">
                              <function symbol="∩">
                               <function symbol="∩">
                                <constant symbol="s_1"/>
                                <constant symbol="s_1"/>
                               </function>
                               <constant symbol="s_2"/>
                              </function>
                              <function symbol="∩">
                               <constant symbol="s_1"/>
                               <function symbol="∩">
                                <constant symbol="s_1"/>
                                <constant symbol="s_2"/>
                               </function>
                              </function>
                             </constantatomformula>
                            </formulalist>
                           </sequent>
                          </rule>
                         </rule>
                        </rule>
                       </rule>
                      </rule>
                     </rule>
                    </rule>
                   </rule>
                  </rule>
                 </rule>
                 <rule type="permr" symbol="\pi:r" param="(1 2)">
                  <sequent>
                   <formulalist>
                    <constantatomformula symbol="L1" definition="L1"/>
                   </formulalist>
                   <formulalist>
                    <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                    <constantatomformula symbol="=">
                     <function symbol="∩">
                      <constant symbol="s_1"/>
                      <constant symbol="s_2"/>
                     </function>
                     <function symbol="∩">
                      <constant symbol="s_1"/>
                      <constant symbol="s_2"/>
                     </function>
                    </constantatomformula>
                   </formulalist>
                  </sequent>
                  <rule type="weakr" symbol="w:r">
                   <sequent>
                    <formulalist>
                     <constantatomformula symbol="L1" definition="L1"/>
                    </formulalist>
                    <formulalist>
                     <constantatomformula symbol="=">
                      <function symbol="∩">
                       <constant symbol="s_1"/>
                       <constant symbol="s_2"/>
                      </function>
                      <function symbol="∩">
                       <constant symbol="s_1"/>
                       <constant symbol="s_2"/>
                      </function>
                     </constantatomformula>
                     <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                    </formulalist>
                   </sequent>
                   <rule type="perml" symbol="\pi:l" param="">
                    <sequent>
                     <formulalist>
                      <constantatomformula symbol="L1" definition="L1"/>
                     </formulalist>
                     <formulalist>
                      <constantatomformula symbol="=">
                       <function symbol="∩">
                        <constant symbol="s_1"/>
                        <constant symbol="s_2"/>
                       </function>
                       <function symbol="∩">
                        <constant symbol="s_1"/>
                        <constant symbol="s_2"/>
                       </function>
                      </constantatomformula>
                     </formulalist>
                    </sequent>
                    <rule type="weakl" symbol="w:l">
                     <sequent>
                      <formulalist>
                       <constantatomformula symbol="L1" definition="L1"/>
                      </formulalist>
                      <formulalist>
                       <constantatomformula symbol="=">
                        <function symbol="∩">
                         <constant symbol="s_1"/>
                         <constant symbol="s_2"/>
                        </function>
                        <function symbol="∩">
                         <constant symbol="s_1"/>
                         <constant symbol="s_2"/>
                        </function>
                       </constantatomformula>
                      </formulalist>
                     </sequent>
                     <rule type="axiom" symbol="axiom">
                      <sequent>
                       <formulalist/>
                       <formulalist>
                        <constantatomformula symbol="=">
                         <function symbol="∩">
                          <constant symbol="s_1"/>
                          <constant symbol="s_2"/>
                         </function>
                         <function symbol="∩">
                          <constant symbol="s_1"/>
                          <constant symbol="s_2"/>
                         </function>
                        </constantatomformula>
                       </formulalist>
                      </sequent>
                     </rule>
                    </rule>
                   </rule>
                  </rule>
                 </rule>
                </rule>
               </rule>
              </rule>
             </rule>
            </rule>
            <rule type="contrr" symbol="c:r" param="2">
             <sequent>
              <formulalist>
               <constantatomformula symbol="=">
                <function symbol="∩">
                 <constant symbol="s_1"/>
                 <function symbol="∩">
                  <constant symbol="s_1"/>
                  <constant symbol="s_2"/>
                 </function>
                </function>
                <function symbol="∩">
                 <constant symbol="s_1"/>
                 <constant symbol="s_2"/>
                </function>
               </constantatomformula>
               <constantatomformula symbol="L1" definition="L1"/>
              </formulalist>
              <formulalist>
               <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
              </formulalist>
             </sequent>
             <rule type="permr" symbol="\pi:r" param="">
              <sequent>
               <formulalist>
                <constantatomformula symbol="=">
                 <function symbol="∩">
                  <constant symbol="s_1"/>
                  <function symbol="∩">
                   <constant symbol="s_1"/>
                   <constant symbol="s_2"/>
                  </function>
                 </function>
                 <function symbol="∩">
                  <constant symbol="s_1"/>
                  <constant symbol="s_2"/>
                 </function>
                </constantatomformula>
                <constantatomformula symbol="L1" definition="L1"/>
               </formulalist>
               <formulalist>
                <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
               </formulalist>
              </sequent>
              <rule type="contrl" symbol="c:l" param="1,2">
               <sequent>
                <formulalist>
                 <constantatomformula symbol="=">
                  <function symbol="∩">
                   <constant symbol="s_1"/>
                   <function symbol="∩">
                    <constant symbol="s_1"/>
                    <constant symbol="s_2"/>
                   </function>
                  </function>
                  <function symbol="∩">
                   <constant symbol="s_1"/>
                   <constant symbol="s_2"/>
                  </function>
                 </constantatomformula>
                 <constantatomformula symbol="L1" definition="L1"/>
                </formulalist>
                <formulalist>
                 <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                 <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                </formulalist>
               </sequent>
               <rule type="perml" symbol="\pi:l" param="">
                <sequent>
                 <formulalist>
                  <constantatomformula symbol="=">
                   <function symbol="∩">
                    <constant symbol="s_1"/>
                    <function symbol="∩">
                     <constant symbol="s_1"/>
                     <constant symbol="s_2"/>
                    </function>
                   </function>
                   <function symbol="∩">
                    <constant symbol="s_1"/>
                    <constant symbol="s_2"/>
                   </function>
                  </constantatomformula>
                  <constantatomformula symbol="L1" definition="L1"/>
                  <constantatomformula symbol="L1" definition="L1"/>
                 </formulalist>
                 <formulalist>
                  <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                  <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                 </formulalist>
                </sequent>
                <rule type="eql1" symbol="=:l1" param="(1)">
                 <sequent>
                  <formulalist>
                   <constantatomformula symbol="=">
                    <function symbol="∩">
                     <constant symbol="s_1"/>
                     <function symbol="∩">
                      <constant symbol="s_1"/>
                      <constant symbol="s_2"/>
                     </function>
                    </function>
                    <function symbol="∩">
                     <constant symbol="s_1"/>
                     <constant symbol="s_2"/>
                    </function>
                   </constantatomformula>
                   <constantatomformula symbol="L1" definition="L1"/>
                   <constantatomformula symbol="L1" definition="L1"/>
                  </formulalist>
                  <formulalist>
                   <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                   <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                  </formulalist>
                 </sequent>
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                  <sequent>
                   <formulalist>
                    <constantatomformula symbol="L1" definition="L1"/>
                   </formulalist>
                   <formulalist>
                    <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                    <constantatomformula symbol="=">
                     <function symbol="∩">
                      <function symbol="∩">
                       <constant symbol="s_1"/>
                       <constant symbol="s_2"/>
                      </function>
                      <constant symbol="s_1"/>
                     </function>
                     <function symbol="∩">
                      <constant symbol="s_1"/>
                      <function symbol="∩">
                       <constant symbol="s_1"/>
                       <constant symbol="s_2"/>
                      </function>
                     </function>
                    </constantatomformula>
                   </formulalist>
                  </sequent>
                  <rule type="weakr" symbol="w:r">
                   <sequent>
                    <formulalist>
                     <constantatomformula symbol="L1" definition="L1"/>
                    </formulalist>
                    <formulalist>
                     <constantatomformula symbol="=">
                      <function symbol="∩">
                       <function symbol="∩">
                        <constant symbol="s_1"/>
                        <constant symbol="s_2"/>
                       </function>
                       <constant symbol="s_1"/>
                      </function>
                      <function symbol="∩">
                       <constant symbol="s_1"/>
                       <function symbol="∩">
                        <constant symbol="s_1"/>
                        <constant symbol="s_2"/>
                       </function>
                      </function>
                     </constantatomformula>
                     <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                    </formulalist>
                   </sequent>
                   <rule type="perml" symbol="\pi:l" param="">
                    <sequent>
                     <formulalist>
                      <constantatomformula symbol="L1" definition="L1"/>
                     </formulalist>
                     <formulalist>
                      <constantatomformula symbol="=">
                       <function symbol="∩">
                        <function symbol="∩">
                         <constant symbol="s_1"/>
                         <constant symbol="s_2"/>
                        </function>
                        <constant symbol="s_1"/>
                       </function>
                       <function symbol="∩">
                        <constant symbol="s_1"/>
                        <function symbol="∩">
                         <constant symbol="s_1"/>
                         <constant symbol="s_2"/>
                        </function>
                       </function>
                      </constantatomformula>
                     </formulalist>
                    </sequent>
                    <rule type="weakl" symbol="w:l">
                     <sequent>
                      <formulalist>
                       <constantatomformula symbol="L1" definition="L1"/>
                      </formulalist>
                      <formulalist>
                       <constantatomformula symbol="=">
                        <function symbol="∩">
                         <function symbol="∩">
                          <constant symbol="s_1"/>
                          <constant symbol="s_2"/>
                         </function>
                         <constant symbol="s_1"/>
                        </function>
                        <function symbol="∩">
                         <constant symbol="s_1"/>
                         <function symbol="∩">
                          <constant symbol="s_1"/>
                          <constant symbol="s_2"/>
                         </function>
                        </function>
                       </constantatomformula>
                      </formulalist>
                     </sequent>
                     <rule type="axiom" symbol="axiom">
                      <sequent>
                       <formulalist/>
                       <formulalist>
                        <constantatomformula symbol="=">
                         <function symbol="∩">
                          <function symbol="∩">
                           <constant symbol="s_1"/>
                           <constant symbol="s_2"/>
                          </function>
                          <constant symbol="s_1"/>
                         </function>
                         <function symbol="∩">
                          <constant symbol="s_1"/>
                          <function symbol="∩">
                           <constant symbol="s_1"/>
                           <constant symbol="s_2"/>
                          </function>
                         </function>
                        </constantatomformula>
                       </formulalist>
                      </sequent>
                     </rule>
                    </rule>
                   </rule>
                  </rule>
                 </rule>
                 <rule type="contrr" symbol="c:r" param="2">
                  <sequent>
                   <formulalist>
                    <constantatomformula symbol="=">
                     <function symbol="∩">
                      <function symbol="∩">
                       <constant symbol="s_1"/>
                       <constant symbol="s_2"/>
                      </function>
                      <constant symbol="s_1"/>
                     </function>
                     <function symbol="∩">
                      <constant symbol="s_1"/>
                      <constant symbol="s_2"/>
                     </function>
                    </constantatomformula>
                    <constantatomformula symbol="L1" definition="L1"/>
                   </formulalist>
                   <formulalist>
                    <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                   </formulalist>
                  </sequent>
                  <rule type="permr" symbol="\pi:r" param="">
                   <sequent>
                    <formulalist>
                     <constantatomformula symbol="=">
                      <function symbol="∩">
                       <function symbol="∩">
                        <constant symbol="s_1"/>
                        <constant symbol="s_2"/>
                       </function>
                       <constant symbol="s_1"/>
                      </function>
                      <function symbol="∩">
                       <constant symbol="s_1"/>
                       <constant symbol="s_2"/>
                      </function>
                     </constantatomformula>
                     <constantatomformula symbol="L1" definition="L1"/>
                    </formulalist>
                    <formulalist>
                     <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                     <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                    </formulalist>
                   </sequent>
                   <rule type="contrl" symbol="c:l" param="1,2">
                    <sequent>
                     <formulalist>
                      <constantatomformula symbol="=">
                       <function symbol="∩">
                        <function symbol="∩">
                         <constant symbol="s_1"/>
                         <constant symbol="s_2"/>
                        </function>
                        <constant symbol="s_1"/>
                       </function>
                       <function symbol="∩">
                        <constant symbol="s_1"/>
                        <constant symbol="s_2"/>
                       </function>
                      </constantatomformula>
                      <constantatomformula symbol="L1" definition="L1"/>
                     </formulalist>
                     <formulalist>
                      <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                      <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                     </formulalist>
                    </sequent>
                    <rule type="perml" symbol="\pi:l" param="">
                     <sequent>
                      <formulalist>
                       <constantatomformula symbol="=">
                        <function symbol="∩">
                         <function symbol="∩">
                          <constant symbol="s_1"/>
                          <constant symbol="s_2"/>
                         </function>
                         <constant symbol="s_1"/>
                        </function>
                        <function symbol="∩">
                         <constant symbol="s_1"/>
                         <constant symbol="s_2"/>
                        </function>
                       </constantatomformula>
                       <constantatomformula symbol="L1" definition="L1"/>
                       <constantatomformula symbol="L1" definition="L1"/>
                      </formulalist>
                      <formulalist>
                       <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                       <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                      </formulalist>
                     </sequent>
                     <rule type="cut" symbol="cut">
                      <sequent>
                       <formulalist>
                        <constantatomformula symbol="=">
                         <function symbol="∩">
                          <function symbol="∩">
                           <constant symbol="s_1"/>
                           <constant symbol="s_2"/>
                          </function>
                          <constant symbol="s_1"/>
                         </function>
                         <function symbol="∩">
                          <constant symbol="s_1"/>
                          <constant symbol="s_2"/>
                         </function>
                        </constantatomformula>
                        <constantatomformula symbol="L1" definition="L1"/>
                        <constantatomformula symbol="L1" definition="L1"/>
                       </formulalist>
                       <formulalist>
                        <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                        <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                       </formulalist>
                      </sequent>
                      <rule type="contrr" symbol="c:r" param="2,1">
                       <sequent>
                        <formulalist>
                         <constantatomformula symbol="=">
                          <function symbol="∩">
                           <function symbol="∩">
                            <constant symbol="s_1"/>
                            <constant symbol="s_2"/>
                           </function>
                           <constant symbol="s_1"/>
                          </function>
                          <function symbol="∩">
                           <constant symbol="s_1"/>
                           <constant symbol="s_2"/>
                          </function>
                         </constantatomformula>
                         <constantatomformula symbol="L1" definition="L1"/>
                        </formulalist>
                        <formulalist>
                         <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                         <constantatomformula symbol="=">
                          <function symbol="∪">
                           <constant symbol="s_1"/>
                           <function symbol="∩">
                            <constant symbol="s_1"/>
                            <constant symbol="s_2"/>
                           </function>
                          </function>
                          <constant symbol="s_1"/>
                         </constantatomformula>
                        </formulalist>
                       </sequent>
                       <rule type="permr" symbol="\pi:r" param="">
                        <sequent>
                         <formulalist>
                          <constantatomformula symbol="=">
                           <function symbol="∩">
                            <function symbol="∩">
                             <constant symbol="s_1"/>
                             <constant symbol="s_2"/>
                            </function>
                            <constant symbol="s_1"/>
                           </function>
                           <function symbol="∩">
                            <constant symbol="s_1"/>
                            <constant symbol="s_2"/>
                           </function>
                          </constantatomformula>
                          <constantatomformula symbol="L1" definition="L1"/>
                         </formulalist>
                         <formulalist>
                          <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                          <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                          <constantatomformula symbol="=">
                           <function symbol="∪">
                            <constant symbol="s_1"/>
                            <function symbol="∩">
                             <constant symbol="s_1"/>
                             <constant symbol="s_2"/>
                            </function>
                           </function>
                           <constant symbol="s_1"/>
                          </constantatomformula>
                         </formulalist>
                        </sequent>
                        <rule type="contrl" symbol="c:l" param="1,2">
                         <sequent>
                          <formulalist>
                           <constantatomformula symbol="=">
                            <function symbol="∩">
                             <function symbol="∩">
                              <constant symbol="s_1"/>
                              <constant symbol="s_2"/>
                             </function>
                             <constant symbol="s_1"/>
                            </function>
                            <function symbol="∩">
                             <constant symbol="s_1"/>
                             <constant symbol="s_2"/>
                            </function>
                           </constantatomformula>
                           <constantatomformula symbol="L1" definition="L1"/>
                          </formulalist>
                          <formulalist>
                           <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                           <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                           <constantatomformula symbol="=">
                            <function symbol="∪">
                             <constant symbol="s_1"/>
                             <function symbol="∩">
                              <constant symbol="s_1"/>
                              <constant symbol="s_2"/>
                             </function>
                            </function>
                            <constant symbol="s_1"/>
                           </constantatomformula>
                          </formulalist>
                         </sequent>
                         <rule type="perml" symbol="\pi:l" param="">
                          <sequent>
                           <formulalist>
                            <constantatomformula symbol="=">
                             <function symbol="∩">
                              <function symbol="∩">
                               <constant symbol="s_1"/>
                               <constant symbol="s_2"/>
                              </function>
                              <constant symbol="s_1"/>
                             </function>
                             <function symbol="∩">
                              <constant symbol="s_1"/>
                              <constant symbol="s_2"/>
                             </function>
                            </constantatomformula>
                            <constantatomformula symbol="L1" definition="L1"/>
                            <constantatomformula symbol="L1" definition="L1"/>
                           </formulalist>
                           <formulalist>
                            <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                            <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                            <constantatomformula symbol="=">
                             <function symbol="∪">
                              <constant symbol="s_1"/>
                              <function symbol="∩">
                               <constant symbol="s_1"/>
                               <constant symbol="s_2"/>
                              </function>
                             </function>
                             <constant symbol="s_1"/>
                            </constantatomformula>
                           </formulalist>
                          </sequent>
                          <rule type="eqr1" symbol="=:r1" param="(1)">
                           <sequent>
                            <formulalist>
                             <constantatomformula symbol="=">
                              <function symbol="∩">
                               <function symbol="∩">
                                <constant symbol="s_1"/>
                                <constant symbol="s_2"/>
                               </function>
                               <constant symbol="s_1"/>
                              </function>
                              <function symbol="∩">
                               <constant symbol="s_1"/>
                               <constant symbol="s_2"/>
                              </function>
                             </constantatomformula>
                             <constantatomformula symbol="L1" definition="L1"/>
                             <constantatomformula symbol="L1" definition="L1"/>
                            </formulalist>
                            <formulalist>
                             <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                             <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                             <constantatomformula symbol="=">
                              <function symbol="∪">
                               <constant symbol="s_1"/>
                               <function symbol="∩">
                                <constant symbol="s_1"/>
                                <constant symbol="s_2"/>
                               </function>
                              </function>
                              <constant symbol="s_1"/>
                             </constantatomformula>
                            </formulalist>
                           </sequent>
                           <rule type="contrr" symbol="c:r" param="2,1">
                            <sequent>
                             <formulalist>
                              <constantatomformula symbol="=">
                               <function symbol="∩">
                                <function symbol="∩">
                                 <constant symbol="s_1"/>
                                 <constant symbol="s_2"/>
                                </function>
                                <constant symbol="s_1"/>
                               </function>
                               <function symbol="∩">
                                <constant symbol="s_1"/>
                                <constant symbol="s_2"/>
                               </function>
                              </constantatomformula>
                              <constantatomformula symbol="L1" definition="L1"/>
                             </formulalist>
                             <formulalist>
                              <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                              <constantatomformula symbol="=">
                               <constant symbol="s_1"/>
                               <function symbol="∪">
                                <constant symbol="s_1"/>
                                <function symbol="∩">
                                 <constant symbol="s_1"/>
                                 <constant symbol="s_2"/>
                                </function>
                               </function>
                              </constantatomformula>
                             </formulalist>
                            </sequent>
                            <rule type="permr" symbol="\pi:r" param="">
                             <sequent>
                              <formulalist>
                               <constantatomformula symbol="=">
                                <function symbol="∩">
                                 <function symbol="∩">
                                  <constant symbol="s_1"/>
                                  <constant symbol="s_2"/>
                                 </function>
                                 <constant symbol="s_1"/>
                                </function>
                                <function symbol="∩">
                                 <constant symbol="s_1"/>
                                 <constant symbol="s_2"/>
                                </function>
                               </constantatomformula>
                               <constantatomformula symbol="L1" definition="L1"/>
                              </formulalist>
                              <formulalist>
                               <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                               <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                               <constantatomformula symbol="=">
                                <constant symbol="s_1"/>
                                <function symbol="∪">
                                 <constant symbol="s_1"/>
                                 <function symbol="∩">
                                  <constant symbol="s_1"/>
                                  <constant symbol="s_2"/>
                                 </function>
                                </function>
                               </constantatomformula>
                              </formulalist>
                             </sequent>
                             <rule type="contrl" symbol="c:l" param="1,2">
                              <sequent>
                               <formulalist>
                                <constantatomformula symbol="=">
                                 <function symbol="∩">
                                  <function symbol="∩">
                                   <constant symbol="s_1"/>
                                   <constant symbol="s_2"/>
                                  </function>
                                  <constant symbol="s_1"/>
                                 </function>
                                 <function symbol="∩">
                                  <constant symbol="s_1"/>
                                  <constant symbol="s_2"/>
                                 </function>
                                </constantatomformula>
                                <constantatomformula symbol="L1" definition="L1"/>
                               </formulalist>
                               <formulalist>
                                <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                                <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                                <constantatomformula symbol="=">
                                 <constant symbol="s_1"/>
                                 <function symbol="∪">
                                  <constant symbol="s_1"/>
                                  <function symbol="∩">
                                   <constant symbol="s_1"/>
                                   <constant symbol="s_2"/>
                                  </function>
                                 </function>
                                </constantatomformula>
                               </formulalist>
                              </sequent>
                              <rule type="perml" symbol="\pi:l" param="(1 2)">
                               <sequent>
                                <formulalist>
                                 <constantatomformula symbol="=">
                                  <function symbol="∩">
                                   <function symbol="∩">
                                    <constant symbol="s_1"/>
                                    <constant symbol="s_2"/>
                                   </function>
                                   <constant symbol="s_1"/>
                                  </function>
                                  <function symbol="∩">
                                   <constant symbol="s_1"/>
                                   <constant symbol="s_2"/>
                                  </function>
                                 </constantatomformula>
                                 <constantatomformula symbol="L1" definition="L1"/>
                                 <constantatomformula symbol="L1" definition="L1"/>
                                </formulalist>
                                <formulalist>
                                 <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                                 <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                                 <constantatomformula symbol="=">
                                  <constant symbol="s_1"/>
                                  <function symbol="∪">
                                   <constant symbol="s_1"/>
                                   <function symbol="∩">
                                    <constant symbol="s_1"/>
                                    <constant symbol="s_2"/>
                                   </function>
                                  </function>
                                 </constantatomformula>
                                </formulalist>
                               </sequent>
                               <rule type="eqr1" symbol="=:r1" param="(1)">
                                <sequent>
                                 <formulalist>
                                  <constantatomformula symbol="L1" definition="L1"/>
                                  <constantatomformula symbol="=">
                                   <function symbol="∩">
                                    <function symbol="∩">
                                     <constant symbol="s_1"/>
                                     <constant symbol="s_2"/>
                                    </function>
                                    <constant symbol="s_1"/>
                                   </function>
                                   <function symbol="∩">
                                    <constant symbol="s_1"/>
                                    <constant symbol="s_2"/>
                                   </function>
                                  </constantatomformula>
                                  <constantatomformula symbol="L1" definition="L1"/>
                                 </formulalist>
                                 <formulalist>
                                  <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                                  <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                                  <constantatomformula symbol="=">
                                   <constant symbol="s_1"/>
                                   <function symbol="∪">
                                    <constant symbol="s_1"/>
                                    <function symbol="∩">
                                     <constant symbol="s_1"/>
                                     <constant symbol="s_2"/>
                                    </function>
                                   </function>
                                  </constantatomformula>
                                 </formulalist>
                                </sequent>
                                <rule type="permr" symbol="\pi:r" param="(1 2)">
                                 <sequent>
                                  <formulalist>
                                   <constantatomformula symbol="L1" definition="L1"/>
                                  </formulalist>
                                  <formulalist>
                                   <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                                   <constantatomformula symbol="=">
                                    <function symbol="∩">
                                     <constant symbol="s_1"/>
                                     <function symbol="∪">
                                      <constant symbol="s_1"/>
                                      <function symbol="∩">
                                       <constant symbol="s_1"/>
                                       <constant symbol="s_2"/>
                                      </function>
                                     </function>
                                    </function>
                                    <constant symbol="s_1"/>
                                   </constantatomformula>
                                  </formulalist>
                                 </sequent>
                                 <rule type="perml" symbol="\pi:l" param="">
                                  <sequent>
                                   <formulalist>
                                    <constantatomformula symbol="L1" definition="L1"/>
                                   </formulalist>
                                   <formulalist>
                                    <constantatomformula symbol="=">
                                     <function symbol="∩">
                                      <constant symbol="s_1"/>
                                      <function symbol="∪">
                                       <constant symbol="s_1"/>
                                       <function symbol="∩">
                                        <constant symbol="s_1"/>
                                        <constant symbol="s_2"/>
                                       </function>
                                      </function>
                                     </function>
                                     <constant symbol="s_1"/>
                                    </constantatomformula>
                                    <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                                   </formulalist>
                                  </sequent>
                                  <rule type="weakr" symbol="w:r">
                                   <sequent>
                                    <formulalist>
                                     <constantatomformula symbol="L1" definition="L1"/>
                                    </formulalist>
                                    <formulalist>
                                     <constantatomformula symbol="=">
                                      <function symbol="∩">
                                       <constant symbol="s_1"/>
                                       <function symbol="∪">
                                        <constant symbol="s_1"/>
                                        <function symbol="∩">
                                         <constant symbol="s_1"/>
                                         <constant symbol="s_2"/>
                                        </function>
                                       </function>
                                      </function>
                                      <constant symbol="s_1"/>
                                     </constantatomformula>
                                     <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                                    </formulalist>
                                   </sequent>
                                   <rule type="contrl" symbol="c:l" param="2">
                                    <sequent>
                                     <formulalist>
                                      <constantatomformula symbol="L1" definition="L1"/>
                                     </formulalist>
                                     <formulalist>
                                      <constantatomformula symbol="=">
                                       <function symbol="∩">
                                        <constant symbol="s_1"/>
                                        <function symbol="∪">
                                         <constant symbol="s_1"/>
                                         <function symbol="∩">
                                          <constant symbol="s_1"/>
                                          <constant symbol="s_2"/>
                                         </function>
                                        </function>
                                       </function>
                                       <constant symbol="s_1"/>
                                      </constantatomformula>
                                     </formulalist>
                                    </sequent>
                                    <rule type="perml" symbol="\pi:l" param="(1 2)">
                                     <sequent>
                                      <formulalist>
                                       <constantatomformula symbol="L1" definition="L1"/>
                                       <constantatomformula symbol="L1" definition="L1"/>
                                      </formulalist>
                                      <formulalist>
                                       <constantatomformula symbol="=">
                                        <function symbol="∩">
                                         <constant symbol="s_1"/>
                                         <function symbol="∪">
                                          <constant symbol="s_1"/>
                                          <function symbol="∩">
                                           <constant symbol="s_1"/>
                                           <constant symbol="s_2"/>
                                          </function>
                                         </function>
                                        </function>
                                        <constant symbol="s_1"/>
                                       </constantatomformula>
                                      </formulalist>
                                     </sequent>
                                     <rule type="weakl" symbol="w:l">
                                      <sequent>
                                       <formulalist>
                                        <constantatomformula symbol="L1" definition="L1"/>
                                        <constantatomformula symbol="L1" definition="L1"/>
                                       </formulalist>
                                       <formulalist>
                                        <constantatomformula symbol="=">
                                         <function symbol="∩">
                                          <constant symbol="s_1"/>
                                          <function symbol="∪">
                                           <constant symbol="s_1"/>
                                           <function symbol="∩">
                                            <constant symbol="s_1"/>
                                            <constant symbol="s_2"/>
                                           </function>
                                          </function>
                                         </function>
                                         <constant symbol="s_1"/>
                                        </constantatomformula>
                                       </formulalist>
                                      </sequent>
                                      <rule type="contrl" symbol="c:l" param="2">
                                       <sequent>
                                        <formulalist>
                                         <constantatomformula symbol="L1" definition="L1"/>
                                        </formulalist>
                                        <formulalist>
                                         <constantatomformula symbol="=">
                                          <function symbol="∩">
                                           <constant symbol="s_1"/>
                                           <function symbol="∪">
                                            <constant symbol="s_1"/>
                                            <function symbol="∩">
                                             <constant symbol="s_1"/>
                                             <constant symbol="s_2"/>
                                            </function>
                                           </function>
                                          </function>
                                          <constant symbol="s_1"/>
                                         </constantatomformula>
                                        </formulalist>
                                       </sequent>
                                       <rule type="perml" symbol="\pi:l" param="(1 2)">
                                        <sequent>
                                         <formulalist>
                                          <constantatomformula symbol="L1" definition="L1"/>
                                          <constantatomformula symbol="L1" definition="L1"/>
                                         </formulalist>
                                         <formulalist>
                                          <constantatomformula symbol="=">
                                           <function symbol="∩">
                                            <constant symbol="s_1"/>
                                            <function symbol="∪">
                                             <constant symbol="s_1"/>
                                             <function symbol="∩">
                                              <constant symbol="s_1"/>
                                              <constant symbol="s_2"/>
                                             </function>
                                            </function>
                                           </function>
                                           <constant symbol="s_1"/>
                                          </constantatomformula>
                                         </formulalist>
                                        </sequent>
                                        <rule type="weakl" symbol="w:l">
                                         <sequent>
                                          <formulalist>
                                           <constantatomformula symbol="L1" definition="L1"/>
                                           <constantatomformula symbol="L1" definition="L1"/>
                                          </formulalist>
                                          <formulalist>
                                           <constantatomformula symbol="=">
                                            <function symbol="∩">
                                             <constant symbol="s_1"/>
                                             <function symbol="∪">
                                              <constant symbol="s_1"/>
                                              <function symbol="∩">
                                               <constant symbol="s_1"/>
                                               <constant symbol="s_2"/>
                                              </function>
                                             </function>
                                            </function>
                                            <constant symbol="s_1"/>
                                           </constantatomformula>
                                          </formulalist>
                                         </sequent>
                                         <rule type="defl" symbol="d:l" param="();[()];()">
                                          <sequent>
                                           <formulalist>
                                            <constantatomformula symbol="L1" definition="L1"/>
                                           </formulalist>
                                           <formulalist>
                                            <constantatomformula symbol="=">
                                             <function symbol="∩">
                                              <constant symbol="s_1"/>
                                              <function symbol="∪">
                                               <constant symbol="s_1"/>
                                               <function symbol="∩">
                                                <constant symbol="s_1"/>
                                                <constant symbol="s_2"/>
                                               </function>
                                              </function>
                                             </function>
                                             <constant symbol="s_1"/>
                                            </constantatomformula>
                                           </formulalist>
                                          </sequent>
                                          <rule type="foralll" symbol="\all:l">
                                           <sequent>
                                            <formulalist>
                                             <quantifiedformula type="all">
                                              <variable symbol="x"/>
                                              <quantifiedformula type="all">
                                               <variable symbol="y"/>
                                               <conjunctiveformula type="and">
                                                <conjunctiveformula type="impl">
                                                 <constantatomformula symbol="=">
                                                  <function symbol="∩">
                                                   <variable symbol="x"/>
                                                   <variable symbol="y"/>
                                                  </function>
                                                  <variable symbol="x"/>
                                                 </constantatomformula>
                                                 <constantatomformula symbol="=">
                                                  <function symbol="∪">
                                                   <variable symbol="x"/>
                                                   <variable symbol="y"/>
                                                  </function>
                                                  <variable symbol="y"/>
                                                 </constantatomformula>
                                                </conjunctiveformula>
                                                <conjunctiveformula type="impl">
                                                 <constantatomformula symbol="=">
                                                  <function symbol="∪">
                                                   <variable symbol="x"/>
                                                   <variable symbol="y"/>
                                                  </function>
                                                  <variable symbol="y"/>
                                                 </constantatomformula>
                                                 <constantatomformula symbol="=">
                                                  <function symbol="∩">
                                                   <variable symbol="x"/>
                                                   <variable symbol="y"/>
                                                  </function>
                                                  <variable symbol="x"/>
                                                 </constantatomformula>
                                                </conjunctiveformula>
                                               </conjunctiveformula>
                                              </quantifiedformula>
                                             </quantifiedformula>
                                            </formulalist>
                                            <formulalist>
                                             <constantatomformula symbol="=">
                                              <function symbol="∩">
                                               <constant symbol="s_1"/>
                                               <function symbol="∪">
                                                <constant symbol="s_1"/>
                                                <function symbol="∩">
                                                 <constant symbol="s_1"/>
                                                 <constant symbol="s_2"/>
                                                </function>
                                               </function>
                                              </function>
                                              <constant symbol="s_1"/>
                                             </constantatomformula>
                                            </formulalist>
                                           </sequent>
                                           <rule type="foralll" symbol="\all:l">
                                            <sequent>
                                             <formulalist>
                                              <quantifiedformula type="all">
                                               <variable symbol="y"/>
                                               <conjunctiveformula type="and">
                                                <conjunctiveformula type="impl">
                                                 <constantatomformula symbol="=">
                                                  <function symbol="∩">
                                                   <constant symbol="s_1"/>
                                                   <variable symbol="y"/>
                                                  </function>
                                                  <constant symbol="s_1"/>
                                                 </constantatomformula>
                                                 <constantatomformula symbol="=">
                                                  <function symbol="∪">
                                                   <constant symbol="s_1"/>
                                                   <variable symbol="y"/>
                                                  </function>
                                                  <variable symbol="y"/>
                                                 </constantatomformula>
                                                </conjunctiveformula>
                                                <conjunctiveformula type="impl">
                                                 <constantatomformula symbol="=">
                                                  <function symbol="∪">
                                                   <constant symbol="s_1"/>
                                                   <variable symbol="y"/>
                                                  </function>
                                                  <variable symbol="y"/>
                                                 </constantatomformula>
                                                 <constantatomformula symbol="=">
                                                  <function symbol="∩">
                                                   <constant symbol="s_1"/>
                                                   <variable symbol="y"/>
                                                  </function>
                                                  <constant symbol="s_1"/>
                                                 </constantatomformula>
                                                </conjunctiveformula>
                                               </conjunctiveformula>
                                              </quantifiedformula>
                                             </formulalist>
                                             <formulalist>
                                              <constantatomformula symbol="=">
                                               <function symbol="∩">
                                                <constant symbol="s_1"/>
                                                <function symbol="∪">
                                                 <constant symbol="s_1"/>
                                                 <function symbol="∩">
                                                  <constant symbol="s_1"/>
                                                  <constant symbol="s_2"/>
                                                 </function>
                                                </function>
                                               </function>
                                               <constant symbol="s_1"/>
                                              </constantatomformula>
                                             </formulalist>
                                            </sequent>
                                            <rule type="andl2" symbol="\and:l2">
                                             <sequent>
                                              <formulalist>
                                               <conjunctiveformula type="and">
                                                <conjunctiveformula type="impl">
                                                 <constantatomformula symbol="=">
                                                  <function symbol="∩">
                                                   <constant symbol="s_1"/>
                                                   <function symbol="∪">
                                                    <constant symbol="s_1"/>
                                                    <function symbol="∩">
                                                     <constant symbol="s_1"/>
                                                     <constant symbol="s_2"/>
                                                    </function>
                                                   </function>
                                                  </function>
                                                  <constant symbol="s_1"/>
                                                 </constantatomformula>
                                                 <constantatomformula symbol="=">
                                                  <function symbol="∪">
                                                   <constant symbol="s_1"/>
                                                   <function symbol="∪">
                                                    <constant symbol="s_1"/>
                                                    <function symbol="∩">
                                                     <constant symbol="s_1"/>
                                                     <constant symbol="s_2"/>
                                                    </function>
                                                   </function>
                                                  </function>
                                                  <function symbol="∪">
                                                   <constant symbol="s_1"/>
                                                   <function symbol="∩">
                                                    <constant symbol="s_1"/>
                                                    <constant symbol="s_2"/>
                                                   </function>
                                                  </function>
                                                 </constantatomformula>
                                                </conjunctiveformula>
                                                <conjunctiveformula type="impl">
                                                 <constantatomformula symbol="=">
                                                  <function symbol="∪">
                                                   <constant symbol="s_1"/>
                                                   <function symbol="∪">
                                                    <constant symbol="s_1"/>
                                                    <function symbol="∩">
                                                     <constant symbol="s_1"/>
                                                     <constant symbol="s_2"/>
                                                    </function>
                                                   </function>
                                                  </function>
                                                  <function symbol="∪">
                                                   <constant symbol="s_1"/>
                                                   <function symbol="∩">
                                                    <constant symbol="s_1"/>
                                                    <constant symbol="s_2"/>
                                                   </function>
                                                  </function>
                                                 </constantatomformula>
                                                 <constantatomformula symbol="=">
                                                  <function symbol="∩">
                                                   <constant symbol="s_1"/>
                                                   <function symbol="∪">
                                                    <constant symbol="s_1"/>
                                                    <function symbol="∩">
                                                     <constant symbol="s_1"/>
                                                     <constant symbol="s_2"/>
                                                    </function>
                                                   </function>
                                                  </function>
                                                  <constant symbol="s_1"/>
                                                 </constantatomformula>
                                                </conjunctiveformula>
                                               </conjunctiveformula>
                                              </formulalist>
                                              <formulalist>
                                               <constantatomformula symbol="=">
                                                <function symbol="∩">
                                                 <constant symbol="s_1"/>
                                                 <function symbol="∪">
                                                  <constant symbol="s_1"/>
                                                  <function symbol="∩">
                                                   <constant symbol="s_1"/>
                                                   <constant symbol="s_2"/>
                                                  </function>
                                                 </function>
                                                </function>
                                                <constant symbol="s_1"/>
                                               </constantatomformula>
                                              </formulalist>
                                             </sequent>
                                             <rule type="impll" symbol="\impl:l">
                                              <sequent>
                                               <formulalist>
                                                <conjunctiveformula type="impl">
                                                 <constantatomformula symbol="=">
                                                  <function symbol="∪">
                                                   <constant symbol="s_1"/>
                                                   <function symbol="∪">
                                                    <constant symbol="s_1"/>
                                                    <function symbol="∩">
                                                     <constant symbol="s_1"/>
                                                     <constant symbol="s_2"/>
                                                    </function>
                                                   </function>
                                                  </function>
                                                  <function symbol="∪">
                                                   <constant symbol="s_1"/>
                                                   <function symbol="∩">
                                                    <constant symbol="s_1"/>
                                                    <constant symbol="s_2"/>
                                                   </function>
                                                  </function>
                                                 </constantatomformula>
                                                 <constantatomformula symbol="=">
                                                  <function symbol="∩">
                                                   <constant symbol="s_1"/>
                                                   <function symbol="∪">
                                                    <constant symbol="s_1"/>
                                                    <function symbol="∩">
                                                     <constant symbol="s_1"/>
                                                     <constant symbol="s_2"/>
                                                    </function>
                                                   </function>
                                                  </function>
                                                  <constant symbol="s_1"/>
                                                 </constantatomformula>
                                                </conjunctiveformula>
                                               </formulalist>
                                               <formulalist>
                                                <constantatomformula symbol="=">
                                                 <function symbol="∩">
                                                  <constant symbol="s_1"/>
                                                  <function symbol="∪">
                                                   <constant symbol="s_1"/>
                                                   <function symbol="∩">
                                                    <constant symbol="s_1"/>
                                                    <constant symbol="s_2"/>
                                                   </function>
                                                  </function>
                                                 </function>
                                                 <constant symbol="s_1"/>
                                                </constantatomformula>
                                               </formulalist>
                                              </sequent>
                                              <rule type="eqr1" symbol="=:r1" param="(1)">
                                               <sequent>
                                                <formulalist/>
                                                <formulalist>
                                                 <constantatomformula symbol="=">
                                                  <function symbol="∪">
                                                   <constant symbol="s_1"/>
                                                   <function symbol="∪">
                                                    <constant symbol="s_1"/>
                                                    <function symbol="∩">
                                                     <constant symbol="s_1"/>
                                                     <constant symbol="s_2"/>
                                                    </function>
                                                   </function>
                                                  </function>
                                                  <function symbol="∪">
                                                   <constant symbol="s_1"/>
                                                   <function symbol="∩">
                                                    <constant symbol="s_1"/>
                                                    <constant symbol="s_2"/>
                                                   </function>
                                                  </function>
                                                 </constantatomformula>
                                                </formulalist>
                                               </sequent>
                                               <rule type="axiom" symbol="axiom">
                                                <sequent>
                                                 <formulalist/>
                                                 <formulalist>
                                                  <constantatomformula symbol="=">
                                                   <function symbol="∪">
                                                    <function symbol="∪">
                                                     <constant symbol="s_1"/>
                                                     <constant symbol="s_1"/>
                                                    </function>
                                                    <function symbol="∩">
                                                     <constant symbol="s_1"/>
                                                     <constant symbol="s_2"/>
                                                    </function>
                                                   </function>
                                                   <function symbol="∪">
                                                    <constant symbol="s_1"/>
                                                    <function symbol="∪">
                                                     <constant symbol="s_1"/>
                                                     <function symbol="∩">
                                                      <constant symbol="s_1"/>
                                                      <constant symbol="s_2"/>
                                                     </function>
                                                    </function>
                                                   </function>
                                                  </constantatomformula>
                                                 </formulalist>
                                                </sequent>
                                               </rule>
                                               <rule type="eqr2" symbol="=:r2" param="(1 1)">
                                                <sequent>
                                                 <formulalist/>
                                                 <formulalist>
                                                  <constantatomformula symbol="=">
                                                   <function symbol="∪">
                                                    <function symbol="∪">
                                                     <constant symbol="s_1"/>
                                                     <constant symbol="s_1"/>
                                                    </function>
                                                    <function symbol="∩">
                                                     <constant symbol="s_1"/>
                                                     <constant symbol="s_2"/>
                                                    </function>
                                                   </function>
                                                   <function symbol="∪">
                                                    <constant symbol="s_1"/>
                                                    <function symbol="∩">
                                                     <constant symbol="s_1"/>
                                                     <constant symbol="s_2"/>
                                                    </function>
                                                   </function>
                                                  </constantatomformula>
                                                 </formulalist>
                                                </sequent>
                                                <rule type="axiom" symbol="axiom">
                                                 <sequent>
                                                  <formulalist/>
                                                  <formulalist>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="∪">
                                                     <constant symbol="s_1"/>
                                                     <constant symbol="s_1"/>
                                                    </function>
                                                    <constant symbol="s_1"/>
                                                   </constantatomformula>
                                                  </formulalist>
                                                 </sequent>
                                                </rule>
                                                <rule type="axiom" symbol="axiom">
                                                 <sequent>
                                                  <formulalist/>
                                                  <formulalist>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="∪">
                                                     <constant symbol="s_1"/>
                                                     <function symbol="∩">
                                                      <constant symbol="s_1"/>
                                                      <constant symbol="s_2"/>
                                                     </function>
                                                    </function>
                                                    <function symbol="∪">
                                                     <constant symbol="s_1"/>
                                                     <function symbol="∩">
                                                      <constant symbol="s_1"/>
                                                      <constant symbol="s_2"/>
                                                     </function>
                                                    </function>
                                                   </constantatomformula>
                                                  </formulalist>
                                                 </sequent>
                                                </rule>
                                               </rule>
                                              </rule>
                                              <rule type="axiom" symbol="axiom">
                                               <sequent>
                                                <formulalist>
                                                 <constantatomformula symbol="=">
                                                  <function symbol="∩">
                                                   <constant symbol="s_1"/>
                                                   <function symbol="∪">
                                                    <constant symbol="s_1"/>
                                                    <function symbol="∩">
                                                     <constant symbol="s_1"/>
                                                     <constant symbol="s_2"/>
                                                    </function>
                                                   </function>
                                                  </function>
                                                  <constant symbol="s_1"/>
                                                 </constantatomformula>
                                                </formulalist>
                                                <formulalist>
                                                 <constantatomformula symbol="=">
                                                  <function symbol="∩">
                                                   <constant symbol="s_1"/>
                                                   <function symbol="∪">
                                                    <constant symbol="s_1"/>
                                                    <function symbol="∩">
                                                     <constant symbol="s_1"/>
                                                     <constant symbol="s_2"/>
                                                    </function>
                                                   </function>
                                                  </function>
                                                  <constant symbol="s_1"/>
                                                 </constantatomformula>
                                                </formulalist>
                                               </sequent>
                                              </rule>
                                             </rule>
                                            </rule>
                                           </rule>
                                          </rule>
                                         </rule>
                                        </rule>
                                       </rule>
                                      </rule>
                                     </rule>
                                    </rule>
                                   </rule>
                                  </rule>
                                 </rule>
                                </rule>
                                <rule type="contrr" symbol="c:r" param="2,1">
                                 <sequent>
                                  <formulalist>
                                   <constantatomformula symbol="=">
                                    <function symbol="∩">
                                     <function symbol="∩">
                                      <constant symbol="s_1"/>
                                      <constant symbol="s_2"/>
                                     </function>
                                     <constant symbol="s_1"/>
                                    </function>
                                    <function symbol="∩">
                                     <constant symbol="s_1"/>
                                     <constant symbol="s_2"/>
                                    </function>
                                   </constantatomformula>
                                   <constantatomformula symbol="L1" definition="L1"/>
                                  </formulalist>
                                  <formulalist>
                                   <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                                   <constantatomformula symbol="=">
                                    <function symbol="∩">
                                     <constant symbol="s_1"/>
                                     <function symbol="∪">
                                      <constant symbol="s_1"/>
                                      <function symbol="∩">
                                       <constant symbol="s_1"/>
                                       <constant symbol="s_2"/>
                                      </function>
                                     </function>
                                    </function>
                                    <function symbol="∪">
                                     <constant symbol="s_1"/>
                                     <function symbol="∩">
                                      <constant symbol="s_1"/>
                                      <constant symbol="s_2"/>
                                     </function>
                                    </function>
                                   </constantatomformula>
                                  </formulalist>
                                 </sequent>
                                 <rule type="permr" symbol="\pi:r" param="">
                                  <sequent>
                                   <formulalist>
                                    <constantatomformula symbol="=">
                                     <function symbol="∩">
                                      <function symbol="∩">
                                       <constant symbol="s_1"/>
                                       <constant symbol="s_2"/>
                                      </function>
                                      <constant symbol="s_1"/>
                                     </function>
                                     <function symbol="∩">
                                      <constant symbol="s_1"/>
                                      <constant symbol="s_2"/>
                                     </function>
                                    </constantatomformula>
                                    <constantatomformula symbol="L1" definition="L1"/>
                                   </formulalist>
                                   <formulalist>
                                    <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                                    <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                                    <constantatomformula symbol="=">
                                     <function symbol="∩">
                                      <constant symbol="s_1"/>
                                      <function symbol="∪">
                                       <constant symbol="s_1"/>
                                       <function symbol="∩">
                                        <constant symbol="s_1"/>
                                        <constant symbol="s_2"/>
                                       </function>
                                      </function>
                                     </function>
                                     <function symbol="∪">
                                      <constant symbol="s_1"/>
                                      <function symbol="∩">
                                       <constant symbol="s_1"/>
                                       <constant symbol="s_2"/>
                                      </function>
                                     </function>
                                    </constantatomformula>
                                   </formulalist>
                                  </sequent>
                                  <rule type="contrl" symbol="c:l" param="1,2">
                                   <sequent>
                                    <formulalist>
                                     <constantatomformula symbol="=">
                                      <function symbol="∩">
                                       <function symbol="∩">
                                        <constant symbol="s_1"/>
                                        <constant symbol="s_2"/>
                                       </function>
                                       <constant symbol="s_1"/>
                                      </function>
                                      <function symbol="∩">
                                       <constant symbol="s_1"/>
                                       <constant symbol="s_2"/>
                                      </function>
                                     </constantatomformula>
                                     <constantatomformula symbol="L1" definition="L1"/>
                                    </formulalist>
                                    <formulalist>
                                     <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                                     <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                                     <constantatomformula symbol="=">
                                      <function symbol="∩">
                                       <constant symbol="s_1"/>
                                       <function symbol="∪">
                                        <constant symbol="s_1"/>
                                        <function symbol="∩">
                                         <constant symbol="s_1"/>
                                         <constant symbol="s_2"/>
                                        </function>
                                       </function>
                                      </function>
                                      <function symbol="∪">
                                       <constant symbol="s_1"/>
                                       <function symbol="∩">
                                        <constant symbol="s_1"/>
                                        <constant symbol="s_2"/>
                                       </function>
                                      </function>
                                     </constantatomformula>
                                    </formulalist>
                                   </sequent>
                                   <rule type="perml" symbol="\pi:l" param="(1 2)">
                                    <sequent>
                                     <formulalist>
                                      <constantatomformula symbol="=">
                                       <function symbol="∩">
                                        <function symbol="∩">
                                         <constant symbol="s_1"/>
                                         <constant symbol="s_2"/>
                                        </function>
                                        <constant symbol="s_1"/>
                                       </function>
                                       <function symbol="∩">
                                        <constant symbol="s_1"/>
                                        <constant symbol="s_2"/>
                                       </function>
                                      </constantatomformula>
                                      <constantatomformula symbol="L1" definition="L1"/>
                                      <constantatomformula symbol="L1" definition="L1"/>
                                     </formulalist>
                                     <formulalist>
                                      <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                                      <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                                      <constantatomformula symbol="=">
                                       <function symbol="∩">
                                        <constant symbol="s_1"/>
                                        <function symbol="∪">
                                         <constant symbol="s_1"/>
                                         <function symbol="∩">
                                          <constant symbol="s_1"/>
                                          <constant symbol="s_2"/>
                                         </function>
                                        </function>
                                       </function>
                                       <function symbol="∪">
                                        <constant symbol="s_1"/>
                                        <function symbol="∩">
                                         <constant symbol="s_1"/>
                                         <constant symbol="s_2"/>
                                        </function>
                                       </function>
                                      </constantatomformula>
                                     </formulalist>
                                    </sequent>
                                    <rule type="eqr1" symbol="=:r1" param="(1)">
                                     <sequent>
                                      <formulalist>
                                       <constantatomformula symbol="L1" definition="L1"/>
                                       <constantatomformula symbol="=">
                                        <function symbol="∩">
                                         <function symbol="∩">
                                          <constant symbol="s_1"/>
                                          <constant symbol="s_2"/>
                                         </function>
                                         <constant symbol="s_1"/>
                                        </function>
                                        <function symbol="∩">
                                         <constant symbol="s_1"/>
                                         <constant symbol="s_2"/>
                                        </function>
                                       </constantatomformula>
                                       <constantatomformula symbol="L1" definition="L1"/>
                                      </formulalist>
                                      <formulalist>
                                       <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                                       <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                                       <constantatomformula symbol="=">
                                        <function symbol="∩">
                                         <constant symbol="s_1"/>
                                         <function symbol="∪">
                                          <constant symbol="s_1"/>
                                          <function symbol="∩">
                                           <constant symbol="s_1"/>
                                           <constant symbol="s_2"/>
                                          </function>
                                         </function>
                                        </function>
                                        <function symbol="∪">
                                         <constant symbol="s_1"/>
                                         <function symbol="∩">
                                          <constant symbol="s_1"/>
                                          <constant symbol="s_2"/>
                                         </function>
                                        </function>
                                       </constantatomformula>
                                      </formulalist>
                                     </sequent>
                                     <rule type="permr" symbol="\pi:r" param="(1 2)">
                                      <sequent>
                                       <formulalist>
                                        <constantatomformula symbol="L1" definition="L1"/>
                                       </formulalist>
                                       <formulalist>
                                        <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                                        <constantatomformula symbol="=">
                                         <function symbol="∩">
                                          <function symbol="∪">
                                           <constant symbol="s_1"/>
                                           <function symbol="∩">
                                            <constant symbol="s_1"/>
                                            <constant symbol="s_2"/>
                                           </function>
                                          </function>
                                          <constant symbol="s_1"/>
                                         </function>
                                         <function symbol="∩">
                                          <constant symbol="s_1"/>
                                          <function symbol="∪">
                                           <constant symbol="s_1"/>
                                           <function symbol="∩">
                                            <constant symbol="s_1"/>
                                            <constant symbol="s_2"/>
                                           </function>
                                          </function>
                                         </function>
                                        </constantatomformula>
                                       </formulalist>
                                      </sequent>
                                      <rule type="weakr" symbol="w:r">
                                       <sequent>
                                        <formulalist>
                                         <constantatomformula symbol="L1" definition="L1"/>
                                        </formulalist>
                                        <formulalist>
                                         <constantatomformula symbol="=">
                                          <function symbol="∩">
                                           <function symbol="∪">
                                            <constant symbol="s_1"/>
                                            <function symbol="∩">
                                             <constant symbol="s_1"/>
                                             <constant symbol="s_2"/>
                                            </function>
                                           </function>
                                           <constant symbol="s_1"/>
                                          </function>
                                          <function symbol="∩">
                                           <constant symbol="s_1"/>
                                           <function symbol="∪">
                                            <constant symbol="s_1"/>
                                            <function symbol="∩">
                                             <constant symbol="s_1"/>
                                             <constant symbol="s_2"/>
                                            </function>
                                           </function>
                                          </function>
                                         </constantatomformula>
                                         <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                                        </formulalist>
                                       </sequent>
                                       <rule type="perml" symbol="\pi:l" param="">
                                        <sequent>
                                         <formulalist>
                                          <constantatomformula symbol="L1" definition="L1"/>
                                         </formulalist>
                                         <formulalist>
                                          <constantatomformula symbol="=">
                                           <function symbol="∩">
                                            <function symbol="∪">
                                             <constant symbol="s_1"/>
                                             <function symbol="∩">
                                              <constant symbol="s_1"/>
                                              <constant symbol="s_2"/>
                                             </function>
                                            </function>
                                            <constant symbol="s_1"/>
                                           </function>
                                           <function symbol="∩">
                                            <constant symbol="s_1"/>
                                            <function symbol="∪">
                                             <constant symbol="s_1"/>
                                             <function symbol="∩">
                                              <constant symbol="s_1"/>
                                              <constant symbol="s_2"/>
                                             </function>
                                            </function>
                                           </function>
                                          </constantatomformula>
                                         </formulalist>
                                        </sequent>
                                        <rule type="weakl" symbol="w:l">
                                         <sequent>
                                          <formulalist>
                                           <constantatomformula symbol="L1" definition="L1"/>
                                          </formulalist>
                                          <formulalist>
                                           <constantatomformula symbol="=">
                                            <function symbol="∩">
                                             <function symbol="∪">
                                              <constant symbol="s_1"/>
                                              <function symbol="∩">
                                               <constant symbol="s_1"/>
                                               <constant symbol="s_2"/>
                                              </function>
                                             </function>
                                             <constant symbol="s_1"/>
                                            </function>
                                            <function symbol="∩">
                                             <constant symbol="s_1"/>
                                             <function symbol="∪">
                                              <constant symbol="s_1"/>
                                              <function symbol="∩">
                                               <constant symbol="s_1"/>
                                               <constant symbol="s_2"/>
                                              </function>
                                             </function>
                                            </function>
                                           </constantatomformula>
                                          </formulalist>
                                         </sequent>
                                         <rule type="axiom" symbol="axiom">
                                          <sequent>
                                           <formulalist/>
                                           <formulalist>
                                            <constantatomformula symbol="=">
                                             <function symbol="∩">
                                              <function symbol="∪">
                                               <constant symbol="s_1"/>
                                               <function symbol="∩">
                                                <constant symbol="s_1"/>
                                                <constant symbol="s_2"/>
                                               </function>
                                              </function>
                                              <constant symbol="s_1"/>
                                             </function>
                                             <function symbol="∩">
                                              <constant symbol="s_1"/>
                                              <function symbol="∪">
                                               <constant symbol="s_1"/>
                                               <function symbol="∩">
                                                <constant symbol="s_1"/>
                                                <constant symbol="s_2"/>
                                               </function>
                                              </function>
                                             </function>
                                            </constantatomformula>
                                           </formulalist>
                                          </sequent>
                                         </rule>
                                        </rule>
                                       </rule>
                                      </rule>
                                     </rule>
                                     <rule type="contrr" symbol="c:r" param="2,1">
                                      <sequent>
                                       <formulalist>
                                        <constantatomformula symbol="=">
                                         <function symbol="∩">
                                          <function symbol="∩">
                                           <constant symbol="s_1"/>
                                           <constant symbol="s_2"/>
                                          </function>
                                          <constant symbol="s_1"/>
                                         </function>
                                         <function symbol="∩">
                                          <constant symbol="s_1"/>
                                          <constant symbol="s_2"/>
                                         </function>
                                        </constantatomformula>
                                        <constantatomformula symbol="L1" definition="L1"/>
                                       </formulalist>
                                       <formulalist>
                                        <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                                        <constantatomformula symbol="=">
                                         <function symbol="∩">
                                          <function symbol="∪">
                                           <constant symbol="s_1"/>
                                           <function symbol="∩">
                                            <constant symbol="s_1"/>
                                            <constant symbol="s_2"/>
                                           </function>
                                          </function>
                                          <constant symbol="s_1"/>
                                         </function>
                                         <function symbol="∪">
                                          <constant symbol="s_1"/>
                                          <function symbol="∩">
                                           <constant symbol="s_1"/>
                                           <constant symbol="s_2"/>
                                          </function>
                                         </function>
                                        </constantatomformula>
                                       </formulalist>
                                      </sequent>
                                      <rule type="permr" symbol="\pi:r" param="">
                                       <sequent>
                                        <formulalist>
                                         <constantatomformula symbol="=">
                                          <function symbol="∩">
                                           <function symbol="∩">
                                            <constant symbol="s_1"/>
                                            <constant symbol="s_2"/>
                                           </function>
                                           <constant symbol="s_1"/>
                                          </function>
                                          <function symbol="∩">
                                           <constant symbol="s_1"/>
                                           <constant symbol="s_2"/>
                                          </function>
                                         </constantatomformula>
                                         <constantatomformula symbol="L1" definition="L1"/>
                                        </formulalist>
                                        <formulalist>
                                         <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                                         <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                                         <constantatomformula symbol="=">
                                          <function symbol="∩">
                                           <function symbol="∪">
                                            <constant symbol="s_1"/>
                                            <function symbol="∩">
                                             <constant symbol="s_1"/>
                                             <constant symbol="s_2"/>
                                            </function>
                                           </function>
                                           <constant symbol="s_1"/>
                                          </function>
                                          <function symbol="∪">
                                           <constant symbol="s_1"/>
                                           <function symbol="∩">
                                            <constant symbol="s_1"/>
                                            <constant symbol="s_2"/>
                                           </function>
                                          </function>
                                         </constantatomformula>
                                        </formulalist>
                                       </sequent>
                                       <rule type="contrl" symbol="c:l" param="1,2">
                                        <sequent>
                                         <formulalist>
                                          <constantatomformula symbol="=">
                                           <function symbol="∩">
                                            <function symbol="∩">
                                             <constant symbol="s_1"/>
                                             <constant symbol="s_2"/>
                                            </function>
                                            <constant symbol="s_1"/>
                                           </function>
                                           <function symbol="∩">
                                            <constant symbol="s_1"/>
                                            <constant symbol="s_2"/>
                                           </function>
                                          </constantatomformula>
                                          <constantatomformula symbol="L1" definition="L1"/>
                                         </formulalist>
                                         <formulalist>
                                          <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                                          <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                                          <constantatomformula symbol="=">
                                           <function symbol="∩">
                                            <function symbol="∪">
                                             <constant symbol="s_1"/>
                                             <function symbol="∩">
                                              <constant symbol="s_1"/>
                                              <constant symbol="s_2"/>
                                             </function>
                                            </function>
                                            <constant symbol="s_1"/>
                                           </function>
                                           <function symbol="∪">
                                            <constant symbol="s_1"/>
                                            <function symbol="∩">
                                             <constant symbol="s_1"/>
                                             <constant symbol="s_2"/>
                                            </function>
                                           </function>
                                          </constantatomformula>
                                         </formulalist>
                                        </sequent>
                                        <rule type="perml" symbol="\pi:l" param="(1 2)">
                                         <sequent>
                                          <formulalist>
                                           <constantatomformula symbol="=">
                                            <function symbol="∩">
                                             <function symbol="∩">
                                              <constant symbol="s_1"/>
                                              <constant symbol="s_2"/>
                                             </function>
                                             <constant symbol="s_1"/>
                                            </function>
                                            <function symbol="∩">
                                             <constant symbol="s_1"/>
                                             <constant symbol="s_2"/>
                                            </function>
                                           </constantatomformula>
                                           <constantatomformula symbol="L1" definition="L1"/>
                                           <constantatomformula symbol="L1" definition="L1"/>
                                          </formulalist>
                                          <formulalist>
                                           <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                                           <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                                           <constantatomformula symbol="=">
                                            <function symbol="∩">
                                             <function symbol="∪">
                                              <constant symbol="s_1"/>
                                              <function symbol="∩">
                                               <constant symbol="s_1"/>
                                               <constant symbol="s_2"/>
                                              </function>
                                             </function>
                                             <constant symbol="s_1"/>
                                            </function>
                                            <function symbol="∪">
                                             <constant symbol="s_1"/>
                                             <function symbol="∩">
                                              <constant symbol="s_1"/>
                                              <constant symbol="s_2"/>
                                             </function>
                                            </function>
                                           </constantatomformula>
                                          </formulalist>
                                         </sequent>
                                         <rule type="cut" symbol="cut">
                                          <sequent>
                                           <formulalist>
                                            <constantatomformula symbol="L1" definition="L1"/>
                                            <constantatomformula symbol="=">
                                             <function symbol="∩">
                                              <function symbol="∩">
                                               <constant symbol="s_1"/>
                                               <constant symbol="s_2"/>
                                              </function>
                                              <constant symbol="s_1"/>
                                             </function>
                                             <function symbol="∩">
                                              <constant symbol="s_1"/>
                                              <constant symbol="s_2"/>
                                             </function>
                                            </constantatomformula>
                                            <constantatomformula symbol="L1" definition="L1"/>
                                           </formulalist>
                                           <formulalist>
                                            <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                                            <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                                            <constantatomformula symbol="=">
                                             <function symbol="∩">
                                              <function symbol="∪">
                                               <constant symbol="s_1"/>
                                               <function symbol="∩">
                                                <constant symbol="s_1"/>
                                                <constant symbol="s_2"/>
                                               </function>
                                              </function>
                                              <constant symbol="s_1"/>
                                             </function>
                                             <function symbol="∪">
                                              <constant symbol="s_1"/>
                                              <function symbol="∩">
                                               <constant symbol="s_1"/>
                                               <constant symbol="s_2"/>
                                              </function>
                                             </function>
                                            </constantatomformula>
                                           </formulalist>
                                          </sequent>
                                          <rule type="permr" symbol="\pi:r" param="(1 2)">
                                           <sequent>
                                            <formulalist>
                                             <constantatomformula symbol="L1" definition="L1"/>
                                            </formulalist>
                                            <formulalist>
                                             <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                                             <constantatomformula symbol="=">
                                              <function symbol="∩">
                                               <constant symbol="s_1"/>
                                               <constant symbol="s_1"/>
                                              </function>
                                              <constant symbol="s_1"/>
                                             </constantatomformula>
                                            </formulalist>
                                           </sequent>
                                           <rule type="weakr" symbol="w:r">
                                            <sequent>
                                             <formulalist>
                                              <constantatomformula symbol="L1" definition="L1"/>
                                             </formulalist>
                                             <formulalist>
                                              <constantatomformula symbol="=">
                                               <function symbol="∩">
                                                <constant symbol="s_1"/>
                                                <constant symbol="s_1"/>
                                               </function>
                                               <constant symbol="s_1"/>
                                              </constantatomformula>
                                              <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                                             </formulalist>
                                            </sequent>
                                            <rule type="perml" symbol="\pi:l" param="">
                                             <sequent>
                                              <formulalist>
                                               <constantatomformula symbol="L1" definition="L1"/>
                                              </formulalist>
                                              <formulalist>
                                               <constantatomformula symbol="=">
                                                <function symbol="∩">
                                                 <constant symbol="s_1"/>
                                                 <constant symbol="s_1"/>
                                                </function>
                                                <constant symbol="s_1"/>
                                               </constantatomformula>
                                              </formulalist>
                                             </sequent>
                                             <rule type="weakl" symbol="w:l">
                                              <sequent>
                                               <formulalist>
                                                <constantatomformula symbol="L1" definition="L1"/>
                                               </formulalist>
                                               <formulalist>
                                                <constantatomformula symbol="=">
                                                 <function symbol="∩">
                                                  <constant symbol="s_1"/>
                                                  <constant symbol="s_1"/>
                                                 </function>
                                                 <constant symbol="s_1"/>
                                                </constantatomformula>
                                               </formulalist>
                                              </sequent>
                                              <rule type="axiom" symbol="axiom">
                                               <sequent>
                                                <formulalist/>
                                                <formulalist>
                                                 <constantatomformula symbol="=">
                                                  <function symbol="∩">
                                                   <constant symbol="s_1"/>
                                                   <constant symbol="s_1"/>
                                                  </function>
                                                  <constant symbol="s_1"/>
                                                 </constantatomformula>
                                                </formulalist>
                                               </sequent>
                                              </rule>
                                             </rule>
                                            </rule>
                                           </rule>
                                          </rule>
                                          <rule type="permr" symbol="\pi:r" param="(1 2)">
                                           <sequent>
                                            <formulalist>
                                             <constantatomformula symbol="=">
                                              <function symbol="∩">
                                               <constant symbol="s_1"/>
                                               <constant symbol="s_1"/>
                                              </function>
                                              <constant symbol="s_1"/>
                                             </constantatomformula>
                                             <constantatomformula symbol="=">
                                              <function symbol="∩">
                                               <function symbol="∩">
                                                <constant symbol="s_1"/>
                                                <constant symbol="s_2"/>
                                               </function>
                                               <constant symbol="s_1"/>
                                              </function>
                                              <function symbol="∩">
                                               <constant symbol="s_1"/>
                                               <constant symbol="s_2"/>
                                              </function>
                                             </constantatomformula>
                                             <constantatomformula symbol="L1" definition="L1"/>
                                            </formulalist>
                                            <formulalist>
                                             <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                                             <constantatomformula symbol="=">
                                              <function symbol="∩">
                                               <function symbol="∪">
                                                <constant symbol="s_1"/>
                                                <function symbol="∩">
                                                 <constant symbol="s_1"/>
                                                 <constant symbol="s_2"/>
                                                </function>
                                               </function>
                                               <constant symbol="s_1"/>
                                              </function>
                                              <function symbol="∪">
                                               <constant symbol="s_1"/>
                                               <function symbol="∩">
                                                <constant symbol="s_1"/>
                                                <constant symbol="s_2"/>
                                               </function>
                                              </function>
                                             </constantatomformula>
                                            </formulalist>
                                           </sequent>
                                           <rule type="perml" symbol="\pi:l" param="(1 2 3)">
                                            <sequent>
                                             <formulalist>
                                              <constantatomformula symbol="=">
                                               <function symbol="∩">
                                                <constant symbol="s_1"/>
                                                <constant symbol="s_1"/>
                                               </function>
                                               <constant symbol="s_1"/>
                                              </constantatomformula>
                                              <constantatomformula symbol="=">
                                               <function symbol="∩">
                                                <function symbol="∩">
                                                 <constant symbol="s_1"/>
                                                 <constant symbol="s_2"/>
                                                </function>
                                                <constant symbol="s_1"/>
                                               </function>
                                               <function symbol="∩">
                                                <constant symbol="s_1"/>
                                                <constant symbol="s_2"/>
                                               </function>
                                              </constantatomformula>
                                              <constantatomformula symbol="L1" definition="L1"/>
                                             </formulalist>
                                             <formulalist>
                                              <constantatomformula symbol="=">
                                               <function symbol="∩">
                                                <function symbol="∪">
                                                 <constant symbol="s_1"/>
                                                 <function symbol="∩">
                                                  <constant symbol="s_1"/>
                                                  <constant symbol="s_2"/>
                                                 </function>
                                                </function>
                                                <constant symbol="s_1"/>
                                               </function>
                                               <function symbol="∪">
                                                <constant symbol="s_1"/>
                                                <function symbol="∩">
                                                 <constant symbol="s_1"/>
                                                 <constant symbol="s_2"/>
                                                </function>
                                               </function>
                                              </constantatomformula>
                                              <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                                             </formulalist>
                                            </sequent>
                                            <rule type="weakr" symbol="w:r">
                                             <sequent>
                                              <formulalist>
                                               <constantatomformula symbol="L1" definition="L1"/>
                                               <constantatomformula symbol="=">
                                                <function symbol="∩">
                                                 <constant symbol="s_1"/>
                                                 <constant symbol="s_1"/>
                                                </function>
                                                <constant symbol="s_1"/>
                                               </constantatomformula>
                                               <constantatomformula symbol="=">
                                                <function symbol="∩">
                                                 <function symbol="∩">
                                                  <constant symbol="s_1"/>
                                                  <constant symbol="s_2"/>
                                                 </function>
                                                 <constant symbol="s_1"/>
                                                </function>
                                                <function symbol="∩">
                                                 <constant symbol="s_1"/>
                                                 <constant symbol="s_2"/>
                                                </function>
                                               </constantatomformula>
                                              </formulalist>
                                              <formulalist>
                                               <constantatomformula symbol="=">
                                                <function symbol="∩">
                                                 <function symbol="∪">
                                                  <constant symbol="s_1"/>
                                                  <function symbol="∩">
                                                   <constant symbol="s_1"/>
                                                   <constant symbol="s_2"/>
                                                  </function>
                                                 </function>
                                                 <constant symbol="s_1"/>
                                                </function>
                                                <function symbol="∪">
                                                 <constant symbol="s_1"/>
                                                 <function symbol="∩">
                                                  <constant symbol="s_1"/>
                                                  <constant symbol="s_2"/>
                                                 </function>
                                                </function>
                                               </constantatomformula>
                                               <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                                              </formulalist>
                                             </sequent>
                                             <rule type="contrl" symbol="c:l" param="2,1,1">
                                              <sequent>
                                               <formulalist>
                                                <constantatomformula symbol="L1" definition="L1"/>
                                                <constantatomformula symbol="=">
                                                 <function symbol="∩">
                                                  <constant symbol="s_1"/>
                                                  <constant symbol="s_1"/>
                                                 </function>
                                                 <constant symbol="s_1"/>
                                                </constantatomformula>
                                                <constantatomformula symbol="=">
                                                 <function symbol="∩">
                                                  <function symbol="∩">
                                                   <constant symbol="s_1"/>
                                                   <constant symbol="s_2"/>
                                                  </function>
                                                  <constant symbol="s_1"/>
                                                 </function>
                                                 <function symbol="∩">
                                                  <constant symbol="s_1"/>
                                                  <constant symbol="s_2"/>
                                                 </function>
                                                </constantatomformula>
                                               </formulalist>
                                               <formulalist>
                                                <constantatomformula symbol="=">
                                                 <function symbol="∩">
                                                  <function symbol="∪">
                                                   <constant symbol="s_1"/>
                                                   <function symbol="∩">
                                                    <constant symbol="s_1"/>
                                                    <constant symbol="s_2"/>
                                                   </function>
                                                  </function>
                                                  <constant symbol="s_1"/>
                                                 </function>
                                                 <function symbol="∪">
                                                  <constant symbol="s_1"/>
                                                  <function symbol="∩">
                                                   <constant symbol="s_1"/>
                                                   <constant symbol="s_2"/>
                                                  </function>
                                                 </function>
                                                </constantatomformula>
                                               </formulalist>
                                              </sequent>
                                              <rule type="weakl" symbol="w:l">
                                               <sequent>
                                                <formulalist>
                                                 <constantatomformula symbol="L1" definition="L1"/>
                                                 <constantatomformula symbol="L1" definition="L1"/>
                                                 <constantatomformula symbol="=">
                                                  <function symbol="∩">
                                                   <constant symbol="s_1"/>
                                                   <constant symbol="s_1"/>
                                                  </function>
                                                  <constant symbol="s_1"/>
                                                 </constantatomformula>
                                                 <constantatomformula symbol="=">
                                                  <function symbol="∩">
                                                   <function symbol="∩">
                                                    <constant symbol="s_1"/>
                                                    <constant symbol="s_2"/>
                                                   </function>
                                                   <constant symbol="s_1"/>
                                                  </function>
                                                  <function symbol="∩">
                                                   <constant symbol="s_1"/>
                                                   <constant symbol="s_2"/>
                                                  </function>
                                                 </constantatomformula>
                                                </formulalist>
                                                <formulalist>
                                                 <constantatomformula symbol="=">
                                                  <function symbol="∩">
                                                   <function symbol="∪">
                                                    <constant symbol="s_1"/>
                                                    <function symbol="∩">
                                                     <constant symbol="s_1"/>
                                                     <constant symbol="s_2"/>
                                                    </function>
                                                   </function>
                                                   <constant symbol="s_1"/>
                                                  </function>
                                                  <function symbol="∪">
                                                   <constant symbol="s_1"/>
                                                   <function symbol="∩">
                                                    <constant symbol="s_1"/>
                                                    <constant symbol="s_2"/>
                                                   </function>
                                                  </function>
                                                 </constantatomformula>
                                                </formulalist>
                                               </sequent>
                                               <rule type="defl" symbol="d:l" param="();[()];()">
                                                <sequent>
                                                 <formulalist>
                                                  <constantatomformula symbol="L1" definition="L1"/>
                                                  <constantatomformula symbol="=">
                                                   <function symbol="∩">
                                                    <constant symbol="s_1"/>
                                                    <constant symbol="s_1"/>
                                                   </function>
                                                   <constant symbol="s_1"/>
                                                  </constantatomformula>
                                                  <constantatomformula symbol="=">
                                                   <function symbol="∩">
                                                    <function symbol="∩">
                                                     <constant symbol="s_1"/>
                                                     <constant symbol="s_2"/>
                                                    </function>
                                                    <constant symbol="s_1"/>
                                                   </function>
                                                   <function symbol="∩">
                                                    <constant symbol="s_1"/>
                                                    <constant symbol="s_2"/>
                                                   </function>
                                                  </constantatomformula>
                                                 </formulalist>
                                                 <formulalist>
                                                  <constantatomformula symbol="=">
                                                   <function symbol="∩">
                                                    <function symbol="∪">
                                                     <constant symbol="s_1"/>
                                                     <function symbol="∩">
                                                      <constant symbol="s_1"/>
                                                      <constant symbol="s_2"/>
                                                     </function>
                                                    </function>
                                                    <constant symbol="s_1"/>
                                                   </function>
                                                   <function symbol="∪">
                                                    <constant symbol="s_1"/>
                                                    <function symbol="∩">
                                                     <constant symbol="s_1"/>
                                                     <constant symbol="s_2"/>
                                                    </function>
                                                   </function>
                                                  </constantatomformula>
                                                 </formulalist>
                                                </sequent>
                                                <rule type="contrl" symbol="c:l" param="2,1,1">
                                                 <sequent>
                                                  <formulalist>
                                                   <quantifiedformula type="all">
                                                    <variable symbol="x"/>
                                                    <quantifiedformula type="all">
                                                     <variable symbol="y"/>
                                                     <conjunctiveformula type="and">
                                                      <conjunctiveformula type="impl">
                                                       <constantatomformula symbol="=">
                                                        <function symbol="∩">
                                                         <variable symbol="x"/>
                                                         <variable symbol="y"/>
                                                        </function>
                                                        <variable symbol="x"/>
                                                       </constantatomformula>
                                                       <constantatomformula symbol="=">
                                                        <function symbol="∪">
                                                         <variable symbol="x"/>
                                                         <variable symbol="y"/>
                                                        </function>
                                                        <variable symbol="y"/>
                                                       </constantatomformula>
                                                      </conjunctiveformula>
                                                      <conjunctiveformula type="impl">
                                                       <constantatomformula symbol="=">
                                                        <function symbol="∪">
                                                         <variable symbol="x"/>
                                                         <variable symbol="y"/>
                                                        </function>
                                                        <variable symbol="y"/>
                                                       </constantatomformula>
                                                       <constantatomformula symbol="=">
                                                        <function symbol="∩">
                                                         <variable symbol="x"/>
                                                         <variable symbol="y"/>
                                                        </function>
                                                        <variable symbol="x"/>
                                                       </constantatomformula>
                                                      </conjunctiveformula>
                                                     </conjunctiveformula>
                                                    </quantifiedformula>
                                                   </quantifiedformula>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="∩">
                                                     <constant symbol="s_1"/>
                                                     <constant symbol="s_1"/>
                                                    </function>
                                                    <constant symbol="s_1"/>
                                                   </constantatomformula>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="∩">
                                                     <function symbol="∩">
                                                      <constant symbol="s_1"/>
                                                      <constant symbol="s_2"/>
                                                     </function>
                                                     <constant symbol="s_1"/>
                                                    </function>
                                                    <function symbol="∩">
                                                     <constant symbol="s_1"/>
                                                     <constant symbol="s_2"/>
                                                    </function>
                                                   </constantatomformula>
                                                  </formulalist>
                                                  <formulalist>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="∩">
                                                     <function symbol="∪">
                                                      <constant symbol="s_1"/>
                                                      <function symbol="∩">
                                                       <constant symbol="s_1"/>
                                                       <constant symbol="s_2"/>
                                                      </function>
                                                     </function>
                                                     <constant symbol="s_1"/>
                                                    </function>
                                                    <function symbol="∪">
                                                     <constant symbol="s_1"/>
                                                     <function symbol="∩">
                                                      <constant symbol="s_1"/>
                                                      <constant symbol="s_2"/>
                                                     </function>
                                                    </function>
                                                   </constantatomformula>
                                                  </formulalist>
                                                 </sequent>
                                                 <rule type="foralll" symbol="\all:l">
                                                  <sequent>
                                                   <formulalist>
                                                    <quantifiedformula type="all">
                                                     <variable symbol="x"/>
                                                     <quantifiedformula type="all">
                                                      <variable symbol="y"/>
                                                      <conjunctiveformula type="and">
                                                       <conjunctiveformula type="impl">
                                                        <constantatomformula symbol="=">
                                                         <function symbol="∩">
                                                          <variable symbol="x"/>
                                                          <variable symbol="y"/>
                                                         </function>
                                                         <variable symbol="x"/>
                                                        </constantatomformula>
                                                        <constantatomformula symbol="=">
                                                         <function symbol="∪">
                                                          <variable symbol="x"/>
                                                          <variable symbol="y"/>
                                                         </function>
                                                         <variable symbol="y"/>
                                                        </constantatomformula>
                                                       </conjunctiveformula>
                                                       <conjunctiveformula type="impl">
                                                        <constantatomformula symbol="=">
                                                         <function symbol="∪">
                                                          <variable symbol="x"/>
                                                          <variable symbol="y"/>
                                                         </function>
                                                         <variable symbol="y"/>
                                                        </constantatomformula>
                                                        <constantatomformula symbol="=">
                                                         <function symbol="∩">
                                                          <variable symbol="x"/>
                                                          <variable symbol="y"/>
                                                         </function>
                                                         <variable symbol="x"/>
                                                        </constantatomformula>
                                                       </conjunctiveformula>
                                                      </conjunctiveformula>
                                                     </quantifiedformula>
                                                    </quantifiedformula>
                                                    <quantifiedformula type="all">
                                                     <variable symbol="x"/>
                                                     <quantifiedformula type="all">
                                                      <variable symbol="y"/>
                                                      <conjunctiveformula type="and">
                                                       <conjunctiveformula type="impl">
                                                        <constantatomformula symbol="=">
                                                         <function symbol="∩">
                                                          <variable symbol="x"/>
                                                          <variable symbol="y"/>
                                                         </function>
                                                         <variable symbol="x"/>
                                                        </constantatomformula>
                                                        <constantatomformula symbol="=">
                                                         <function symbol="∪">
                                                          <variable symbol="x"/>
                                                          <variable symbol="y"/>
                                                         </function>
                                                         <variable symbol="y"/>
                                                        </constantatomformula>
                                                       </conjunctiveformula>
                                                       <conjunctiveformula type="impl">
                                                        <constantatomformula symbol="=">
                                                         <function symbol="∪">
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                                                       <constant symbol="s_2"/>
                                                      </function>
                                                     </function>
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                                                        <constantatomformula symbol="=">
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                                                         <constantatomformula symbol="=">
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                                                         <constantatomformula symbol="=">
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                                                         <constantatomformula symbol="=">
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                                                     <constantatomformula symbol="=">
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                                                     <constantatomformula symbol="=">
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                                                     <constantatomformula symbol="=">
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                                                        <constant symbol="s_1"/>
                                                        <constant symbol="s_2"/>
                                                       </function>
                                                      </function>
                                                     </constantatomformula>
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                                                   </sequent>
                                                   <rule type="andl1" symbol="\and:l1">
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                                                        <constantatomformula symbol="=">
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                                                        <constantatomformula symbol="=">
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                                                        <constantatomformula symbol="=">
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                                                          <constant symbol="s_1"/>
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                                                        <constantatomformula symbol="=">
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                                                          <constant symbol="s_1"/>
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                                                          <constantatomformula symbol="=">
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                                                          <constantatomformula symbol="=">
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                                                               <constantatomformula symbol="=">
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                                                               <constantatomformula symbol="=">
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                                                             </conjunctiveformula>
                                                            </quantifiedformula>
                                                           </quantifiedformula>
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                                                            <quantifiedformula type="all">
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                                                               <constantatomformula symbol="=">
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                                                               <constantatomformula symbol="=">
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                                                               <constantatomformula symbol="=">
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                                                           <constantatomformula symbol="=">
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                                                             <constant symbol="s_1"/>
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                                                           <constantatomformula symbol="=">
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                                                             <function symbol="∩">
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                                                              <constant symbol="s_2"/>
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                                                           <constantatomformula symbol="=">
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                                                              <constant symbol="s_2"/>
                                                             </function>
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                                                                <variable symbol="y"/>
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                                                                <variable symbol="y"/>
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                                                               <constantatomformula symbol="=">
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                                                            <quantifiedformula type="all">
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                                                                <constantatomformula symbol="=">
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                                                                 <variable symbol="x"/>
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                                                                <constantatomformula symbol="=">
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                                                               <conjunctiveformula type="impl">
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                                                                <constantatomformula symbol="=">
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                                                                 <variable symbol="x"/>
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                                                            <constantatomformula symbol="=">
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                                                            <constantatomformula symbol="=">
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                                                              <function symbol="∩">
                                                               <constant symbol="s_1"/>
                                                               <constant symbol="s_2"/>
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                                                             </function>
                                                             <function symbol="∩">
                                                              <constant symbol="s_1"/>
                                                              <constant symbol="s_2"/>
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                                                           <formulalist>
                                                            <constantatomformula symbol="=">
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                                                               <constant symbol="s_1"/>
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                                                                <constant symbol="s_2"/>
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                                                              </function>
                                                              <constant symbol="s_1"/>
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                                                             <function symbol="∪">
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                                                              <function symbol="∩">
                                                               <constant symbol="s_1"/>
                                                               <constant symbol="s_2"/>
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                                                             </function>
                                                            </constantatomformula>
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                                                          </sequent>
                                                          <rule type="andl1" symbol="\and:l1">
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                                                               <constantatomformula symbol="=">
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                                                                   <constant symbol="s_1"/>
                                                                   <constant symbol="s_2"/>
                                                                  </function>
                                                                 </function>
                                                                </constantatomformula>
                                                               </formulalist>
                                                              </sequent>
                                                              <rule type="foralll" symbol="\all:l">
                                                               <sequent>
                                                                <formulalist>
                                                                 <quantifiedformula type="all">
                                                                  <variable symbol="x"/>
                                                                  <quantifiedformula type="all">
                                                                   <variable symbol="y"/>
                                                                   <conjunctiveformula type="and">
                                                                    <conjunctiveformula type="impl">
                                                                     <constantatomformula symbol="=">
                                                                      <function symbol="∩">
                                                                       <variable symbol="x"/>
                                                                       <variable symbol="y"/>
                                                                      </function>
                                                                      <variable symbol="x"/>
                                                                     </constantatomformula>
                                                                     <constantatomformula symbol="=">
                                                                      <function symbol="∪">
                                                                       <variable symbol="x"/>
                                                                       <variable symbol="y"/>
                                                                      </function>
                                                                      <variable symbol="y"/>
                                                                     </constantatomformula>
                                                                    </conjunctiveformula>
                                                                    <conjunctiveformula type="impl">
                                                                     <constantatomformula symbol="=">
                                                                      <function symbol="∪">
                                                                       <variable symbol="x"/>
                                                                       <variable symbol="y"/>
                                                                      </function>
                                                                      <variable symbol="y"/>
                                                                     </constantatomformula>
                                                                     <constantatomformula symbol="=">
                                                                      <function symbol="∩">
                                                                       <variable symbol="x"/>
                                                                       <variable symbol="y"/>
                                                                      </function>
                                                                      <variable symbol="x"/>
                                                                     </constantatomformula>
                                                                    </conjunctiveformula>
                                                                   </conjunctiveformula>
                                                                  </quantifiedformula>
                                                                 </quantifiedformula>
                                                                 <constantatomformula symbol="=">
                                                                  <function symbol="∪">
                                                                   <function symbol="∩">
                                                                    <constant symbol="s_1"/>
                                                                    <constant symbol="s_2"/>
                                                                   </function>
                                                                   <constant symbol="s_1"/>
                                                                  </function>
                                                                  <constant symbol="s_1"/>
                                                                 </constantatomformula>
                                                                 <constantatomformula symbol="=">
                                                                  <function symbol="∪">
                                                                   <constant symbol="s_1"/>
                                                                   <constant symbol="s_1"/>
                                                                  </function>
                                                                  <constant symbol="s_1"/>
                                                                 </constantatomformula>
                                                                </formulalist>
                                                                <formulalist>
                                                                 <constantatomformula symbol="=">
                                                                  <function symbol="∩">
                                                                   <function symbol="∪">
                                                                    <constant symbol="s_1"/>
                                                                    <function symbol="∩">
                                                                     <constant symbol="s_1"/>
                                                                     <constant symbol="s_2"/>
                                                                    </function>
                                                                   </function>
                                                                   <constant symbol="s_1"/>
                                                                  </function>
                                                                  <function symbol="∪">
                                                                   <constant symbol="s_1"/>
                                                                   <function symbol="∩">
                                                                    <constant symbol="s_1"/>
                                                                    <constant symbol="s_2"/>
                                                                   </function>
                                                                  </function>
                                                                 </constantatomformula>
                                                                </formulalist>
                                                               </sequent>
                                                               <rule type="foralll" symbol="\all:l">
                                                                <sequent>
                                                                 <formulalist>
                                                                  <quantifiedformula type="all">
                                                                   <variable symbol="y"/>
                                                                   <conjunctiveformula type="and">
                                                                    <conjunctiveformula type="impl">
                                                                     <constantatomformula symbol="=">
                                                                      <function symbol="∩">
                                                                       <function symbol="∪">
                                                                        <constant symbol="s_1"/>
                                                                        <function symbol="∩">
                                                                         <constant symbol="s_1"/>
                                                                         <constant symbol="s_2"/>
                                                                        </function>
                                                                       </function>
                                                                       <variable symbol="y"/>
                                                                      </function>
                                                                      <function symbol="∪">
                                                                       <constant symbol="s_1"/>
                                                                       <function symbol="∩">
                                                                        <constant symbol="s_1"/>
                                                                        <constant symbol="s_2"/>
                                                                       </function>
                                                                      </function>
                                                                     </constantatomformula>
                                                                     <constantatomformula symbol="=">
                                                                      <function symbol="∪">
                                                                       <function symbol="∪">
                                                                        <constant symbol="s_1"/>
                                                                        <function symbol="∩">
                                                                         <constant symbol="s_1"/>
                                                                         <constant symbol="s_2"/>
                                                                        </function>
                                                                       </function>
                                                                       <variable symbol="y"/>
                                                                      </function>
                                                                      <variable symbol="y"/>
                                                                     </constantatomformula>
                                                                    </conjunctiveformula>
                                                                    <conjunctiveformula type="impl">
                                                                     <constantatomformula symbol="=">
                                                                      <function symbol="∪">
                                                                       <function symbol="∪">
                                                                        <constant symbol="s_1"/>
                                                                        <function symbol="∩">
                                                                         <constant symbol="s_1"/>
                                                                         <constant symbol="s_2"/>
                                                                        </function>
                                                                       </function>
                                                                       <variable symbol="y"/>
                                                                      </function>
                                                                      <variable symbol="y"/>
                                                                     </constantatomformula>
                                                                     <constantatomformula symbol="=">
                                                                      <function symbol="∩">
                                                                       <function symbol="∪">
                                                                        <constant symbol="s_1"/>
                                                                        <function symbol="∩">
                                                                         <constant symbol="s_1"/>
                                                                         <constant symbol="s_2"/>
                                                                        </function>
                                                                       </function>
                                                                       <variable symbol="y"/>
                                                                      </function>
                                                                      <function symbol="∪">
                                                                       <constant symbol="s_1"/>
                                                                       <function symbol="∩">
                                                                        <constant symbol="s_1"/>
                                                                        <constant symbol="s_2"/>
                                                                       </function>
                                                                      </function>
                                                                     </constantatomformula>
                                                                    </conjunctiveformula>
                                                                   </conjunctiveformula>
                                                                  </quantifiedformula>
                                                                  <constantatomformula symbol="=">
                                                                   <function symbol="∪">
                                                                    <function symbol="∩">
                                                                     <constant symbol="s_1"/>
                                                                     <constant symbol="s_2"/>
                                                                    </function>
                                                                    <constant symbol="s_1"/>
                                                                   </function>
                                                                   <constant symbol="s_1"/>
                                                                  </constantatomformula>
                                                                  <constantatomformula symbol="=">
                                                                   <function symbol="∪">
                                                                    <constant symbol="s_1"/>
                                                                    <constant symbol="s_1"/>
                                                                   </function>
                                                                   <constant symbol="s_1"/>
                                                                  </constantatomformula>
                                                                 </formulalist>
                                                                 <formulalist>
                                                                  <constantatomformula symbol="=">
                                                                   <function symbol="∩">
                                                                    <function symbol="∪">
                                                                     <constant symbol="s_1"/>
                                                                     <function symbol="∩">
                                                                      <constant symbol="s_1"/>
                                                                      <constant symbol="s_2"/>
                                                                     </function>
                                                                    </function>
                                                                    <constant symbol="s_1"/>
                                                                   </function>
                                                                   <function symbol="∪">
                                                                    <constant symbol="s_1"/>
                                                                    <function symbol="∩">
                                                                     <constant symbol="s_1"/>
                                                                     <constant symbol="s_2"/>
                                                                    </function>
                                                                   </function>
                                                                  </constantatomformula>
                                                                 </formulalist>
                                                                </sequent>
                                                                <rule type="andl2" symbol="\and:l2">
                                                                 <sequent>
                                                                  <formulalist>
                                                                   <conjunctiveformula type="and">
                                                                    <conjunctiveformula type="impl">
                                                                     <constantatomformula symbol="=">
                                                                      <function symbol="∩">
                                                                       <function symbol="∪">
                                                                        <constant symbol="s_1"/>
                                                                        <function symbol="∩">
                                                                         <constant symbol="s_1"/>
                                                                         <constant symbol="s_2"/>
                                                                        </function>
                                                                       </function>
                                                                       <constant symbol="s_1"/>
                                                                      </function>
                                                                      <function symbol="∪">
                                                                       <constant symbol="s_1"/>
                                                                       <function symbol="∩">
                                                                        <constant symbol="s_1"/>
                                                                        <constant symbol="s_2"/>
                                                                       </function>
                                                                      </function>
                                                                     </constantatomformula>
                                                                     <constantatomformula symbol="=">
                                                                      <function symbol="∪">
                                                                       <function symbol="∪">
                                                                        <constant symbol="s_1"/>
                                                                        <function symbol="∩">
                                                                         <constant symbol="s_1"/>
                                                                         <constant symbol="s_2"/>
                                                                        </function>
                                                                       </function>
                                                                       <constant symbol="s_1"/>
                                                                      </function>
                                                                      <constant symbol="s_1"/>
                                                                     </constantatomformula>
                                                                    </conjunctiveformula>
                                                                    <conjunctiveformula type="impl">
                                                                     <constantatomformula symbol="=">
                                                                      <function symbol="∪">
                                                                       <function symbol="∪">
                                                                        <constant symbol="s_1"/>
                                                                        <function symbol="∩">
                                                                         <constant symbol="s_1"/>
                                                                         <constant symbol="s_2"/>
                                                                        </function>
                                                                       </function>
                                                                       <constant symbol="s_1"/>
                                                                      </function>
                                                                      <constant symbol="s_1"/>
                                                                     </constantatomformula>
                                                                     <constantatomformula symbol="=">
                                                                      <function symbol="∩">
                                                                       <function symbol="∪">
                                                                        <constant symbol="s_1"/>
                                                                        <function symbol="∩">
                                                                         <constant symbol="s_1"/>
                                                                         <constant symbol="s_2"/>
                                                                        </function>
                                                                       </function>
                                                                       <constant symbol="s_1"/>
                                                                      </function>
                                                                      <function symbol="∪">
                                                                       <constant symbol="s_1"/>
                                                                       <function symbol="∩">
                                                                        <constant symbol="s_1"/>
                                                                        <constant symbol="s_2"/>
                                                                       </function>
                                                                      </function>
                                                                     </constantatomformula>
                                                                    </conjunctiveformula>
                                                                   </conjunctiveformula>
                                                                   <constantatomformula symbol="=">
                                                                    <function symbol="∪">
                                                                     <function symbol="∩">
                                                                      <constant symbol="s_1"/>
                                                                      <constant symbol="s_2"/>
                                                                     </function>
                                                                     <constant symbol="s_1"/>
                                                                    </function>
                                                                    <constant symbol="s_1"/>
                                                                   </constantatomformula>
                                                                   <constantatomformula symbol="=">
                                                                    <function symbol="∪">
                                                                     <constant symbol="s_1"/>
                                                                     <constant symbol="s_1"/>
                                                                    </function>
                                                                    <constant symbol="s_1"/>
                                                                   </constantatomformula>
                                                                  </formulalist>
                                                                  <formulalist>
                                                                   <constantatomformula symbol="=">
                                                                    <function symbol="∩">
                                                                     <function symbol="∪">
                                                                      <constant symbol="s_1"/>
                                                                      <function symbol="∩">
                                                                       <constant symbol="s_1"/>
                                                                       <constant symbol="s_2"/>
                                                                      </function>
                                                                     </function>
                                                                     <constant symbol="s_1"/>
                                                                    </function>
                                                                    <function symbol="∪">
                                                                     <constant symbol="s_1"/>
                                                                     <function symbol="∩">
                                                                      <constant symbol="s_1"/>
                                                                      <constant symbol="s_2"/>
                                                                     </function>
                                                                    </function>
                                                                   </constantatomformula>
                                                                  </formulalist>
                                                                 </sequent>
                                                                 <rule type="impll" symbol="\impl:l">
                                                                  <sequent>
                                                                   <formulalist>
                                                                    <conjunctiveformula type="impl">
                                                                     <constantatomformula symbol="=">
                                                                      <function symbol="∪">
                                                                       <function symbol="∪">
                                                                        <constant symbol="s_1"/>
                                                                        <function symbol="∩">
                                                                         <constant symbol="s_1"/>
                                                                         <constant symbol="s_2"/>
                                                                        </function>
                                                                       </function>
                                                                       <constant symbol="s_1"/>
                                                                      </function>
                                                                      <constant symbol="s_1"/>
                                                                     </constantatomformula>
                                                                     <constantatomformula symbol="=">
                                                                      <function symbol="∩">
                                                                       <function symbol="∪">
                                                                        <constant symbol="s_1"/>
                                                                        <function symbol="∩">
                                                                         <constant symbol="s_1"/>
                                                                         <constant symbol="s_2"/>
                                                                        </function>
                                                                       </function>
                                                                       <constant symbol="s_1"/>
                                                                      </function>
                                                                      <function symbol="∪">
                                                                       <constant symbol="s_1"/>
                                                                       <function symbol="∩">
                                                                        <constant symbol="s_1"/>
                                                                        <constant symbol="s_2"/>
                                                                       </function>
                                                                      </function>
                                                                     </constantatomformula>
                                                                    </conjunctiveformula>
                                                                    <constantatomformula symbol="=">
                                                                     <function symbol="∪">
                                                                      <function symbol="∩">
                                                                       <constant symbol="s_1"/>
                                                                       <constant symbol="s_2"/>
                                                                      </function>
                                                                      <constant symbol="s_1"/>
                                                                     </function>
                                                                     <constant symbol="s_1"/>
                                                                    </constantatomformula>
                                                                    <constantatomformula symbol="=">
                                                                     <function symbol="∪">
                                                                      <constant symbol="s_1"/>
                                                                      <constant symbol="s_1"/>
                                                                     </function>
                                                                     <constant symbol="s_1"/>
                                                                    </constantatomformula>
                                                                   </formulalist>
                                                                   <formulalist>
                                                                    <constantatomformula symbol="=">
                                                                     <function symbol="∩">
                                                                      <function symbol="∪">
                                                                       <constant symbol="s_1"/>
                                                                       <function symbol="∩">
                                                                        <constant symbol="s_1"/>
                                                                        <constant symbol="s_2"/>
                                                                       </function>
                                                                      </function>
                                                                      <constant symbol="s_1"/>
                                                                     </function>
                                                                     <function symbol="∪">
                                                                      <constant symbol="s_1"/>
                                                                      <function symbol="∩">
                                                                       <constant symbol="s_1"/>
                                                                       <constant symbol="s_2"/>
                                                                      </function>
                                                                     </function>
                                                                    </constantatomformula>
                                                                   </formulalist>
                                                                  </sequent>
                                                                  <rule type="eqr2" symbol="=:r2" param="(1)">
                                                                   <sequent>
                                                                    <formulalist>
                                                                     <constantatomformula symbol="=">
                                                                      <function symbol="∪">
                                                                       <function symbol="∩">
                                                                        <constant symbol="s_1"/>
                                                                        <constant symbol="s_2"/>
                                                                       </function>
                                                                       <constant symbol="s_1"/>
                                                                      </function>
                                                                      <constant symbol="s_1"/>
                                                                     </constantatomformula>
                                                                     <constantatomformula symbol="=">
                                                                      <function symbol="∪">
                                                                       <constant symbol="s_1"/>
                                                                       <constant symbol="s_1"/>
                                                                      </function>
                                                                      <constant symbol="s_1"/>
                                                                     </constantatomformula>
                                                                    </formulalist>
                                                                    <formulalist>
                                                                     <constantatomformula symbol="=">
                                                                      <function symbol="∪">
                                                                       <function symbol="∪">
                                                                        <constant symbol="s_1"/>
                                                                        <function symbol="∩">
                                                                         <constant symbol="s_1"/>
                                                                         <constant symbol="s_2"/>
                                                                        </function>
                                                                       </function>
                                                                       <constant symbol="s_1"/>
                                                                      </function>
                                                                      <constant symbol="s_1"/>
                                                                     </constantatomformula>
                                                                    </formulalist>
                                                                   </sequent>
                                                                   <rule type="axiom" symbol="axiom">
                                                                    <sequent>
                                                                     <formulalist/>
                                                                     <formulalist>
                                                                      <constantatomformula symbol="=">
                                                                       <function symbol="∪">
                                                                        <function symbol="∪">
                                                                         <constant symbol="s_1"/>
                                                                         <function symbol="∩">
                                                                          <constant symbol="s_1"/>
                                                                          <constant symbol="s_2"/>
                                                                         </function>
                                                                        </function>
                                                                        <constant symbol="s_1"/>
                                                                       </function>
                                                                       <function symbol="∪">
                                                                        <constant symbol="s_1"/>
                                                                        <function symbol="∪">
                                                                         <function symbol="∩">
                                                                          <constant symbol="s_1"/>
                                                                          <constant symbol="s_2"/>
                                                                         </function>
                                                                         <constant symbol="s_1"/>
                                                                        </function>
                                                                       </function>
                                                                      </constantatomformula>
                                                                     </formulalist>
                                                                    </sequent>
                                                                   </rule>
                                                                   <rule type="eqr2" symbol="=:r2" param="(1 2)">
                                                                    <sequent>
                                                                     <formulalist>
                                                                      <constantatomformula symbol="=">
                                                                       <function symbol="∪">
                                                                        <function symbol="∩">
                                                                         <constant symbol="s_1"/>
                                                                         <constant symbol="s_2"/>
                                                                        </function>
                                                                        <constant symbol="s_1"/>
                                                                       </function>
                                                                       <constant symbol="s_1"/>
                                                                      </constantatomformula>
                                                                      <constantatomformula symbol="=">
                                                                       <function symbol="∪">
                                                                        <constant symbol="s_1"/>
                                                                        <constant symbol="s_1"/>
                                                                       </function>
                                                                       <constant symbol="s_1"/>
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                                                                     </formulalist>
                                                                     <formulalist>
                                                                      <constantatomformula symbol="=">
                                                                       <function symbol="∪">
                                                                        <constant symbol="s_1"/>
                                                                        <function symbol="∪">
                                                                         <function symbol="∩">
                                                                          <constant symbol="s_1"/>
                                                                          <constant symbol="s_2"/>
                                                                         </function>
                                                                         <constant symbol="s_1"/>
                                                                        </function>
                                                                       </function>
                                                                       <constant symbol="s_1"/>
                                                                      </constantatomformula>
                                                                     </formulalist>
                                                                    </sequent>
                                                                    <rule type="axiom" symbol="axiom">
                                                                     <sequent>
                                                                      <formulalist>
                                                                       <constantatomformula symbol="=">
                                                                        <function symbol="∪">
                                                                         <function symbol="∩">
                                                                          <constant symbol="s_1"/>
                                                                          <constant symbol="s_2"/>
                                                                         </function>
                                                                         <constant symbol="s_1"/>
                                                                        </function>
                                                                        <constant symbol="s_1"/>
                                                                       </constantatomformula>
                                                                      </formulalist>
                                                                      <formulalist>
                                                                       <constantatomformula symbol="=">
                                                                        <function symbol="∪">
                                                                         <function symbol="∩">
                                                                          <constant symbol="s_1"/>
                                                                          <constant symbol="s_2"/>
                                                                         </function>
                                                                         <constant symbol="s_1"/>
                                                                        </function>
                                                                        <constant symbol="s_1"/>
                                                                       </constantatomformula>
                                                                      </formulalist>
                                                                     </sequent>
                                                                    </rule>
                                                                    <rule type="contrr" symbol="c:r" param="2">
                                                                     <sequent>
                                                                      <formulalist>
                                                                       <constantatomformula symbol="=">
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                                                                         <constant symbol="s_1"/>
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                                                                        <constant symbol="s_1"/>
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                                                                      </formulalist>
                                                                      <formulalist>
                                                                       <constantatomformula symbol="=">
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                                                                         <constant symbol="s_1"/>
                                                                         <constant symbol="s_1"/>
                                                                        </function>
                                                                        <constant symbol="s_1"/>
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                                                                      </formulalist>
                                                                     </sequent>
                                                                     <rule type="contrl" symbol="c:l" param="2">
                                                                      <sequent>
                                                                       <formulalist>
                                                                        <constantatomformula symbol="=">
                                                                         <function symbol="∪">
                                                                          <constant symbol="s_1"/>
                                                                          <constant symbol="s_1"/>
                                                                         </function>
                                                                         <constant symbol="s_1"/>
                                                                        </constantatomformula>
                                                                       </formulalist>
                                                                       <formulalist>
                                                                        <constantatomformula symbol="=">
                                                                         <function symbol="∪">
                                                                          <constant symbol="s_1"/>
                                                                          <constant symbol="s_1"/>
                                                                         </function>
                                                                         <constant symbol="s_1"/>
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                                                                        <constantatomformula symbol="=">
                                                                         <function symbol="∪">
                                                                          <constant symbol="s_1"/>
                                                                          <constant symbol="s_1"/>
                                                                         </function>
                                                                         <constant symbol="s_1"/>
                                                                        </constantatomformula>
                                                                       </formulalist>
                                                                      </sequent>
                                                                      <rule type="eqr2" symbol="=:r2" param="(1)">
                                                                       <sequent>
                                                                        <formulalist>
                                                                         <constantatomformula symbol="=">
                                                                          <function symbol="∪">
                                                                           <constant symbol="s_1"/>
                                                                           <constant symbol="s_1"/>
                                                                          </function>
                                                                          <constant symbol="s_1"/>
                                                                         </constantatomformula>
                                                                         <constantatomformula symbol="=">
                                                                          <function symbol="∪">
                                                                           <constant symbol="s_1"/>
                                                                           <constant symbol="s_1"/>
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                                                                          <constant symbol="s_1"/>
                                                                         </constantatomformula>
                                                                        </formulalist>
                                                                        <formulalist>
                                                                         <constantatomformula symbol="=">
                                                                          <function symbol="∪">
                                                                           <constant symbol="s_1"/>
                                                                           <constant symbol="s_1"/>
                                                                          </function>
                                                                          <constant symbol="s_1"/>
                                                                         </constantatomformula>
                                                                         <constantatomformula symbol="=">
                                                                          <function symbol="∪">
                                                                           <constant symbol="s_1"/>
                                                                           <constant symbol="s_1"/>
                                                                          </function>
                                                                          <constant symbol="s_1"/>
                                                                         </constantatomformula>
                                                                        </formulalist>
                                                                       </sequent>
                                                                       <rule type="axiom" symbol="axiom">
                                                                        <sequent>
                                                                         <formulalist>
                                                                          <constantatomformula symbol="=">
                                                                           <function symbol="∪">
                                                                            <constant symbol="s_1"/>
                                                                            <constant symbol="s_1"/>
                                                                           </function>
                                                                           <constant symbol="s_1"/>
                                                                          </constantatomformula>
                                                                         </formulalist>
                                                                         <formulalist>
                                                                          <constantatomformula symbol="=">
                                                                           <function symbol="∪">
                                                                            <constant symbol="s_1"/>
                                                                            <constant symbol="s_1"/>
                                                                           </function>
                                                                           <constant symbol="s_1"/>
                                                                          </constantatomformula>
                                                                         </formulalist>
                                                                        </sequent>
                                                                       </rule>
                                                                       <rule type="weakr" symbol="w:r">
                                                                        <sequent>
                                                                         <formulalist>
                                                                          <constantatomformula symbol="=">
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                                                                            <constant symbol="s_1"/>
                                                                            <constant symbol="s_1"/>
                                                                           </function>
                                                                           <constant symbol="s_1"/>
                                                                          </constantatomformula>
                                                                         </formulalist>
                                                                         <formulalist>
                                                                          <constantatomformula symbol="=">
                                                                           <function symbol="∪">
                                                                            <constant symbol="s_1"/>
                                                                            <constant symbol="s_1"/>
                                                                           </function>
                                                                           <constant symbol="s_1"/>
                                                                          </constantatomformula>
                                                                          <constantatomformula symbol="=">
                                                                           <constant symbol="s_1"/>
                                                                           <constant symbol="s_1"/>
                                                                          </constantatomformula>
                                                                         </formulalist>
                                                                        </sequent>
                                                                        <rule type="axiom" symbol="axiom">
                                                                         <sequent>
                                                                          <formulalist>
                                                                           <constantatomformula symbol="=">
                                                                            <function symbol="∪">
                                                                             <constant symbol="s_1"/>
                                                                             <constant symbol="s_1"/>
                                                                            </function>
                                                                            <constant symbol="s_1"/>
                                                                           </constantatomformula>
                                                                          </formulalist>
                                                                          <formulalist>
                                                                           <constantatomformula symbol="=">
                                                                            <function symbol="∪">
                                                                             <constant symbol="s_1"/>
                                                                             <constant symbol="s_1"/>
                                                                            </function>
                                                                            <constant symbol="s_1"/>
                                                                           </constantatomformula>
                                                                          </formulalist>
                                                                         </sequent>
                                                                        </rule>
                                                                       </rule>
                                                                      </rule>
                                                                     </rule>
                                                                    </rule>
                                                                   </rule>
                                                                  </rule>
                                                                  <rule type="axiom" symbol="axiom">
                                                                   <sequent>
                                                                    <formulalist>
                                                                     <constantatomformula symbol="=">
                                                                      <function symbol="∩">
                                                                       <function symbol="∪">
                                                                        <constant symbol="s_1"/>
                                                                        <function symbol="∩">
                                                                         <constant symbol="s_1"/>
                                                                         <constant symbol="s_2"/>
                                                                        </function>
                                                                       </function>
                                                                       <constant symbol="s_1"/>
                                                                      </function>
                                                                      <function symbol="∪">
                                                                       <constant symbol="s_1"/>
                                                                       <function symbol="∩">
                                                                        <constant symbol="s_1"/>
                                                                        <constant symbol="s_2"/>
                                                                       </function>
                                                                      </function>
                                                                     </constantatomformula>
                                                                    </formulalist>
                                                                    <formulalist>
                                                                     <constantatomformula symbol="=">
                                                                      <function symbol="∩">
                                                                       <function symbol="∪">
                                                                        <constant symbol="s_1"/>
                                                                        <function symbol="∩">
                                                                         <constant symbol="s_1"/>
                                                                         <constant symbol="s_2"/>
                                                                        </function>
                                                                       </function>
                                                                       <constant symbol="s_1"/>
                                                                      </function>
                                                                      <function symbol="∪">
                                                                       <constant symbol="s_1"/>
                                                                       <function symbol="∩">
                                                                        <constant symbol="s_1"/>
                                                                        <constant symbol="s_2"/>
                                                                       </function>
                                                                      </function>
                                                                     </constantatomformula>
                                                                    </formulalist>
                                                                   </sequent>
                                                                  </rule>
                                                                 </rule>
                                                                </rule>
                                                               </rule>
                                                              </rule>
                                                             </rule>
                                                            </rule>
                                                           </rule>
                                                          </rule>
                                                         </rule>
                                                        </rule>
                                                       </rule>
                                                      </rule>
                                                     </rule>
                                                    </rule>
                                                   </rule>
                                                  </rule>
                                                 </rule>
                                                </rule>
                                               </rule>
                                              </rule>
                                             </rule>
                                            </rule>
                                           </rule>
                                          </rule>
                                         </rule>
                                        </rule>
                                       </rule>
                                      </rule>
                                     </rule>
                                    </rule>
                                   </rule>
                                  </rule>
                                 </rule>
                                </rule>
                               </rule>
                              </rule>
                             </rule>
                            </rule>
                           </rule>
                           <rule type="permr" symbol="\pi:r" param="(1 2)">
                            <sequent>
                             <formulalist>
                              <constantatomformula symbol="L1" definition="L1"/>
                             </formulalist>
                             <formulalist>
                              <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                              <constantatomformula symbol="=">
                               <constant symbol="s_1"/>
                               <constant symbol="s_1"/>
                              </constantatomformula>
                             </formulalist>
                            </sequent>
                            <rule type="weakr" symbol="w:r">
                             <sequent>
                              <formulalist>
                               <constantatomformula symbol="L1" definition="L1"/>
                              </formulalist>
                              <formulalist>
                               <constantatomformula symbol="=">
                                <constant symbol="s_1"/>
                                <constant symbol="s_1"/>
                               </constantatomformula>
                               <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                              </formulalist>
                             </sequent>
                             <rule type="perml" symbol="\pi:l" param="">
                              <sequent>
                               <formulalist>
                                <constantatomformula symbol="L1" definition="L1"/>
                               </formulalist>
                               <formulalist>
                                <constantatomformula symbol="=">
                                 <constant symbol="s_1"/>
                                 <constant symbol="s_1"/>
                                </constantatomformula>
                               </formulalist>
                              </sequent>
                              <rule type="weakl" symbol="w:l">
                               <sequent>
                                <formulalist>
                                 <constantatomformula symbol="L1" definition="L1"/>
                                </formulalist>
                                <formulalist>
                                 <constantatomformula symbol="=">
                                  <constant symbol="s_1"/>
                                  <constant symbol="s_1"/>
                                 </constantatomformula>
                                </formulalist>
                               </sequent>
                               <rule type="axiom" symbol="axiom">
                                <sequent>
                                 <formulalist/>
                                 <formulalist>
                                  <constantatomformula symbol="=">
                                   <constant symbol="s_1"/>
                                   <constant symbol="s_1"/>
                                  </constantatomformula>
                                 </formulalist>
                                </sequent>
                               </rule>
                              </rule>
                             </rule>
                            </rule>
                           </rule>
                          </rule>
                         </rule>
                        </rule>
                       </rule>
                      </rule>
                      <rule type="contrr" symbol="c:r" param="2">
                       <sequent>
                        <formulalist>
                         <constantatomformula symbol="=">
                          <function symbol="∪">
                           <constant symbol="s_1"/>
                           <function symbol="∩">
                            <constant symbol="s_1"/>
                            <constant symbol="s_2"/>
                           </function>
                          </function>
                          <constant symbol="s_1"/>
                         </constantatomformula>
                         <constantatomformula symbol="L1" definition="L1"/>
                        </formulalist>
                        <formulalist>
                         <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                        </formulalist>
                       </sequent>
                       <rule type="permr" symbol="\pi:r" param="">
                        <sequent>
                         <formulalist>
                          <constantatomformula symbol="=">
                           <function symbol="∪">
                            <constant symbol="s_1"/>
                            <function symbol="∩">
                             <constant symbol="s_1"/>
                             <constant symbol="s_2"/>
                            </function>
                           </function>
                           <constant symbol="s_1"/>
                          </constantatomformula>
                          <constantatomformula symbol="L1" definition="L1"/>
                         </formulalist>
                         <formulalist>
                          <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                          <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                         </formulalist>
                        </sequent>
                        <rule type="contrl" symbol="c:l" param="1,2">
                         <sequent>
                          <formulalist>
                           <constantatomformula symbol="=">
                            <function symbol="∪">
                             <constant symbol="s_1"/>
                             <function symbol="∩">
                              <constant symbol="s_1"/>
                              <constant symbol="s_2"/>
                             </function>
                            </function>
                            <constant symbol="s_1"/>
                           </constantatomformula>
                           <constantatomformula symbol="L1" definition="L1"/>
                          </formulalist>
                          <formulalist>
                           <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                           <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                          </formulalist>
                         </sequent>
                         <rule type="perml" symbol="\pi:l" param="(1 2)">
                          <sequent>
                           <formulalist>
                            <constantatomformula symbol="=">
                             <function symbol="∪">
                              <constant symbol="s_1"/>
                              <function symbol="∩">
                               <constant symbol="s_1"/>
                               <constant symbol="s_2"/>
                              </function>
                             </function>
                             <constant symbol="s_1"/>
                            </constantatomformula>
                            <constantatomformula symbol="L1" definition="L1"/>
                            <constantatomformula symbol="L1" definition="L1"/>
                           </formulalist>
                           <formulalist>
                            <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                            <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                           </formulalist>
                          </sequent>
                          <rule type="cut" symbol="cut">
                           <sequent>
                            <formulalist>
                             <constantatomformula symbol="L1" definition="L1"/>
                             <constantatomformula symbol="=">
                              <function symbol="∪">
                               <constant symbol="s_1"/>
                               <function symbol="∩">
                                <constant symbol="s_1"/>
                                <constant symbol="s_2"/>
                               </function>
                              </function>
                              <constant symbol="s_1"/>
                             </constantatomformula>
                             <constantatomformula symbol="L1" definition="L1"/>
                            </formulalist>
                            <formulalist>
                             <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                             <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                            </formulalist>
                           </sequent>
                           <rule type="permr" symbol="\pi:r" param="(1 2)">
                            <sequent>
                             <formulalist>
                              <constantatomformula symbol="L1" definition="L1"/>
                             </formulalist>
                             <formulalist>
                              <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                              <constantatomformula symbol="=">
                               <constant symbol="s_3"/>
                               <constant symbol="s_3"/>
                              </constantatomformula>
                             </formulalist>
                            </sequent>
                            <rule type="weakr" symbol="w:r">
                             <sequent>
                              <formulalist>
                               <constantatomformula symbol="L1" definition="L1"/>
                              </formulalist>
                              <formulalist>
                               <constantatomformula symbol="=">
                                <constant symbol="s_3"/>
                                <constant symbol="s_3"/>
                               </constantatomformula>
                               <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                              </formulalist>
                             </sequent>
                             <rule type="perml" symbol="\pi:l" param="">
                              <sequent>
                               <formulalist>
                                <constantatomformula symbol="L1" definition="L1"/>
                               </formulalist>
                               <formulalist>
                                <constantatomformula symbol="=">
                                 <constant symbol="s_3"/>
                                 <constant symbol="s_3"/>
                                </constantatomformula>
                               </formulalist>
                              </sequent>
                              <rule type="weakl" symbol="w:l">
                               <sequent>
                                <formulalist>
                                 <constantatomformula symbol="L1" definition="L1"/>
                                </formulalist>
                                <formulalist>
                                 <constantatomformula symbol="=">
                                  <constant symbol="s_3"/>
                                  <constant symbol="s_3"/>
                                 </constantatomformula>
                                </formulalist>
                               </sequent>
                               <rule type="axiom" symbol="axiom">
                                <sequent>
                                 <formulalist/>
                                 <formulalist>
                                  <constantatomformula symbol="=">
                                   <constant symbol="s_3"/>
                                   <constant symbol="s_3"/>
                                  </constantatomformula>
                                 </formulalist>
                                </sequent>
                               </rule>
                              </rule>
                             </rule>
                            </rule>
                           </rule>
                           <rule type="perml" symbol="\pi:l" param="(1 2)">
                            <sequent>
                             <formulalist>
                              <constantatomformula symbol="=">
                               <constant symbol="s_3"/>
                               <constant symbol="s_3"/>
                              </constantatomformula>
                              <constantatomformula symbol="=">
                               <function symbol="∪">
                                <constant symbol="s_1"/>
                                <function symbol="∩">
                                 <constant symbol="s_1"/>
                                 <constant symbol="s_2"/>
                                </function>
                               </function>
                               <constant symbol="s_1"/>
                              </constantatomformula>
                              <constantatomformula symbol="L1" definition="L1"/>
                             </formulalist>
                             <formulalist>
                              <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                             </formulalist>
                            </sequent>
                            <rule type="perml" symbol="\pi:l" param="(1 2)">
                             <sequent>
                              <formulalist>
                               <constantatomformula symbol="=">
                                <function symbol="∪">
                                 <constant symbol="s_1"/>
                                 <function symbol="∩">
                                  <constant symbol="s_1"/>
                                  <constant symbol="s_2"/>
                                 </function>
                                </function>
                                <constant symbol="s_1"/>
                               </constantatomformula>
                               <constantatomformula symbol="=">
                                <constant symbol="s_3"/>
                                <constant symbol="s_3"/>
                               </constantatomformula>
                               <constantatomformula symbol="L1" definition="L1"/>
                              </formulalist>
                              <formulalist>
                               <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                              </formulalist>
                             </sequent>
                             <rule type="contrr" symbol="c:r" param="2">
                              <sequent>
                               <formulalist>
                                <constantatomformula symbol="=">
                                 <constant symbol="s_3"/>
                                 <constant symbol="s_3"/>
                                </constantatomformula>
                                <constantatomformula symbol="=">
                                 <function symbol="∪">
                                  <constant symbol="s_1"/>
                                  <function symbol="∩">
                                   <constant symbol="s_1"/>
                                   <constant symbol="s_2"/>
                                  </function>
                                 </function>
                                 <constant symbol="s_1"/>
                                </constantatomformula>
                                <constantatomformula symbol="L1" definition="L1"/>
                               </formulalist>
                               <formulalist>
                                <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                               </formulalist>
                              </sequent>
                              <rule type="permr" symbol="\pi:r" param="">
                               <sequent>
                                <formulalist>
                                 <constantatomformula symbol="=">
                                  <constant symbol="s_3"/>
                                  <constant symbol="s_3"/>
                                 </constantatomformula>
                                 <constantatomformula symbol="=">
                                  <function symbol="∪">
                                   <constant symbol="s_1"/>
                                   <function symbol="∩">
                                    <constant symbol="s_1"/>
                                    <constant symbol="s_2"/>
                                   </function>
                                  </function>
                                  <constant symbol="s_1"/>
                                 </constantatomformula>
                                 <constantatomformula symbol="L1" definition="L1"/>
                                </formulalist>
                                <formulalist>
                                 <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                                 <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                                </formulalist>
                               </sequent>
                               <rule type="contrl" symbol="c:l" param="1,1,2">
                                <sequent>
                                 <formulalist>
                                  <constantatomformula symbol="=">
                                   <constant symbol="s_3"/>
                                   <constant symbol="s_3"/>
                                  </constantatomformula>
                                  <constantatomformula symbol="=">
                                   <function symbol="∪">
                                    <constant symbol="s_1"/>
                                    <function symbol="∩">
                                     <constant symbol="s_1"/>
                                     <constant symbol="s_2"/>
                                    </function>
                                   </function>
                                   <constant symbol="s_1"/>
                                  </constantatomformula>
                                  <constantatomformula symbol="L1" definition="L1"/>
                                 </formulalist>
                                 <formulalist>
                                  <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                                  <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                                 </formulalist>
                                </sequent>
                                <rule type="perml" symbol="\pi:l" param="(2 3)">
                                 <sequent>
                                  <formulalist>
                                   <constantatomformula symbol="=">
                                    <constant symbol="s_3"/>
                                    <constant symbol="s_3"/>
                                   </constantatomformula>
                                   <constantatomformula symbol="=">
                                    <function symbol="∪">
                                     <constant symbol="s_1"/>
                                     <function symbol="∩">
                                      <constant symbol="s_1"/>
                                      <constant symbol="s_2"/>
                                     </function>
                                    </function>
                                    <constant symbol="s_1"/>
                                   </constantatomformula>
                                   <constantatomformula symbol="L1" definition="L1"/>
                                   <constantatomformula symbol="L1" definition="L1"/>
                                  </formulalist>
                                  <formulalist>
                                   <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                                   <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                                  </formulalist>
                                 </sequent>
                                 <rule type="eql1" symbol="=:l1" param="(1)">
                                  <sequent>
                                   <formulalist>
                                    <constantatomformula symbol="=">
                                     <constant symbol="s_3"/>
                                     <constant symbol="s_3"/>
                                    </constantatomformula>
                                    <constantatomformula symbol="L1" definition="L1"/>
                                    <constantatomformula symbol="=">
                                     <function symbol="∪">
                                      <constant symbol="s_1"/>
                                      <function symbol="∩">
                                       <constant symbol="s_1"/>
                                       <constant symbol="s_2"/>
                                      </function>
                                     </function>
                                     <constant symbol="s_1"/>
                                    </constantatomformula>
                                    <constantatomformula symbol="L1" definition="L1"/>
                                   </formulalist>
                                   <formulalist>
                                    <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                                    <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                                   </formulalist>
                                  </sequent>
                                  <rule type="permr" symbol="\pi:r" param="(1 2)">
                                   <sequent>
                                    <formulalist>
                                     <constantatomformula symbol="L1" definition="L1"/>
                                    </formulalist>
                                    <formulalist>
                                     <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                                     <constantatomformula symbol="=">
                                      <function symbol="∩">
                                       <constant symbol="s_3"/>
                                       <function symbol="∪">
                                        <constant symbol="s_3"/>
                                        <constant symbol="s_4"/>
                                       </function>
                                      </function>
                                      <constant symbol="s_3"/>
                                     </constantatomformula>
                                    </formulalist>
                                   </sequent>
                                   <rule type="perml" symbol="\pi:l" param="">
                                    <sequent>
                                     <formulalist>
                                      <constantatomformula symbol="L1" definition="L1"/>
                                     </formulalist>
                                     <formulalist>
                                      <constantatomformula symbol="=">
                                       <function symbol="∩">
                                        <constant symbol="s_3"/>
                                        <function symbol="∪">
                                         <constant symbol="s_3"/>
                                         <constant symbol="s_4"/>
                                        </function>
                                       </function>
                                       <constant symbol="s_3"/>
                                      </constantatomformula>
                                      <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                                     </formulalist>
                                    </sequent>
                                    <rule type="weakr" symbol="w:r">
                                     <sequent>
                                      <formulalist>
                                       <constantatomformula symbol="L1" definition="L1"/>
                                      </formulalist>
                                      <formulalist>
                                       <constantatomformula symbol="=">
                                        <function symbol="∩">
                                         <constant symbol="s_3"/>
                                         <function symbol="∪">
                                          <constant symbol="s_3"/>
                                          <constant symbol="s_4"/>
                                         </function>
                                        </function>
                                        <constant symbol="s_3"/>
                                       </constantatomformula>
                                       <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                                      </formulalist>
                                     </sequent>
                                     <rule type="contrl" symbol="c:l" param="2">
                                      <sequent>
                                       <formulalist>
                                        <constantatomformula symbol="L1" definition="L1"/>
                                       </formulalist>
                                       <formulalist>
                                        <constantatomformula symbol="=">
                                         <function symbol="∩">
                                          <constant symbol="s_3"/>
                                          <function symbol="∪">
                                           <constant symbol="s_3"/>
                                           <constant symbol="s_4"/>
                                          </function>
                                         </function>
                                         <constant symbol="s_3"/>
                                        </constantatomformula>
                                       </formulalist>
                                      </sequent>
                                      <rule type="perml" symbol="\pi:l" param="(1 2)">
                                       <sequent>
                                        <formulalist>
                                         <constantatomformula symbol="L1" definition="L1"/>
                                         <constantatomformula symbol="L1" definition="L1"/>
                                        </formulalist>
                                        <formulalist>
                                         <constantatomformula symbol="=">
                                          <function symbol="∩">
                                           <constant symbol="s_3"/>
                                           <function symbol="∪">
                                            <constant symbol="s_3"/>
                                            <constant symbol="s_4"/>
                                           </function>
                                          </function>
                                          <constant symbol="s_3"/>
                                         </constantatomformula>
                                        </formulalist>
                                       </sequent>
                                       <rule type="weakl" symbol="w:l">
                                        <sequent>
                                         <formulalist>
                                          <constantatomformula symbol="L1" definition="L1"/>
                                          <constantatomformula symbol="L1" definition="L1"/>
                                         </formulalist>
                                         <formulalist>
                                          <constantatomformula symbol="=">
                                           <function symbol="∩">
                                            <constant symbol="s_3"/>
                                            <function symbol="∪">
                                             <constant symbol="s_3"/>
                                             <constant symbol="s_4"/>
                                            </function>
                                           </function>
                                           <constant symbol="s_3"/>
                                          </constantatomformula>
                                         </formulalist>
                                        </sequent>
                                        <rule type="contrl" symbol="c:l" param="2">
                                         <sequent>
                                          <formulalist>
                                           <constantatomformula symbol="L1" definition="L1"/>
                                          </formulalist>
                                          <formulalist>
                                           <constantatomformula symbol="=">
                                            <function symbol="∩">
                                             <constant symbol="s_3"/>
                                             <function symbol="∪">
                                              <constant symbol="s_3"/>
                                              <constant symbol="s_4"/>
                                             </function>
                                            </function>
                                            <constant symbol="s_3"/>
                                           </constantatomformula>
                                          </formulalist>
                                         </sequent>
                                         <rule type="perml" symbol="\pi:l" param="(1 2)">
                                          <sequent>
                                           <formulalist>
                                            <constantatomformula symbol="L1" definition="L1"/>
                                            <constantatomformula symbol="L1" definition="L1"/>
                                           </formulalist>
                                           <formulalist>
                                            <constantatomformula symbol="=">
                                             <function symbol="∩">
                                              <constant symbol="s_3"/>
                                              <function symbol="∪">
                                               <constant symbol="s_3"/>
                                               <constant symbol="s_4"/>
                                              </function>
                                             </function>
                                             <constant symbol="s_3"/>
                                            </constantatomformula>
                                           </formulalist>
                                          </sequent>
                                          <rule type="weakl" symbol="w:l">
                                           <sequent>
                                            <formulalist>
                                             <constantatomformula symbol="L1" definition="L1"/>
                                             <constantatomformula symbol="L1" definition="L1"/>
                                            </formulalist>
                                            <formulalist>
                                             <constantatomformula symbol="=">
                                              <function symbol="∩">
                                               <constant symbol="s_3"/>
                                               <function symbol="∪">
                                                <constant symbol="s_3"/>
                                                <constant symbol="s_4"/>
                                               </function>
                                              </function>
                                              <constant symbol="s_3"/>
                                             </constantatomformula>
                                            </formulalist>
                                           </sequent>
                                           <rule type="defl" symbol="d:l" param="();[()];()">
                                            <sequent>
                                             <formulalist>
                                              <constantatomformula symbol="L1" definition="L1"/>
                                             </formulalist>
                                             <formulalist>
                                              <constantatomformula symbol="=">
                                               <function symbol="∩">
                                                <constant symbol="s_3"/>
                                                <function symbol="∪">
                                                 <constant symbol="s_3"/>
                                                 <constant symbol="s_4"/>
                                                </function>
                                               </function>
                                               <constant symbol="s_3"/>
                                              </constantatomformula>
                                             </formulalist>
                                            </sequent>
                                            <rule type="foralll" symbol="\all:l">
                                             <sequent>
                                              <formulalist>
                                               <quantifiedformula type="all">
                                                <variable symbol="x"/>
                                                <quantifiedformula type="all">
                                                 <variable symbol="y"/>
                                                 <conjunctiveformula type="and">
                                                  <conjunctiveformula type="impl">
                                                   <constantatomformula symbol="=">
                                                    <function symbol="∩">
                                                     <variable symbol="x"/>
                                                     <variable symbol="y"/>
                                                    </function>
                                                    <variable symbol="x"/>
                                                   </constantatomformula>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="∪">
                                                     <variable symbol="x"/>
                                                     <variable symbol="y"/>
                                                    </function>
                                                    <variable symbol="y"/>
                                                   </constantatomformula>
                                                  </conjunctiveformula>
                                                  <conjunctiveformula type="impl">
                                                   <constantatomformula symbol="=">
                                                    <function symbol="∪">
                                                     <variable symbol="x"/>
                                                     <variable symbol="y"/>
                                                    </function>
                                                    <variable symbol="y"/>
                                                   </constantatomformula>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="∩">
                                                     <variable symbol="x"/>
                                                     <variable symbol="y"/>
                                                    </function>
                                                    <variable symbol="x"/>
                                                   </constantatomformula>
                                                  </conjunctiveformula>
                                                 </conjunctiveformula>
                                                </quantifiedformula>
                                               </quantifiedformula>
                                              </formulalist>
                                              <formulalist>
                                               <constantatomformula symbol="=">
                                                <function symbol="∩">
                                                 <constant symbol="s_3"/>
                                                 <function symbol="∪">
                                                  <constant symbol="s_3"/>
                                                  <constant symbol="s_4"/>
                                                 </function>
                                                </function>
                                                <constant symbol="s_3"/>
                                               </constantatomformula>
                                              </formulalist>
                                             </sequent>
                                             <rule type="foralll" symbol="\all:l">
                                              <sequent>
                                               <formulalist>
                                                <quantifiedformula type="all">
                                                 <variable symbol="y"/>
                                                 <conjunctiveformula type="and">
                                                  <conjunctiveformula type="impl">
                                                   <constantatomformula symbol="=">
                                                    <function symbol="∩">
                                                     <constant symbol="s_3"/>
                                                     <variable symbol="y"/>
                                                    </function>
                                                    <constant symbol="s_3"/>
                                                   </constantatomformula>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="∪">
                                                     <constant symbol="s_3"/>
                                                     <variable symbol="y"/>
                                                    </function>
                                                    <variable symbol="y"/>
                                                   </constantatomformula>
                                                  </conjunctiveformula>
                                                  <conjunctiveformula type="impl">
                                                   <constantatomformula symbol="=">
                                                    <function symbol="∪">
                                                     <constant symbol="s_3"/>
                                                     <variable symbol="y"/>
                                                    </function>
                                                    <variable symbol="y"/>
                                                   </constantatomformula>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="∩">
                                                     <constant symbol="s_3"/>
                                                     <variable symbol="y"/>
                                                    </function>
                                                    <constant symbol="s_3"/>
                                                   </constantatomformula>
                                                  </conjunctiveformula>
                                                 </conjunctiveformula>
                                                </quantifiedformula>
                                               </formulalist>
                                               <formulalist>
                                                <constantatomformula symbol="=">
                                                 <function symbol="∩">
                                                  <constant symbol="s_3"/>
                                                  <function symbol="∪">
                                                   <constant symbol="s_3"/>
                                                   <constant symbol="s_4"/>
                                                  </function>
                                                 </function>
                                                 <constant symbol="s_3"/>
                                                </constantatomformula>
                                               </formulalist>
                                              </sequent>
                                              <rule type="andl2" symbol="\and:l2">
                                               <sequent>
                                                <formulalist>
                                                 <conjunctiveformula type="and">
                                                  <conjunctiveformula type="impl">
                                                   <constantatomformula symbol="=">
                                                    <function symbol="∩">
                                                     <constant symbol="s_3"/>
                                                     <function symbol="∪">
                                                      <constant symbol="s_3"/>
                                                      <constant symbol="s_4"/>
                                                     </function>
                                                    </function>
                                                    <constant symbol="s_3"/>
                                                   </constantatomformula>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="∪">
                                                     <constant symbol="s_3"/>
                                                     <function symbol="∪">
                                                      <constant symbol="s_3"/>
                                                      <constant symbol="s_4"/>
                                                     </function>
                                                    </function>
                                                    <function symbol="∪">
                                                     <constant symbol="s_3"/>
                                                     <constant symbol="s_4"/>
                                                    </function>
                                                   </constantatomformula>
                                                  </conjunctiveformula>
                                                  <conjunctiveformula type="impl">
                                                   <constantatomformula symbol="=">
                                                    <function symbol="∪">
                                                     <constant symbol="s_3"/>
                                                     <function symbol="∪">
                                                      <constant symbol="s_3"/>
                                                      <constant symbol="s_4"/>
                                                     </function>
                                                    </function>
                                                    <function symbol="∪">
                                                     <constant symbol="s_3"/>
                                                     <constant symbol="s_4"/>
                                                    </function>
                                                   </constantatomformula>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="∩">
                                                     <constant symbol="s_3"/>
                                                     <function symbol="∪">
                                                      <constant symbol="s_3"/>
                                                      <constant symbol="s_4"/>
                                                     </function>
                                                    </function>
                                                    <constant symbol="s_3"/>
                                                   </constantatomformula>
                                                  </conjunctiveformula>
                                                 </conjunctiveformula>
                                                </formulalist>
                                                <formulalist>
                                                 <constantatomformula symbol="=">
                                                  <function symbol="∩">
                                                   <constant symbol="s_3"/>
                                                   <function symbol="∪">
                                                    <constant symbol="s_3"/>
                                                    <constant symbol="s_4"/>
                                                   </function>
                                                  </function>
                                                  <constant symbol="s_3"/>
                                                 </constantatomformula>
                                                </formulalist>
                                               </sequent>
                                               <rule type="impll" symbol="\impl:l">
                                                <sequent>
                                                 <formulalist>
                                                  <conjunctiveformula type="impl">
                                                   <constantatomformula symbol="=">
                                                    <function symbol="∪">
                                                     <constant symbol="s_3"/>
                                                     <function symbol="∪">
                                                      <constant symbol="s_3"/>
                                                      <constant symbol="s_4"/>
                                                     </function>
                                                    </function>
                                                    <function symbol="∪">
                                                     <constant symbol="s_3"/>
                                                     <constant symbol="s_4"/>
                                                    </function>
                                                   </constantatomformula>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="∩">
                                                     <constant symbol="s_3"/>
                                                     <function symbol="∪">
                                                      <constant symbol="s_3"/>
                                                      <constant symbol="s_4"/>
                                                     </function>
                                                    </function>
                                                    <constant symbol="s_3"/>
                                                   </constantatomformula>
                                                  </conjunctiveformula>
                                                 </formulalist>
                                                 <formulalist>
                                                  <constantatomformula symbol="=">
                                                   <function symbol="∩">
                                                    <constant symbol="s_3"/>
                                                    <function symbol="∪">
                                                     <constant symbol="s_3"/>
                                                     <constant symbol="s_4"/>
                                                    </function>
                                                   </function>
                                                   <constant symbol="s_3"/>
                                                  </constantatomformula>
                                                 </formulalist>
                                                </sequent>
                                                <rule type="eqr1" symbol="=:r1" param="(1)">
                                                 <sequent>
                                                  <formulalist/>
                                                  <formulalist>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="∪">
                                                     <constant symbol="s_3"/>
                                                     <function symbol="∪">
                                                      <constant symbol="s_3"/>
                                                      <constant symbol="s_4"/>
                                                     </function>
                                                    </function>
                                                    <function symbol="∪">
                                                     <constant symbol="s_3"/>
                                                     <constant symbol="s_4"/>
                                                    </function>
                                                   </constantatomformula>
                                                  </formulalist>
                                                 </sequent>
                                                 <rule type="axiom" symbol="axiom">
                                                  <sequent>
                                                   <formulalist/>
                                                   <formulalist>
                                                    <constantatomformula symbol="=">
                                                     <function symbol="∪">
                                                      <function symbol="∪">
                                                       <constant symbol="s_3"/>
                                                       <constant symbol="s_3"/>
                                                      </function>
                                                      <constant symbol="s_4"/>
                                                     </function>
                                                     <function symbol="∪">
                                                      <constant symbol="s_3"/>
                                                      <function symbol="∪">
                                                       <constant symbol="s_3"/>
                                                       <constant symbol="s_4"/>
                                                      </function>
                                                     </function>
                                                    </constantatomformula>
                                                   </formulalist>
                                                  </sequent>
                                                 </rule>
                                                 <rule type="eqr2" symbol="=:r2" param="(1 1)">
                                                  <sequent>
                                                   <formulalist/>
                                                   <formulalist>
                                                    <constantatomformula symbol="=">
                                                     <function symbol="∪">
                                                      <function symbol="∪">
                                                       <constant symbol="s_3"/>
                                                       <constant symbol="s_3"/>
                                                      </function>
                                                      <constant symbol="s_4"/>
                                                     </function>
                                                     <function symbol="∪">
                                                      <constant symbol="s_3"/>
                                                      <constant symbol="s_4"/>
                                                     </function>
                                                    </constantatomformula>
                                                   </formulalist>
                                                  </sequent>
                                                  <rule type="axiom" symbol="axiom">
                                                   <sequent>
                                                    <formulalist/>
                                                    <formulalist>
                                                     <constantatomformula symbol="=">
                                                      <function symbol="∪">
                                                       <constant symbol="s_3"/>
                                                       <constant symbol="s_3"/>
                                                      </function>
                                                      <constant symbol="s_3"/>
                                                     </constantatomformula>
                                                    </formulalist>
                                                   </sequent>
                                                  </rule>
                                                  <rule type="axiom" symbol="axiom">
                                                   <sequent>
                                                    <formulalist/>
                                                    <formulalist>
                                                     <constantatomformula symbol="=">
                                                      <function symbol="∪">
                                                       <constant symbol="s_3"/>
                                                       <constant symbol="s_4"/>
                                                      </function>
                                                      <function symbol="∪">
                                                       <constant symbol="s_3"/>
                                                       <constant symbol="s_4"/>
                                                      </function>
                                                     </constantatomformula>
                                                    </formulalist>
                                                   </sequent>
                                                  </rule>
                                                 </rule>
                                                </rule>
                                                <rule type="axiom" symbol="axiom">
                                                 <sequent>
                                                  <formulalist>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="∩">
                                                     <constant symbol="s_3"/>
                                                     <function symbol="∪">
                                                      <constant symbol="s_3"/>
                                                      <constant symbol="s_4"/>
                                                     </function>
                                                    </function>
                                                    <constant symbol="s_3"/>
                                                   </constantatomformula>
                                                  </formulalist>
                                                  <formulalist>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="∩">
                                                     <constant symbol="s_3"/>
                                                     <function symbol="∪">
                                                      <constant symbol="s_3"/>
                                                      <constant symbol="s_4"/>
                                                     </function>
                                                    </function>
                                                    <constant symbol="s_3"/>
                                                   </constantatomformula>
                                                  </formulalist>
                                                 </sequent>
                                                </rule>
                                               </rule>
                                              </rule>
                                             </rule>
                                            </rule>
                                           </rule>
                                          </rule>
                                         </rule>
                                        </rule>
                                       </rule>
                                      </rule>
                                     </rule>
                                    </rule>
                                   </rule>
                                  </rule>
                                  <rule type="perml" symbol="\pi:l" param="(1 2)">
                                   <sequent>
                                    <formulalist>
                                     <constantatomformula symbol="=">
                                      <function symbol="∩">
                                       <constant symbol="s_3"/>
                                       <function symbol="∪">
                                        <constant symbol="s_3"/>
                                        <constant symbol="s_4"/>
                                       </function>
                                      </function>
                                      <constant symbol="s_3"/>
                                     </constantatomformula>
                                     <constantatomformula symbol="=">
                                      <function symbol="∪">
                                       <constant symbol="s_1"/>
                                       <function symbol="∩">
                                        <constant symbol="s_1"/>
                                        <constant symbol="s_2"/>
                                       </function>
                                      </function>
                                      <constant symbol="s_1"/>
                                     </constantatomformula>
                                     <constantatomformula symbol="L1" definition="L1"/>
                                    </formulalist>
                                    <formulalist>
                                     <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                                    </formulalist>
                                   </sequent>
                                   <rule type="perml" symbol="\pi:l" param="(1 2)">
                                    <sequent>
                                     <formulalist>
                                      <constantatomformula symbol="=">
                                       <function symbol="∪">
                                        <constant symbol="s_1"/>
                                        <function symbol="∩">
                                         <constant symbol="s_1"/>
                                         <constant symbol="s_2"/>
                                        </function>
                                       </function>
                                       <constant symbol="s_1"/>
                                      </constantatomformula>
                                      <constantatomformula symbol="=">
                                       <function symbol="∩">
                                        <constant symbol="s_3"/>
                                        <function symbol="∪">
                                         <constant symbol="s_3"/>
                                         <constant symbol="s_4"/>
                                        </function>
                                       </function>
                                       <constant symbol="s_3"/>
                                      </constantatomformula>
                                      <constantatomformula symbol="L1" definition="L1"/>
                                     </formulalist>
                                     <formulalist>
                                      <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                                     </formulalist>
                                    </sequent>
                                    <rule type="contrr" symbol="c:r" param="2">
                                     <sequent>
                                      <formulalist>
                                       <constantatomformula symbol="=">
                                        <function symbol="∩">
                                         <constant symbol="s_3"/>
                                         <function symbol="∪">
                                          <constant symbol="s_3"/>
                                          <constant symbol="s_4"/>
                                         </function>
                                        </function>
                                        <constant symbol="s_3"/>
                                       </constantatomformula>
                                       <constantatomformula symbol="=">
                                        <function symbol="∪">
                                         <constant symbol="s_1"/>
                                         <function symbol="∩">
                                          <constant symbol="s_1"/>
                                          <constant symbol="s_2"/>
                                         </function>
                                        </function>
                                        <constant symbol="s_1"/>
                                       </constantatomformula>
                                       <constantatomformula symbol="L1" definition="L1"/>
                                      </formulalist>
                                      <formulalist>
                                       <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                                      </formulalist>
                                     </sequent>
                                     <rule type="permr" symbol="\pi:r" param="">
                                      <sequent>
                                       <formulalist>
                                        <constantatomformula symbol="=">
                                         <function symbol="∩">
                                          <constant symbol="s_3"/>
                                          <function symbol="∪">
                                           <constant symbol="s_3"/>
                                           <constant symbol="s_4"/>
                                          </function>
                                         </function>
                                         <constant symbol="s_3"/>
                                        </constantatomformula>
                                        <constantatomformula symbol="=">
                                         <function symbol="∪">
                                          <constant symbol="s_1"/>
                                          <function symbol="∩">
                                           <constant symbol="s_1"/>
                                           <constant symbol="s_2"/>
                                          </function>
                                         </function>
                                         <constant symbol="s_1"/>
                                        </constantatomformula>
                                        <constantatomformula symbol="L1" definition="L1"/>
                                       </formulalist>
                                       <formulalist>
                                        <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                                        <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                                       </formulalist>
                                      </sequent>
                                      <rule type="contrl" symbol="c:l" param="1,1,2">
                                       <sequent>
                                        <formulalist>
                                         <constantatomformula symbol="=">
                                          <function symbol="∩">
                                           <constant symbol="s_3"/>
                                           <function symbol="∪">
                                            <constant symbol="s_3"/>
                                            <constant symbol="s_4"/>
                                           </function>
                                          </function>
                                          <constant symbol="s_3"/>
                                         </constantatomformula>
                                         <constantatomformula symbol="=">
                                          <function symbol="∪">
                                           <constant symbol="s_1"/>
                                           <function symbol="∩">
                                            <constant symbol="s_1"/>
                                            <constant symbol="s_2"/>
                                           </function>
                                          </function>
                                          <constant symbol="s_1"/>
                                         </constantatomformula>
                                         <constantatomformula symbol="L1" definition="L1"/>
                                        </formulalist>
                                        <formulalist>
                                         <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                                         <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                                        </formulalist>
                                       </sequent>
                                       <rule type="perml" symbol="\pi:l" param="(2 3)">
                                        <sequent>
                                         <formulalist>
                                          <constantatomformula symbol="=">
                                           <function symbol="∩">
                                            <constant symbol="s_3"/>
                                            <function symbol="∪">
                                             <constant symbol="s_3"/>
                                             <constant symbol="s_4"/>
                                            </function>
                                           </function>
                                           <constant symbol="s_3"/>
                                          </constantatomformula>
                                          <constantatomformula symbol="=">
                                           <function symbol="∪">
                                            <constant symbol="s_1"/>
                                            <function symbol="∩">
                                             <constant symbol="s_1"/>
                                             <constant symbol="s_2"/>
                                            </function>
                                           </function>
                                           <constant symbol="s_1"/>
                                          </constantatomformula>
                                          <constantatomformula symbol="L1" definition="L1"/>
                                          <constantatomformula symbol="L1" definition="L1"/>
                                         </formulalist>
                                         <formulalist>
                                          <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                                          <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                                         </formulalist>
                                        </sequent>
                                        <rule type="eql1" symbol="=:l1" param="(1)">
                                         <sequent>
                                          <formulalist>
                                           <constantatomformula symbol="=">
                                            <function symbol="∩">
                                             <constant symbol="s_3"/>
                                             <function symbol="∪">
                                              <constant symbol="s_3"/>
                                              <constant symbol="s_4"/>
                                             </function>
                                            </function>
                                            <constant symbol="s_3"/>
                                           </constantatomformula>
                                           <constantatomformula symbol="L1" definition="L1"/>
                                           <constantatomformula symbol="=">
                                            <function symbol="∪">
                                             <constant symbol="s_1"/>
                                             <function symbol="∩">
                                              <constant symbol="s_1"/>
                                              <constant symbol="s_2"/>
                                             </function>
                                            </function>
                                            <constant symbol="s_1"/>
                                           </constantatomformula>
                                           <constantatomformula symbol="L1" definition="L1"/>
                                          </formulalist>
                                          <formulalist>
                                           <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                                           <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                                          </formulalist>
                                         </sequent>
                                         <rule type="permr" symbol="\pi:r" param="(1 2)">
                                          <sequent>
                                           <formulalist>
                                            <constantatomformula symbol="L1" definition="L1"/>
                                           </formulalist>
                                           <formulalist>
                                            <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                                            <constantatomformula symbol="=">
                                             <function symbol="∩">
                                              <function symbol="∪">
                                               <constant symbol="s_3"/>
                                               <constant symbol="s_4"/>
                                              </function>
                                              <constant symbol="s_3"/>
                                             </function>
                                             <function symbol="∩">
                                              <constant symbol="s_3"/>
                                              <function symbol="∪">
                                               <constant symbol="s_3"/>
                                               <constant symbol="s_4"/>
                                              </function>
                                             </function>
                                            </constantatomformula>
                                           </formulalist>
                                          </sequent>
                                          <rule type="weakr" symbol="w:r">
                                           <sequent>
                                            <formulalist>
                                             <constantatomformula symbol="L1" definition="L1"/>
                                            </formulalist>
                                            <formulalist>
                                             <constantatomformula symbol="=">
                                              <function symbol="∩">
                                               <function symbol="∪">
                                                <constant symbol="s_3"/>
                                                <constant symbol="s_4"/>
                                               </function>
                                               <constant symbol="s_3"/>
                                              </function>
                                              <function symbol="∩">
                                               <constant symbol="s_3"/>
                                               <function symbol="∪">
                                                <constant symbol="s_3"/>
                                                <constant symbol="s_4"/>
                                               </function>
                                              </function>
                                             </constantatomformula>
                                             <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                                            </formulalist>
                                           </sequent>
                                           <rule type="perml" symbol="\pi:l" param="">
                                            <sequent>
                                             <formulalist>
                                              <constantatomformula symbol="L1" definition="L1"/>
                                             </formulalist>
                                             <formulalist>
                                              <constantatomformula symbol="=">
                                               <function symbol="∩">
                                                <function symbol="∪">
                                                 <constant symbol="s_3"/>
                                                 <constant symbol="s_4"/>
                                                </function>
                                                <constant symbol="s_3"/>
                                               </function>
                                               <function symbol="∩">
                                                <constant symbol="s_3"/>
                                                <function symbol="∪">
                                                 <constant symbol="s_3"/>
                                                 <constant symbol="s_4"/>
                                                </function>
                                               </function>
                                              </constantatomformula>
                                             </formulalist>
                                            </sequent>
                                            <rule type="weakl" symbol="w:l">
                                             <sequent>
                                              <formulalist>
                                               <constantatomformula symbol="L1" definition="L1"/>
                                              </formulalist>
                                              <formulalist>
                                               <constantatomformula symbol="=">
                                                <function symbol="∩">
                                                 <function symbol="∪">
                                                  <constant symbol="s_3"/>
                                                  <constant symbol="s_4"/>
                                                 </function>
                                                 <constant symbol="s_3"/>
                                                </function>
                                                <function symbol="∩">
                                                 <constant symbol="s_3"/>
                                                 <function symbol="∪">
                                                  <constant symbol="s_3"/>
                                                  <constant symbol="s_4"/>
                                                 </function>
                                                </function>
                                               </constantatomformula>
                                              </formulalist>
                                             </sequent>
                                             <rule type="axiom" symbol="axiom">
                                              <sequent>
                                               <formulalist/>
                                               <formulalist>
                                                <constantatomformula symbol="=">
                                                 <function symbol="∩">
                                                  <function symbol="∪">
                                                   <constant symbol="s_3"/>
                                                   <constant symbol="s_4"/>
                                                  </function>
                                                  <constant symbol="s_3"/>
                                                 </function>
                                                 <function symbol="∩">
                                                  <constant symbol="s_3"/>
                                                  <function symbol="∪">
                                                   <constant symbol="s_3"/>
                                                   <constant symbol="s_4"/>
                                                  </function>
                                                 </function>
                                                </constantatomformula>
                                               </formulalist>
                                              </sequent>
                                             </rule>
                                            </rule>
                                           </rule>
                                          </rule>
                                         </rule>
                                         <rule type="perml" symbol="\pi:l" param="(1 2)">
                                          <sequent>
                                           <formulalist>
                                            <constantatomformula symbol="=">
                                             <function symbol="∩">
                                              <function symbol="∪">
                                               <constant symbol="s_3"/>
                                               <constant symbol="s_4"/>
                                              </function>
                                              <constant symbol="s_3"/>
                                             </function>
                                             <constant symbol="s_3"/>
                                            </constantatomformula>
                                            <constantatomformula symbol="=">
                                             <function symbol="∪">
                                              <constant symbol="s_1"/>
                                              <function symbol="∩">
                                               <constant symbol="s_1"/>
                                               <constant symbol="s_2"/>
                                              </function>
                                             </function>
                                             <constant symbol="s_1"/>
                                            </constantatomformula>
                                            <constantatomformula symbol="L1" definition="L1"/>
                                           </formulalist>
                                           <formulalist>
                                            <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                                           </formulalist>
                                          </sequent>
                                          <rule type="contrr" symbol="c:r" param="2">
                                           <sequent>
                                            <formulalist>
                                             <constantatomformula symbol="=">
                                              <function symbol="∪">
                                               <constant symbol="s_1"/>
                                               <function symbol="∩">
                                                <constant symbol="s_1"/>
                                                <constant symbol="s_2"/>
                                               </function>
                                              </function>
                                              <constant symbol="s_1"/>
                                             </constantatomformula>
                                             <constantatomformula symbol="=">
                                              <function symbol="∩">
                                               <function symbol="∪">
                                                <constant symbol="s_3"/>
                                                <constant symbol="s_4"/>
                                               </function>
                                               <constant symbol="s_3"/>
                                              </function>
                                              <constant symbol="s_3"/>
                                             </constantatomformula>
                                             <constantatomformula symbol="L1" definition="L1"/>
                                            </formulalist>
                                            <formulalist>
                                             <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                                            </formulalist>
                                           </sequent>
                                           <rule type="permr" symbol="\pi:r" param="">
                                            <sequent>
                                             <formulalist>
                                              <constantatomformula symbol="=">
                                               <function symbol="∪">
                                                <constant symbol="s_1"/>
                                                <function symbol="∩">
                                                 <constant symbol="s_1"/>
                                                 <constant symbol="s_2"/>
                                                </function>
                                               </function>
                                               <constant symbol="s_1"/>
                                              </constantatomformula>
                                              <constantatomformula symbol="=">
                                               <function symbol="∩">
                                                <function symbol="∪">
                                                 <constant symbol="s_3"/>
                                                 <constant symbol="s_4"/>
                                                </function>
                                                <constant symbol="s_3"/>
                                               </function>
                                               <constant symbol="s_3"/>
                                              </constantatomformula>
                                              <constantatomformula symbol="L1" definition="L1"/>
                                             </formulalist>
                                             <formulalist>
                                              <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                                              <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                                             </formulalist>
                                            </sequent>
                                            <rule type="contrl" symbol="c:l" param="1,1,2">
                                             <sequent>
                                              <formulalist>
                                               <constantatomformula symbol="=">
                                                <function symbol="∪">
                                                 <constant symbol="s_1"/>
                                                 <function symbol="∩">
                                                  <constant symbol="s_1"/>
                                                  <constant symbol="s_2"/>
                                                 </function>
                                                </function>
                                                <constant symbol="s_1"/>
                                               </constantatomformula>
                                               <constantatomformula symbol="=">
                                                <function symbol="∩">
                                                 <function symbol="∪">
                                                  <constant symbol="s_3"/>
                                                  <constant symbol="s_4"/>
                                                 </function>
                                                 <constant symbol="s_3"/>
                                                </function>
                                                <constant symbol="s_3"/>
                                               </constantatomformula>
                                               <constantatomformula symbol="L1" definition="L1"/>
                                              </formulalist>
                                              <formulalist>
                                               <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                                               <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                                              </formulalist>
                                             </sequent>
                                             <rule type="perml" symbol="\pi:l" param="(2 3)">
                                              <sequent>
                                               <formulalist>
                                                <constantatomformula symbol="=">
                                                 <function symbol="∪">
                                                  <constant symbol="s_1"/>
                                                  <function symbol="∩">
                                                   <constant symbol="s_1"/>
                                                   <constant symbol="s_2"/>
                                                  </function>
                                                 </function>
                                                 <constant symbol="s_1"/>
                                                </constantatomformula>
                                                <constantatomformula symbol="=">
                                                 <function symbol="∩">
                                                  <function symbol="∪">
                                                   <constant symbol="s_3"/>
                                                   <constant symbol="s_4"/>
                                                  </function>
                                                  <constant symbol="s_3"/>
                                                 </function>
                                                 <constant symbol="s_3"/>
                                                </constantatomformula>
                                                <constantatomformula symbol="L1" definition="L1"/>
                                                <constantatomformula symbol="L1" definition="L1"/>
                                               </formulalist>
                                               <formulalist>
                                                <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                                                <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                                               </formulalist>
                                              </sequent>
                                              <rule type="eql1" symbol="=:l1" param="(1)">
                                               <sequent>
                                                <formulalist>
                                                 <constantatomformula symbol="=">
                                                  <function symbol="∪">
                                                   <constant symbol="s_1"/>
                                                   <function symbol="∩">
                                                    <constant symbol="s_1"/>
                                                    <constant symbol="s_2"/>
                                                   </function>
                                                  </function>
                                                  <constant symbol="s_1"/>
                                                 </constantatomformula>
                                                 <constantatomformula symbol="L1" definition="L1"/>
                                                 <constantatomformula symbol="=">
                                                  <function symbol="∩">
                                                   <function symbol="∪">
                                                    <constant symbol="s_3"/>
                                                    <constant symbol="s_4"/>
                                                   </function>
                                                   <constant symbol="s_3"/>
                                                  </function>
                                                  <constant symbol="s_3"/>
                                                 </constantatomformula>
                                                 <constantatomformula symbol="L1" definition="L1"/>
                                                </formulalist>
                                                <formulalist>
                                                 <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                                                 <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                                                </formulalist>
                                               </sequent>
                                               <rule type="permr" symbol="\pi:r" param="(1 2)">
                                                <sequent>
                                                 <formulalist>
                                                  <constantatomformula symbol="L1" definition="L1"/>
                                                 </formulalist>
                                                 <formulalist>
                                                  <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                                                  <constantatomformula symbol="=">
                                                   <function symbol="∪">
                                                    <function symbol="∩">
                                                     <constant symbol="s_1"/>
                                                     <constant symbol="s_2"/>
                                                    </function>
                                                    <constant symbol="s_1"/>
                                                   </function>
                                                   <function symbol="∪">
                                                    <constant symbol="s_1"/>
                                                    <function symbol="∩">
                                                     <constant symbol="s_1"/>
                                                     <constant symbol="s_2"/>
                                                    </function>
                                                   </function>
                                                  </constantatomformula>
                                                 </formulalist>
                                                </sequent>
                                                <rule type="weakr" symbol="w:r">
                                                 <sequent>
                                                  <formulalist>
                                                   <constantatomformula symbol="L1" definition="L1"/>
                                                  </formulalist>
                                                  <formulalist>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="∪">
                                                     <function symbol="∩">
                                                      <constant symbol="s_1"/>
                                                      <constant symbol="s_2"/>
                                                     </function>
                                                     <constant symbol="s_1"/>
                                                    </function>
                                                    <function symbol="∪">
                                                     <constant symbol="s_1"/>
                                                     <function symbol="∩">
                                                      <constant symbol="s_1"/>
                                                      <constant symbol="s_2"/>
                                                     </function>
                                                    </function>
                                                   </constantatomformula>
                                                   <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                                                  </formulalist>
                                                 </sequent>
                                                 <rule type="perml" symbol="\pi:l" param="">
                                                  <sequent>
                                                   <formulalist>
                                                    <constantatomformula symbol="L1" definition="L1"/>
                                                   </formulalist>
                                                   <formulalist>
                                                    <constantatomformula symbol="=">
                                                     <function symbol="∪">
                                                      <function symbol="∩">
                                                       <constant symbol="s_1"/>
                                                       <constant symbol="s_2"/>
                                                      </function>
                                                      <constant symbol="s_1"/>
                                                     </function>
                                                     <function symbol="∪">
                                                      <constant symbol="s_1"/>
                                                      <function symbol="∩">
                                                       <constant symbol="s_1"/>
                                                       <constant symbol="s_2"/>
                                                      </function>
                                                     </function>
                                                    </constantatomformula>
                                                   </formulalist>
                                                  </sequent>
                                                  <rule type="weakl" symbol="w:l">
                                                   <sequent>
                                                    <formulalist>
                                                     <constantatomformula symbol="L1" definition="L1"/>
                                                    </formulalist>
                                                    <formulalist>
                                                     <constantatomformula symbol="=">
                                                      <function symbol="∪">
                                                       <function symbol="∩">
                                                        <constant symbol="s_1"/>
                                                        <constant symbol="s_2"/>
                                                       </function>
                                                       <constant symbol="s_1"/>
                                                      </function>
                                                      <function symbol="∪">
                                                       <constant symbol="s_1"/>
                                                       <function symbol="∩">
                                                        <constant symbol="s_1"/>
                                                        <constant symbol="s_2"/>
                                                       </function>
                                                      </function>
                                                     </constantatomformula>
                                                    </formulalist>
                                                   </sequent>
                                                   <rule type="axiom" symbol="axiom">
                                                    <sequent>
                                                     <formulalist/>
                                                     <formulalist>
                                                      <constantatomformula symbol="=">
                                                       <function symbol="∪">
                                                        <function symbol="∩">
                                                         <constant symbol="s_1"/>
                                                         <constant symbol="s_2"/>
                                                        </function>
                                                        <constant symbol="s_1"/>
                                                       </function>
                                                       <function symbol="∪">
                                                        <constant symbol="s_1"/>
                                                        <function symbol="∩">
                                                         <constant symbol="s_1"/>
                                                         <constant symbol="s_2"/>
                                                        </function>
                                                       </function>
                                                      </constantatomformula>
                                                     </formulalist>
                                                    </sequent>
                                                   </rule>
                                                  </rule>
                                                 </rule>
                                                </rule>
                                               </rule>
                                               <rule type="permr" symbol="\pi:r" param="">
                                                <sequent>
                                                 <formulalist>
                                                  <constantatomformula symbol="=">
                                                   <function symbol="∪">
                                                    <function symbol="∩">
                                                     <constant symbol="s_1"/>
                                                     <constant symbol="s_2"/>
                                                    </function>
                                                    <constant symbol="s_1"/>
                                                   </function>
                                                   <constant symbol="s_1"/>
                                                  </constantatomformula>
                                                  <constantatomformula symbol="=">
                                                   <function symbol="∩">
                                                    <function symbol="∪">
                                                     <constant symbol="s_3"/>
                                                     <constant symbol="s_4"/>
                                                    </function>
                                                    <constant symbol="s_3"/>
                                                   </function>
                                                   <constant symbol="s_3"/>
                                                  </constantatomformula>
                                                  <constantatomformula symbol="L1" definition="L1"/>
                                                 </formulalist>
                                                 <formulalist>
                                                  <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                                                 </formulalist>
                                                </sequent>
                                                <rule type="perml" symbol="\pi:l" param="(1 2 3)">
                                                 <sequent>
                                                  <formulalist>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="∪">
                                                     <function symbol="∩">
                                                      <constant symbol="s_1"/>
                                                      <constant symbol="s_2"/>
                                                     </function>
                                                     <constant symbol="s_1"/>
                                                    </function>
                                                    <constant symbol="s_1"/>
                                                   </constantatomformula>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="∩">
                                                     <function symbol="∪">
                                                      <constant symbol="s_3"/>
                                                      <constant symbol="s_4"/>
                                                     </function>
                                                     <constant symbol="s_3"/>
                                                    </function>
                                                    <constant symbol="s_3"/>
                                                   </constantatomformula>
                                                   <constantatomformula symbol="L1" definition="L1"/>
                                                  </formulalist>
                                                  <formulalist>
                                                   <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                                                  </formulalist>
                                                 </sequent>
                                                 <rule type="weakl" symbol="w:l">
                                                  <sequent>
                                                   <formulalist>
                                                    <constantatomformula symbol="L1" definition="L1"/>
                                                    <constantatomformula symbol="=">
                                                     <function symbol="∪">
                                                      <function symbol="∩">
                                                       <constant symbol="s_1"/>
                                                       <constant symbol="s_2"/>
                                                      </function>
                                                      <constant symbol="s_1"/>
                                                     </function>
                                                     <constant symbol="s_1"/>
                                                    </constantatomformula>
                                                    <constantatomformula symbol="=">
                                                     <function symbol="∩">
                                                      <function symbol="∪">
                                                       <constant symbol="s_3"/>
                                                       <constant symbol="s_4"/>
                                                      </function>
                                                      <constant symbol="s_3"/>
                                                     </function>
                                                     <constant symbol="s_3"/>
                                                    </constantatomformula>
                                                   </formulalist>
                                                   <formulalist>
                                                    <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                                                   </formulalist>
                                                  </sequent>
                                                  <rule type="defr" symbol="d:r" param="();[()];()">
                                                   <sequent>
                                                    <formulalist>
                                                     <constantatomformula symbol="=">
                                                      <function symbol="∪">
                                                       <function symbol="∩">
                                                        <constant symbol="s_1"/>
                                                        <constant symbol="s_2"/>
                                                       </function>
                                                       <constant symbol="s_1"/>
                                                      </function>
                                                      <constant symbol="s_1"/>
                                                     </constantatomformula>
                                                     <constantatomformula symbol="=">
                                                      <function symbol="∩">
                                                       <function symbol="∪">
                                                        <constant symbol="s_3"/>
                                                        <constant symbol="s_4"/>
                                                       </function>
                                                       <constant symbol="s_3"/>
                                                      </function>
                                                      <constant symbol="s_3"/>
                                                     </constantatomformula>
                                                    </formulalist>
                                                    <formulalist>
                                                     <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
                                                    </formulalist>
                                                   </sequent>
                                                   <rule type="andr" symbol="\and:r">
                                                    <sequent>
                                                     <formulalist>
                                                      <constantatomformula symbol="=">
                                                       <function symbol="∪">
                                                        <function symbol="∩">
                                                         <constant symbol="s_1"/>
                                                         <constant symbol="s_2"/>
                                                        </function>
                                                        <constant symbol="s_1"/>
                                                       </function>
                                                       <constant symbol="s_1"/>
                                                      </constantatomformula>
                                                      <constantatomformula symbol="=">
                                                       <function symbol="∩">
                                                        <function symbol="∪">
                                                         <constant symbol="s_3"/>
                                                         <constant symbol="s_4"/>
                                                        </function>
                                                        <constant symbol="s_3"/>
                                                       </function>
                                                       <constant symbol="s_3"/>
                                                      </constantatomformula>
                                                     </formulalist>
                                                     <formulalist>
                                                      <conjunctiveformula type="and">
                                                       <constantatomformula symbol="=">
                                                        <function symbol="∪">
                                                         <function symbol="∩">
                                                          <constant symbol="s_1"/>
                                                          <constant symbol="s_2"/>
                                                         </function>
                                                         <constant symbol="s_1"/>
                                                        </function>
                                                        <constant symbol="s_1"/>
                                                       </constantatomformula>
                                                       <constantatomformula symbol="=">
                                                        <function symbol="∩">
                                                         <function symbol="∪">
                                                          <constant symbol="s_3"/>
                                                          <constant symbol="s_4"/>
                                                         </function>
                                                         <constant symbol="s_3"/>
                                                        </function>
                                                        <constant symbol="s_3"/>
                                                       </constantatomformula>
                                                      </conjunctiveformula>
                                                     </formulalist>
                                                    </sequent>
                                                    <rule type="axiom" symbol="axiom">
                                                     <sequent>
                                                      <formulalist>
                                                       <constantatomformula symbol="=">
                                                        <function symbol="∪">
                                                         <function symbol="∩">
                                                          <constant symbol="s_1"/>
                                                          <constant symbol="s_2"/>
                                                         </function>
                                                         <constant symbol="s_1"/>
                                                        </function>
                                                        <constant symbol="s_1"/>
                                                       </constantatomformula>
                                                      </formulalist>
                                                      <formulalist>
                                                       <constantatomformula symbol="=">
                                                        <function symbol="∪">
                                                         <function symbol="∩">
                                                          <constant symbol="s_1"/>
                                                          <constant symbol="s_2"/>
                                                         </function>
                                                         <constant symbol="s_1"/>
                                                        </function>
                                                        <constant symbol="s_1"/>
                                                       </constantatomformula>
                                                      </formulalist>
                                                     </sequent>
                                                    </rule>
                                                    <rule type="axiom" symbol="axiom">
                                                     <sequent>
                                                      <formulalist>
                                                       <constantatomformula symbol="=">
                                                        <function symbol="∩">
                                                         <function symbol="∪">
                                                          <constant symbol="s_3"/>
                                                          <constant symbol="s_4"/>
                                                         </function>
                                                         <constant symbol="s_3"/>
                                                        </function>
                                                        <constant symbol="s_3"/>
                                                       </constantatomformula>
                                                      </formulalist>
                                                      <formulalist>
                                                       <constantatomformula symbol="=">
                                                        <function symbol="∩">
                                                         <function symbol="∪">
                                                          <constant symbol="s_3"/>
                                                          <constant symbol="s_4"/>
                                                         </function>
                                                         <constant symbol="s_3"/>
                                                        </function>
                                                        <constant symbol="s_3"/>
                                                       </constantatomformula>
                                                      </formulalist>
                                                     </sequent>
                                                    </rule>
                                                   </rule>
                                                  </rule>
                                                 </rule>
                                                </rule>
                                               </rule>
                                              </rule>
                                             </rule>
                                            </rule>
                                           </rule>
                                          </rule>
                                         </rule>
                                        </rule>
                                       </rule>
                                      </rule>
                                     </rule>
                                    </rule>
                                   </rule>
                                  </rule>
                                 </rule>
                                </rule>
                               </rule>
                              </rule>
                             </rule>
                            </rule>
                           </rule>
                          </rule>
                         </rule>
                        </rule>
                       </rule>
                      </rule>
                     </rule>
                    </rule>
                   </rule>
                  </rule>
                 </rule>
                </rule>
               </rule>
              </rule>
             </rule>
            </rule>
           </rule>
          </rule>
         </rule>
        </rule>
       </rule>
      </rule>
     </rule>
    </rule>
   </rule>
  </rule>
 </proof>
 <proof symbol="herbrand sequent" calculus="LK">
  <rule type="herbrand sequent">
   <sequent>
    <formulalist>
     <conjunctiveformula type="impl">
      <constantatomformula symbol="=">
       <function symbol="∪">
        <constant symbol="s_1"/>
        <function symbol="∪">
         <constant symbol="s_1"/>
         <function symbol="∩">
          <constant symbol="s_1"/>
          <constant symbol="s_2"/>
         </function>
        </function>
       </function>
       <function symbol="∪">
        <constant symbol="s_1"/>
        <function symbol="∩">
         <constant symbol="s_1"/>
         <constant symbol="s_2"/>
        </function>
       </function>
      </constantatomformula>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <constant symbol="s_1"/>
        <function symbol="∪">
         <constant symbol="s_1"/>
         <function symbol="∩">
          <constant symbol="s_1"/>
          <constant symbol="s_2"/>
         </function>
        </function>
       </function>
       <constant symbol="s_1"/>
      </constantatomformula>
     </conjunctiveformula>
     <conjunctiveformula type="impl">
      <constantatomformula symbol="=">
       <function symbol="∩">
        <constant symbol="s_1"/>
        <constant symbol="s_1"/>
       </function>
       <constant symbol="s_1"/>
      </constantatomformula>
      <constantatomformula symbol="=">
       <function symbol="∪">
        <constant symbol="s_1"/>
        <constant symbol="s_1"/>
       </function>
       <constant symbol="s_1"/>
      </constantatomformula>
     </conjunctiveformula>
     <conjunctiveformula type="impl">
      <constantatomformula symbol="=">
       <function symbol="∩">
        <function symbol="∩">
         <constant symbol="s_1"/>
         <constant symbol="s_2"/>
        </function>
        <constant symbol="s_1"/>
       </function>
       <function symbol="∩">
        <constant symbol="s_1"/>
        <constant symbol="s_2"/>
       </function>
      </constantatomformula>
      <constantatomformula symbol="=">
       <function symbol="∪">
        <function symbol="∩">
         <constant symbol="s_1"/>
         <constant symbol="s_2"/>
        </function>
        <constant symbol="s_1"/>
       </function>
       <constant symbol="s_1"/>
      </constantatomformula>
     </conjunctiveformula>
     <conjunctiveformula type="impl">
      <constantatomformula symbol="=">
       <function symbol="∪">
        <function symbol="∪">
         <constant symbol="s_1"/>
         <function symbol="∩">
          <constant symbol="s_1"/>
          <constant symbol="s_2"/>
         </function>
        </function>
        <constant symbol="s_1"/>
       </function>
       <constant symbol="s_1"/>
      </constantatomformula>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <function symbol="∪">
         <constant symbol="s_1"/>
         <function symbol="∩">
          <constant symbol="s_1"/>
          <constant symbol="s_2"/>
         </function>
        </function>
        <constant symbol="s_1"/>
       </function>
       <function symbol="∪">
        <constant symbol="s_1"/>
        <function symbol="∩">
         <constant symbol="s_1"/>
         <constant symbol="s_2"/>
        </function>
       </function>
      </constantatomformula>
     </conjunctiveformula>
     <conjunctiveformula type="impl">
      <constantatomformula symbol="=">
       <function symbol="∪">
        <constant symbol="s_3"/>
        <function symbol="∪">
         <constant symbol="s_3"/>
         <constant symbol="s_4"/>
        </function>
       </function>
       <function symbol="∪">
        <constant symbol="s_3"/>
        <constant symbol="s_4"/>
       </function>
      </constantatomformula>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <constant symbol="s_3"/>
        <function symbol="∪">
         <constant symbol="s_3"/>
         <constant symbol="s_4"/>
        </function>
       </function>
       <constant symbol="s_3"/>
      </constantatomformula>
     </conjunctiveformula>
    </formulalist>
    <formulalist>
     <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
     <conjunctiveformula type="and">
      <constantatomformula symbol="=">
       <function symbol="∪">
        <function symbol="∩">
         <constant symbol="s_1"/>
         <constant symbol="s_2"/>
        </function>
        <constant symbol="s_1"/>
       </function>
       <constant symbol="s_1"/>
      </constantatomformula>
      <constantatomformula symbol="=">
       <function symbol="∩">
        <function symbol="∪">
         <constant symbol="s_3"/>
         <constant symbol="s_4"/>
        </function>
        <constant symbol="s_3"/>
       </function>
       <constant symbol="s_3"/>
      </constantatomformula>
     </conjunctiveformula>
    </formulalist>
   </sequent>
  </rule>
 </proof>
 <sequentlist symbol="simplified herbrand sequent in term level ">
  <sequent>
   <formulalist>
    <conjunctiveformula type="impl">
     <constantatomformula symbol="=">
      <function symbol="∪">
       <constant symbol="s_1"/>
       <function symbol="∪">
        <constant symbol="s_1"/>
        <function symbol="∩">
         <constant symbol="s_1"/>
         <constant symbol="s_2"/>
        </function>
       </function>
      </function>
      <function symbol="∪">
       <constant symbol="s_1"/>
       <function symbol="∩">
        <constant symbol="s_1"/>
        <constant symbol="s_2"/>
       </function>
      </function>
     </constantatomformula>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <constant symbol="s_1"/>
       <function symbol="∪">
        <constant symbol="s_1"/>
        <function symbol="∩">
         <constant symbol="s_1"/>
         <constant symbol="s_2"/>
        </function>
       </function>
      </function>
      <constant symbol="s_1"/>
     </constantatomformula>
    </conjunctiveformula>
    <conjunctiveformula type="impl">
     <constantatomformula symbol="=">
      <function symbol="∩">
       <constant symbol="s_1"/>
       <constant symbol="s_1"/>
      </function>
      <constant symbol="s_1"/>
     </constantatomformula>
     <constantatomformula symbol="=">
      <function symbol="∪">
       <constant symbol="s_1"/>
       <constant symbol="s_1"/>
      </function>
      <constant symbol="s_1"/>
     </constantatomformula>
    </conjunctiveformula>
    <conjunctiveformula type="impl">
     <constantatomformula symbol="=">
      <function symbol="∩">
       <function symbol="∩">
        <constant symbol="s_1"/>
        <constant symbol="s_2"/>
       </function>
       <constant symbol="s_1"/>
      </function>
      <function symbol="∩">
       <constant symbol="s_1"/>
       <constant symbol="s_2"/>
      </function>
     </constantatomformula>
     <constantatomformula symbol="=">
      <function symbol="∪">
       <function symbol="∩">
        <constant symbol="s_1"/>
        <constant symbol="s_2"/>
       </function>
       <constant symbol="s_1"/>
      </function>
      <constant symbol="s_1"/>
     </constantatomformula>
    </conjunctiveformula>
    <conjunctiveformula type="impl">
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   <formulalist>
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      <constant symbol="s_1"/>
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      <constant symbol="s_1"/>
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       <constant symbol="s_3"/>
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   <formulalist>
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      <constant symbol="s_1"/>
      <constant symbol="s_1"/>
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       <constant symbol="s_3"/>
       <constant symbol="s_4"/>
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   <formulalist>
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       <constant symbol="s_3"/>
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  <sequent>
   <formulalist>
    <constantatomformula symbol="=">
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      <function symbol="∩">
       <constant symbol="s_1"/>
       <constant symbol="s_2"/>
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   <formulalist>
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  <sequent>
   <formulalist>
    <constantatomformula symbol="=">
     <function symbol="∩">
      <function symbol="∩">
       <constant symbol="s_1"/>
       <constant symbol="s_2"/>
      </function>
      <constant symbol="s_1"/>
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   <formulalist>
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      <constant symbol="s_1"/>
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    <constantatomformula symbol="=">
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       <function symbol="∪">
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  <sequent>
   <formulalist>
    <constantatomformula symbol="=">
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      <function symbol="∩">
       <constant symbol="s_1"/>
       <constant symbol="s_2"/>
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    <constantatomformula symbol="=">
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   <formulalist>
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  </sequent>
  <sequent>
   <formulalist>
    <constantatomformula symbol="=">
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       <constant symbol="s_1"/>
       <constant symbol="s_2"/>
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    <constantatomformula symbol="=">
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   <formulalist>
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  <sequent>
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    <constantatomformula symbol="=">
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      <function symbol="∩">
       <constant symbol="s_1"/>
       <constant symbol="s_2"/>
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    <constantatomformula symbol="=">
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     <function symbol="∩">
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       <constant symbol="s_4"/>
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   <formulalist>
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      <constant symbol="s_1"/>
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     <function symbol="∩">
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    <constantatomformula symbol="=">
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     <function symbol="∩">
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       <constant symbol="s_3"/>
       <constant symbol="s_4"/>
      </function>
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   </formulalist>
  </sequent>
  <sequent>
   <formulalist>
    <constantatomformula symbol="=">
     <function symbol="∩">
      <variable symbol="x_0"/>
      <variable symbol="z_0"/>
     </function>
     <variable symbol="x_0"/>
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    <constantatomformula symbol="=">
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      <variable symbol="y_0"/>
      <variable symbol="z_0"/>
     </function>
     <variable symbol="y_0"/>
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   <formulalist>
    <constantatomformula symbol="=">
     <function symbol="∩">
      <function symbol="∪">
       <variable symbol="x_0"/>
       <variable symbol="y_0"/>
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      <variable symbol="z_0"/>
     </function>
     <function symbol="∪">
      <variable symbol="x_0"/>
      <variable symbol="y_0"/>
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  </sequent>
  <sequent>
   <formulalist/>
   <formulalist>
    <constantatomformula symbol="=">
     <function symbol="∩">
      <variable symbol="y_0"/>
      <function symbol="∪">
       <variable symbol="y_0"/>
       <variable symbol="x_0"/>
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  </sequent>
  <sequent>
   <formulalist/>
   <formulalist>
    <constantatomformula symbol="=">
     <function symbol="∪">
      <variable symbol="x_0"/>
      <variable symbol="y_0"/>
     </function>
     <function symbol="∪">
      <variable symbol="y_0"/>
      <variable symbol="x_0"/>
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    </constantatomformula>
   </formulalist>
  </sequent>
  <sequent>
   <formulalist/>
   <formulalist>
    <constantatomformula symbol="=">
     <function symbol="∩">
      <variable symbol="x_0"/>
      <function symbol="∪">
       <variable symbol="x_0"/>
       <variable symbol="y_0"/>
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     <variable symbol="x_0"/>
    </constantatomformula>
   </formulalist>
  </sequent>
  <sequent>
   <formulalist>
    <constantatomformula symbol="=">
     <function symbol="∩">
      <variable symbol="z_0"/>
      <variable symbol="y_0"/>
     </function>
     <variable symbol="z_0"/>
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   <formulalist>
    <constantatomformula symbol="=">
     <function symbol="∩">
      <variable symbol="z_0"/>
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     <variable symbol="z_0"/>
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   </formulalist>
  </sequent>
  <sequent>
   <formulalist>
    <constantatomformula symbol="=">
     <function symbol="∩">
      <variable symbol="z_0"/>
      <variable symbol="x_0"/>
     </function>
     <variable symbol="z_0"/>
    </constantatomformula>
   </formulalist>
   <formulalist>
    <constantatomformula symbol="=">
     <function symbol="∩">
      <variable symbol="z_0"/>
      <variable symbol="x_0"/>
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     <variable symbol="z_0"/>
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   </formulalist>
  </sequent>
  <sequent>
   <formulalist/>
   <formulalist>
    <constantatomformula symbol="=">
     <function symbol="∩">
      <function symbol="∩">
       <variable symbol="z_0"/>
       <variable symbol="x_0"/>
      </function>
      <variable symbol="y_0"/>
     </function>
     <function symbol="∩">
      <variable symbol="z_0"/>
      <function symbol="∩">
       <variable symbol="x_0"/>
       <variable symbol="y_0"/>
      </function>
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    </constantatomformula>
   </formulalist>
  </sequent>
  <sequent>
   <formulalist/>
   <formulalist>
    <constantatomformula symbol="=">
     <function symbol="∩">
      <variable symbol="x_0"/>
      <variable symbol="y_0"/>
     </function>
     <function symbol="∩">
      <variable symbol="x_0"/>
      <variable symbol="y_0"/>
     </function>
    </constantatomformula>
   </formulalist>
  </sequent>
  <sequent>
   <formulalist/>
   <formulalist>
    <constantatomformula symbol="=">
     <function symbol="∩">
      <variable symbol="y_0"/>
      <variable symbol="y_0"/>
     </function>
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  </sequent>
  <sequent>
   <formulalist/>
   <formulalist>
    <constantatomformula symbol="=">
     <function symbol="∩">
      <function symbol="∩">
       <variable symbol="x_0"/>
       <variable symbol="y_0"/>
      </function>
      <variable symbol="y_0"/>
     </function>
     <function symbol="∩">
      <variable symbol="x_0"/>
      <function symbol="∩">
       <variable symbol="y_0"/>
       <variable symbol="y_0"/>
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    </constantatomformula>
   </formulalist>
  </sequent>
  <sequent>
   <formulalist/>
   <formulalist>
    <constantatomformula symbol="=">
     <function symbol="∩">
      <variable symbol="x_0"/>
      <variable symbol="y_0"/>
     </function>
     <function symbol="∩">
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      <variable symbol="y_0"/>
     </function>
    </constantatomformula>
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  </sequent>
  <sequent>
   <formulalist/>
   <formulalist>
    <constantatomformula symbol="=">
     <function symbol="∩">
      <variable symbol="x_0"/>
      <variable symbol="x_0"/>
     </function>
     <variable symbol="x_0"/>
    </constantatomformula>
   </formulalist>
  </sequent>
  <sequent>
   <formulalist/>
   <formulalist>
    <constantatomformula symbol="=">
     <function symbol="∩">
      <function symbol="∩">
       <variable symbol="x_0"/>
       <variable symbol="x_0"/>
      </function>
      <variable symbol="y_0"/>
     </function>
     <function symbol="∩">
      <variable symbol="x_0"/>
      <function symbol="∩">
       <variable symbol="x_0"/>
       <variable symbol="y_0"/>
      </function>
     </function>
    </constantatomformula>
   </formulalist>
  </sequent>
  <sequent>
   <formulalist/>
   <formulalist>
    <constantatomformula symbol="=">
     <function symbol="∩">
      <function symbol="∩">
       <variable symbol="x_0"/>
       <variable symbol="y_0"/>
      </function>
      <variable symbol="x_0"/>
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     <function symbol="∩">
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      <function symbol="∩">
       <variable symbol="x_0"/>
       <variable symbol="y_0"/>
      </function>
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    </constantatomformula>
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  </sequent>
  <sequent>
   <formulalist/>
   <formulalist>
    <constantatomformula symbol="=">
     <function symbol="∩">
      <function symbol="∩">
       <variable symbol="x_0"/>
       <variable symbol="y_0"/>
      </function>
      <variable symbol="z_0"/>
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     <function symbol="∩">
      <variable symbol="x_0"/>
      <function symbol="∩">
       <variable symbol="y_0"/>
       <variable symbol="z_0"/>
      </function>
     </function>
    </constantatomformula>
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  </sequent>
  <sequent>
   <formulalist>
    <constantatomformula symbol="=">
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      <variable symbol="y_0"/>
      <variable symbol="z_0"/>
     </function>
     <variable symbol="y_0"/>
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   <formulalist>
    <constantatomformula symbol="=">
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  </sequent>
  <sequent>
   <formulalist>
    <constantatomformula symbol="=">
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      <variable symbol="x_0"/>
      <variable symbol="y_0"/>
     </function>
     <variable symbol="x_0"/>
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   <formulalist>
    <constantatomformula symbol="=">
     <function symbol="∩">
      <variable symbol="x_0"/>
      <variable symbol="y_0"/>
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  </sequent>
  <sequent>
   <formulalist>
    <constantatomformula symbol="=">
     <variable symbol="x_0"/>
     <variable symbol="y_0"/>
    </constantatomformula>
   </formulalist>
   <formulalist>
    <constantatomformula symbol="=">
     <variable symbol="x_0"/>
     <variable symbol="y_0"/>
    </constantatomformula>
   </formulalist>
  </sequent>
  <sequent>
   <formulalist>
    <constantatomformula symbol="=">
     <function symbol="∩">
      <variable symbol="x_0"/>
      <variable symbol="y_0"/>
     </function>
     <variable symbol="x_0"/>
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   <formulalist>
    <constantatomformula symbol="=">
     <function symbol="∩">
      <variable symbol="x_0"/>
      <variable symbol="y_0"/>
     </function>
     <variable symbol="x_0"/>
    </constantatomformula>
   </formulalist>
  </sequent>
  <sequent>
   <formulalist/>
   <formulalist>
    <constantatomformula symbol="=">
     <function symbol="∩">
      <variable symbol="x_0"/>
      <variable symbol="y_0"/>
     </function>
     <function symbol="∩">
      <variable symbol="y_0"/>
      <variable symbol="x_0"/>
     </function>
    </constantatomformula>
   </formulalist>
  </sequent>
  <sequent>
   <formulalist/>
   <formulalist>
    <constantatomformula symbol="=">
     <function symbol="∩">
      <variable symbol="x_0"/>
      <variable symbol="x_0"/>
     </function>
     <variable symbol="x_0"/>
    </constantatomformula>
   </formulalist>
  </sequent>
 </sequentlist>
 <sequentlist symbol="herbrand sequent">
  <sequent>
   <formulalist>
    <conjunctiveformula type="impl">
     <constantatomformula symbol="=">
      <function symbol="∪">
       <constant symbol="s_1"/>
       <function symbol="∪">
        <constant symbol="s_1"/>
        <function symbol="∩">
         <constant symbol="s_1"/>
         <constant symbol="s_2"/>
        </function>
       </function>
      </function>
      <function symbol="∪">
       <constant symbol="s_1"/>
       <function symbol="∩">
        <constant symbol="s_1"/>
        <constant symbol="s_2"/>
       </function>
      </function>
     </constantatomformula>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <constant symbol="s_1"/>
       <function symbol="∪">
        <constant symbol="s_1"/>
        <function symbol="∩">
         <constant symbol="s_1"/>
         <constant symbol="s_2"/>
        </function>
       </function>
      </function>
      <constant symbol="s_1"/>
     </constantatomformula>
    </conjunctiveformula>
    <conjunctiveformula type="impl">
     <constantatomformula symbol="=">
      <function symbol="∩">
       <constant symbol="s_1"/>
       <constant symbol="s_1"/>
      </function>
      <constant symbol="s_1"/>
     </constantatomformula>
     <constantatomformula symbol="=">
      <function symbol="∪">
       <constant symbol="s_1"/>
       <constant symbol="s_1"/>
      </function>
      <constant symbol="s_1"/>
     </constantatomformula>
    </conjunctiveformula>
    <conjunctiveformula type="impl">
     <constantatomformula symbol="=">
      <function symbol="∩">
       <function symbol="∩">
        <constant symbol="s_1"/>
        <constant symbol="s_2"/>
       </function>
       <constant symbol="s_1"/>
      </function>
      <function symbol="∩">
       <constant symbol="s_1"/>
       <constant symbol="s_2"/>
      </function>
     </constantatomformula>
     <constantatomformula symbol="=">
      <function symbol="∪">
       <function symbol="∩">
        <constant symbol="s_1"/>
        <constant symbol="s_2"/>
       </function>
       <constant symbol="s_1"/>
      </function>
      <constant symbol="s_1"/>
     </constantatomformula>
    </conjunctiveformula>
    <conjunctiveformula type="impl">
     <constantatomformula symbol="=">
      <function symbol="∪">
       <function symbol="∪">
        <constant symbol="s_1"/>
        <function symbol="∩">
         <constant symbol="s_1"/>
         <constant symbol="s_2"/>
        </function>
       </function>
       <constant symbol="s_1"/>
      </function>
      <constant symbol="s_1"/>
     </constantatomformula>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <function symbol="∪">
        <constant symbol="s_1"/>
        <function symbol="∩">
         <constant symbol="s_1"/>
         <constant symbol="s_2"/>
        </function>
       </function>
       <constant symbol="s_1"/>
      </function>
      <function symbol="∪">
       <constant symbol="s_1"/>
       <function symbol="∩">
        <constant symbol="s_1"/>
        <constant symbol="s_2"/>
       </function>
      </function>
     </constantatomformula>
    </conjunctiveformula>
    <conjunctiveformula type="impl">
     <constantatomformula symbol="=">
      <function symbol="∪">
       <constant symbol="s_3"/>
       <function symbol="∪">
        <constant symbol="s_3"/>
        <constant symbol="s_4"/>
       </function>
      </function>
      <function symbol="∪">
       <constant symbol="s_3"/>
       <constant symbol="s_4"/>
      </function>
     </constantatomformula>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <constant symbol="s_3"/>
       <function symbol="∪">
        <constant symbol="s_3"/>
        <constant symbol="s_4"/>
       </function>
      </function>
      <constant symbol="s_3"/>
     </constantatomformula>
    </conjunctiveformula>
   </formulalist>
   <formulalist>
    <constantatomformula symbol="L2_{s_1,s_2,s_3,s_4}" definition="L2"/>
    <conjunctiveformula type="and">
     <constantatomformula symbol="=">
      <function symbol="∪">
       <function symbol="∩">
        <constant symbol="s_1"/>
        <constant symbol="s_2"/>
       </function>
       <constant symbol="s_1"/>
      </function>
      <constant symbol="s_1"/>
     </constantatomformula>
     <constantatomformula symbol="=">
      <function symbol="∩">
       <function symbol="∪">
        <constant symbol="s_3"/>
        <constant symbol="s_4"/>
       </function>
       <constant symbol="s_3"/>
      </function>
      <constant symbol="s_3"/>
     </constantatomformula>
    </conjunctiveformula>
   </formulalist>
  </sequent>
 </sequentlist>
 <variabledefinitions/>
</proofdatabase>
