<?xml version="1.0" encoding="UTF-8"?>
<!DOCTYPE proofdatabase
SYSTEM "http://www.logic.at/ceres/xml/4.0/proofdatabase.dtd">
<proofdatabase>
 <definitionlist/>
 <axiomset/>
 <proof symbol="p'_5(x_0, y_0)" calculus="LK">
  <rule symbol="d:r" type="defr" param="();[(1)(1 1)(2)][(2)(1 2)];()">
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      <rule symbol="⋀:l2" type="andl2">
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        <formulalist>
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           <formulalist/>
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           <formulalist/>
           <formulalist>
            <constantatomformula symbol="=">
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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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          <rule type="axiom">
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            <formulalist/>
            <formulalist>
             <constantatomformula symbol="=">
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               <variable symbol="x_0"/>
               <variable symbol="x_0"/>
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              <variable symbol="x_0"/>
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          <rule type="axiom">
           <sequent>
            <formulalist/>
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             <constantatomformula symbol="=">
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         <sequent>
          <formulalist>
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         </sequent>
        </rule>
       </rule>
      </rule>
     </rule>
    </rule>
   </rule>
  </rule>
 </proof>
 <proof symbol="p_1" calculus="LK">
  <rule symbol="⋀:r" type="andr">
   <sequent>
    <formulalist/>
    <formulalist>
     <conjunctiveformula type="and">
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   <rule symbol="d:r" type="defr" param="();;()">
    <sequent>
     <formulalist/>
     <formulalist>
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    <rule symbol="∀:r" type="forallr">
     <sequent>
      <formulalist/>
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     <rule symbol="∀:r" type="forallr">
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       <rule symbol="c:l" type="contrl" param="2">
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         <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 symbol="∀:r" type="forallr">
      <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 symbol="⊃:r" type="implr">
       <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>
       </sequent>
       <rule symbol="c:l" type="contrl" 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 symbol="⋀:l1" type="andl1">
         <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 symbol="π:l" type="perml" param="(2 1)">
          <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 symbol="⋀:l2" type="andl2">
           <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 symbol="π:l" type="perml" param="(2 1)">
            <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"/>
              </constantatomformula>
             </formulalist>
            </sequent>
            <rule symbol="d:l" type="defl" 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="z_0"/>
               </constantatomformula>
              </formulalist>
              <formulalist>
               <constantatomformula symbol="≤" definition="≤">
                <variable symbol="x_0"/>
                <variable symbol="z_0"/>
               </constantatomformula>
              </formulalist>
             </sequent>
             <rule symbol="π:l" type="perml" param="(2 1)">
              <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 symbol="d:l" type="defl" 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 symbol="π:l" type="perml" param="(2 1)">
                <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 symbol="d:r" type="defr" 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 symbol="=:r1" type="eqr1" param="(1 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="=">
                     <function symbol="∩">
                      <variable symbol="x_0"/>
                      <variable symbol="z_0"/>
                     </function>
                     <variable symbol="x_0"/>
                    </constantatomformula>
                   </formulalist>
                  </sequent>
                  <rule type="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 symbol="=:r1" type="eqr1" 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">
                    <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">
                    <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>
 </proof>
 <proof symbol="p_3" calculus="LK">
  <rule symbol="⋀:r" type="andr">
   <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 symbol="d:r" type="defr" param="();;()">
    <sequent>
     <formulalist/>
     <formulalist>
      <constantatomformula symbol="GLB" definition="GLB"/>
     </formulalist>
    </sequent>
    <rule symbol="∀:r" type="forallr">
     <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 symbol="∀:r" type="forallr">
      <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>
      </sequent>
      <rule symbol="⋀:r" type="andr">
       <sequent>
        <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 symbol="⋀:r" type="andr">
        <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 symbol="d:r" type="defr" 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>
         <prooflink symbol="h1(x_0, y_0)"/>
        </rule>
        <prooflink symbol="p'_3"/>
       </rule>
       <prooflink symbol="p_4"/>
      </rule>
     </rule>
    </rule>
   </rule>
   <prooflink symbol="p_5"/>
  </rule>
 </proof>
 <proof symbol="p_4" calculus="LK">
  <rule symbol="∀:r" type="forallr">
   <sequent>
    <formulalist/>
    <formulalist>
     <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>
    </formulalist>
   </sequent>
   <rule symbol="⊃:r" type="implr">
    <sequent>
     <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>
    <rule symbol="c:l" type="contrl" param="2">
     <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>
      </formulalist>
      <formulalist>
       <constantatomformula symbol="≤" definition="≤">
        <variable symbol="z_0"/>
        <function symbol="∩">
         <variable symbol="x_0"/>
         <variable symbol="y_0"/>
        </function>
       </constantatomformula>
      </formulalist>
     </sequent>
     <rule symbol="⋀:l1" type="andl1">
      <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 symbol="π:l" type="perml" param="(2 1)">
       <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 symbol="⋀:l2" type="andl2">
        <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 symbol="π:l" type="perml" param="(2 1)">
         <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 symbol="d:l" type="defl" 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 symbol="π:l" type="perml" param="(2 1)">
           <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 symbol="d:l" type="defl" 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 symbol="π:l" type="perml" param="(2 1)">
             <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 symbol="d:r" type="defr" 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 symbol="=:r1" type="eqr1" 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">
                <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 symbol="=:r2" type="eqr2" 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">
                 <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">
                 <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>
 </proof>
 <proof symbol="p_5" calculus="LK">
  <rule symbol="d:r" type="defr" param="();;()">
   <sequent>
    <formulalist>
     <constantatomformula symbol="L1" definition="L1"/>
    </formulalist>
    <formulalist>
     <constantatomformula symbol="LUB" definition="LUB"/>
    </formulalist>
   </sequent>
   <rule symbol="∀:r" type="forallr">
    <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 symbol="∀:r" type="forallr">
     <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 symbol="c:l" type="contrl" 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 symbol="⋀:r" type="andr">
       <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 symbol="c:l" type="contrl" 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 symbol="⋀:r" type="andr">
         <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>
         <prooflink symbol="p'_5(x_0, y_0)"/>
         <rule symbol="=:r2" type="eqr2" 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">
           <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>
          <prooflink symbol="p'_5(y_0, x_0)"/>
         </rule>
        </rule>
       </rule>
       <prooflink symbol="p_6"/>
      </rule>
     </rule>
    </rule>
   </rule>
  </rule>
 </proof>
 <proof symbol="p_6" calculus="LK">
  <rule symbol="∀:r" type="forallr">
   <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 symbol="⊃:r" type="implr">
    <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 symbol="c:l" type="contrl" 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 symbol="π:l" type="perml" 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 symbol="⋀:l1" type="andl1">
       <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 symbol="π:l" type="perml" param="(3 1)">
        <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 symbol="⋀:l2" type="andl2">
         <sequent>
          <formulalist>
           <conjunctiveformula type="and">
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                                </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 symbol="∀:l" type="foralll">
                               <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 symbol="∀:l" type="foralll">
                                <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 symbol="⋀:l2" type="andl2">
                                 <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 symbol="⊃:l" type="impll">
                                  <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 symbol="=:r2" type="eqr2" 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">
                                    <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 symbol="=:r2" type="eqr2" 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">
                                     <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 symbol="c:r" type="contrr" 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 symbol="c:l" type="contrl" 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 symbol="=:r2" type="eqr2" 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">
                                        <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 symbol="w:r" type="weakr">
                                        <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">
                                         <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">
                                   <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>
 </proof>
 <proof symbol="p'_3" calculus="LK">
  <rule symbol="d:r" type="defr" 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>
   <prooflink symbol="h2(x_0, y_0)"/>
  </rule>
 </proof>
 <proof symbol="q_1" calculus="LK">
  <rule symbol="∀:r" type="forallr">
   <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 symbol="∀:r" type="forallr">
    <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 symbol="d:l" type="defl" 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 symbol="∀:l" type="foralll">
      <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 symbol="∀:l" type="foralll">
       <sequent>
        <formulalist>
         <quantifiedformula type="all">
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                   <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>
                  </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="∩">
                       <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 symbol="⋀:l2" type="andl2">
                 <sequent>
                  <formulalist>
                   <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>
                    </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="∩">
                        <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 symbol="π:l" type="perml" param="(4 1)">
                  <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 symbol="⋀:r" type="andr">
                   <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>
                   <prooflink symbol="r_1"/>
                   <rule symbol="d:l" type="defl" 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 symbol="d:r" type="defr" 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">
                      <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">
         <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>
 </proof>
 <proof symbol="q_2" calculus="LK">
  <rule symbol="∀:r" type="forallr">
   <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 symbol="∀:r" type="forallr">
    <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 symbol="d:l" type="defl" 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 symbol="∀:l" type="foralll">
      <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 symbol="∀:l" type="foralll">
       <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 symbol="⊃:l" type="impll">
        <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 symbol="d:l" type="defl" 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 symbol="∀:l" type="foralll">
          <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 symbol="∀:l" type="foralll">
           <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"/>
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                    <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 symbol="c:l" type="contrl" 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 symbol="⋀:l2" type="andl2">
             <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 symbol="π:l" type="perml" param="(2 1)">
              <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 symbol="⋀:l1" type="andl1">
               <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="≤">
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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"/>
                    </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="≤">
                   <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 symbol="⋀:l2" type="andl2">
                <sequent>
                 <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>
                    <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>
                  <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 symbol="⋀:r" type="andr">
                 <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>
                   <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="≤">
                     <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 symbol="d:l" type="defl" 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 symbol="d:r" type="defr" 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">
                    <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>
                 <prooflink symbol="r_2"/>
                </rule>
               </rule>
              </rule>
             </rule>
            </rule>
           </rule>
          </rule>
         </rule>
        </rule>
        <rule type="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>
 </proof>
 <proof symbol="h1(x_0, y_0)" calculus="LK">
  <rule symbol="=:r2" type="eqr2" param="(1)">
   <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"/>
       <variable symbol="y_0"/>
      </function>
     </constantatomformula>
    </formulalist>
   </sequent>
   <rule type="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 symbol="=:r1" type="eqr1" 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">
     <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 symbol="=:r2" type="eqr2" 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">
      <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">
      <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>
 </proof>
 <proof symbol="r_1" calculus="LK">
  <rule symbol="∀:l" type="foralll">
   <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 symbol="⊃:l" type="impll">
    <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 symbol="⋀:r" type="andr">
     <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>
     <prooflink symbol="r'_1"/>
     <rule symbol="d:l" type="defl" 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 symbol="∀:l" type="foralll">
       <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 symbol="d:l" type="defl" 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 symbol="d:r" type="defr" 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">
          <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 symbol="d:l" type="defl" 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 symbol="d:r" type="defr" 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">
       <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>
 </proof>
 <proof symbol="r_2" calculus="LK">
  <rule symbol="∀:l" type="foralll">
   <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>
     <constantatomformula symbol="LUB" definition="LUB"/>
     <constantatomformula symbol="R" definition="R"/>
    </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 symbol="⊃:l" type="impll">
    <sequent>
     <formulalist>
      <conjunctiveformula type="impl">
       <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="x_0"/>
         <variable symbol="x_0"/>
        </constantatomformula>
       </conjunctiveformula>
       <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="LUB" definition="LUB"/>
      <constantatomformula symbol="R" definition="R"/>
     </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 symbol="⋀:r" type="andr">
     <sequent>
      <formulalist>
       <constantatomformula symbol="LUB" definition="LUB"/>
       <constantatomformula symbol="R" definition="R"/>
      </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="x_0"/>
         <variable symbol="x_0"/>
        </constantatomformula>
       </conjunctiveformula>
      </formulalist>
     </sequent>
     <prooflink symbol="r'_2"/>
     <rule symbol="d:l" type="defl" 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 symbol="∀:l" type="foralll">
       <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 symbol="d:l" type="defl" 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 symbol="d:r" type="defr" 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">
          <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 symbol="d:l" type="defl" 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 symbol="d:r" type="defr" 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">
       <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>
 </proof>
 <proof symbol="h2(x_0, y_0)" calculus="LK">
  <rule symbol="=:r2" type="eqr2" 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">
    <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 symbol="=:r2" type="eqr2" 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">
     <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">
     <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="p'_5(y_0, x_0)" calculus="LK">
  <rule symbol="d:r" type="defr" 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 symbol="d:l" type="defl" 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 symbol="∀:l" type="foralll">
     <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 symbol="∀:l" type="foralll">
      <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 symbol="⋀:l2" type="andl2">
       <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 symbol="⊃:l" type="impll">
        <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 symbol="=:r1" type="eqr1" 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">
          <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 symbol="=:r2" type="eqr2" 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">
           <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">
           <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">
         <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>
 </proof>
 <proof symbol="p" calculus="LK">
  <rule symbol="cut" type="cut">
   <sequent>
    <formulalist>
     <constantatomformula symbol="L1" definition="L1"/>
    </formulalist>
    <formulalist>
     <constantatomformula symbol="L2" definition="L2"/>
    </formulalist>
   </sequent>
   <prooflink symbol="p^1_3"/>
   <prooflink symbol="p^3_2"/>
  </rule>
 </proof>
 <proof symbol="r'_1" calculus="LK">
  <rule symbol="d:l" type="defl" 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 symbol="∀:l" type="foralll">
    <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 symbol="∀:l" type="foralll">
     <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 symbol="⋀:l1" type="andl1">
      <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 symbol="⋀:l1" type="andl1">
       <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 symbol="d:l" type="defl" 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 symbol="d:r" type="defr" 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">
          <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>
 </proof>
 <proof symbol="r'_2" calculus="LK">
  <rule symbol="d:l" type="defl" param="();;()">
   <sequent>
    <formulalist>
     <constantatomformula symbol="LUB" definition="LUB"/>
    </formulalist>
    <formulalist>
     <constantatomformula symbol="≤" definition="≤">
      <variable symbol="x_0"/>
      <function symbol="∪">
       <variable symbol="x_0"/>
       <variable symbol="y_0"/>
      </function>
     </constantatomformula>
    </formulalist>
   </sequent>
   <rule symbol="∀:l" type="foralll">
    <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="≤">
       <variable symbol="x_0"/>
       <function symbol="∪">
        <variable symbol="x_0"/>
        <variable symbol="y_0"/>
       </function>
      </constantatomformula>
     </formulalist>
    </sequent>
    <rule symbol="∀:l" type="foralll">
     <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 symbol="⋀:l1" type="andl1">
      <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 symbol="⋀:l1" type="andl1">
       <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 symbol="d:l" type="defl" 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 symbol="d:r" type="defr" 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">
          <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>
 </proof>
 <proof symbol="p^1_3" calculus="LK">
  <rule symbol="d:r" type="defr" param="();;()">
   <sequent>
    <formulalist>
     <constantatomformula symbol="L1" definition="L1"/>
    </formulalist>
    <formulalist>
     <constantatomformula symbol="L3" definition="L3"/>
    </formulalist>
   </sequent>
   <rule symbol="⋀:r" type="andr">
    <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 symbol="d:r" type="defr" param="();;()">
     <sequent>
      <formulalist/>
      <formulalist>
       <constantatomformula symbol="POSET" definition="POSET"/>
      </formulalist>
     </sequent>
     <rule symbol="⋀:r" type="andr">
      <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 symbol="d:r" type="defr" param="();;()">
       <sequent>
        <formulalist/>
        <formulalist>
         <constantatomformula symbol="R" definition="R"/>
        </formulalist>
       </sequent>
       <rule symbol="∀:r" type="forallr">
        <sequent>
         <formulalist/>
         <formulalist>
          <quantifiedformula type="all">
           <variable symbol="x"/>
           <constantatomformula symbol="≤" definition="≤">
            <variable symbol="x"/>
            <variable symbol="x"/>
           </constantatomformula>
          </quantifiedformula>
         </formulalist>
        </sequent>
        <rule symbol="d:r" type="defr" 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">
          <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>
      <prooflink symbol="p_1"/>
     </rule>
    </rule>
    <prooflink symbol="p_3"/>
   </rule>
  </rule>
 </proof>
 <proof symbol="p^3_2" calculus="LK">
  <rule symbol="d:l" type="defl" param="();;()">
   <sequent>
    <formulalist>
     <constantatomformula symbol="L3" definition="L3"/>
    </formulalist>
    <formulalist>
     <constantatomformula symbol="L2" definition="L2"/>
    </formulalist>
   </sequent>
   <rule symbol="c:l" type="contrl" 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 symbol="⋀:l2" type="andl2">
     <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 symbol="c:l" type="contrl" 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 symbol="π:l" type="perml" 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 symbol="⋀:l1" type="andl1">
        <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 symbol="π:l" type="perml" param="(3 1)">
         <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 symbol="⋀:l2" type="andl2">
          <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 symbol="π:l" type="perml" param="(2 1)">
           <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 symbol="⋀:l1" type="andl1">
            <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 symbol="d:l" type="defl" 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 symbol="c:l" type="contrl" 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 symbol="⋀:l1" type="andl1">
               <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 symbol="π:l" type="perml" param="(2 1)">
                <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 symbol="⋀:l2" type="andl2">
                 <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 symbol="⋀:l1" type="andl1">
                  <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 symbol="d:r" type="defr" 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 symbol="c:l" type="contrl" 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 symbol="π:l" type="perml" 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 symbol="⋀:r" type="andr">
                      <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>
                      <prooflink symbol="q_1"/>
                      <prooflink symbol="q_2"/>
                     </rule>
                    </rule>
                   </rule>
                  </rule>
                 </rule>
                </rule>
               </rule>
              </rule>
             </rule>
            </rule>
           </rule>
          </rule>
         </rule>
        </rule>
       </rule>
      </rule>
     </rule>
    </rule>
   </rule>
  </rule>
 </proof>
 <variabledefinitions/>
</proofdatabase>
