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<b>let </b><font color="Maroon">S</font> be  non <a href="struct_0.html#V3">empty</a> non <a href="msualg_1.html#V2">void</a>  <a href="msualg_1.html#L1">ManySortedSign</a> ;<br/><b>let </b><font color="Maroon">V</font> be  <a href="relat_1.html#V2">V5</a>  <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">X</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="msaterm.html#K1">-Terms</a> <font color="Maroon">V</font></span>)</span>;<br/>





<b>set </b><font color="Maroon">G</font> = <font color="Maroon">X</font> <a href="circtrm1.html#K1" target="_self">-CircuitStr</a> ;<br/>
<b>set </b><font color="Maroon">C</font> = <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/>
<a NAME="E1:20"/>
 <a href="relat_1.html#K2">rng</a> the <a href="msualg_1.html#U3">ResultSort</a> of <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="circtrm1.html#K1" target="_self">-CircuitStr</a> </span>)</span> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="circtrm1.html#K1" target="_self">-CircuitStr</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T71">RELAT_1:71</a></i>;<br/>
<a NAME="E2:20"/><b>then </b><i><font color="Green">E36</font></i>: 
 <a href="msafree2.html#K2">InputVertices</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="circtrm1.html#K1" target="_self">-CircuitStr</a> </span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="circtrm1.html#K1" target="_self">-CircuitStr</a> </span>)</span> <a href="subset_1.html#K6">\</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="circtrm1.html#K1" target="_self">-CircuitStr</a> </span>)</span>
 
;<br/>
<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:20_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K2">InputVertices</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="circtrm1.html#K1" target="_self">-CircuitStr</a> </span>)</span>
 ;<br/><a NAME="E2:20_1"/><b>then </b><i><font color="Green">E42</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="circtrm1.html#K1" target="_self">-CircuitStr</a> </span>)</span> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="circtrm1.html#K1" target="_self">-CircuitStr</a> </span>)</span> )
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>thus </b><a NAME="E3:20_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="trees_9.html#K9">Subtrees</a> <font color="Maroon">X</font>
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">t</font> = <font color="Maroon">x</font> as   <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">S</font>,<font color="Maroon">V</font> <b>by </b><i>, </i>;<br/><a NAME="E5:20_1"/>
(  ex <font color="Olive">s</font> being  <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font> ex <font color="Olive">v</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <font color="Olive">s</font> st <font color="Maroon">t</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">v</font>,<font color="Olive">s</font></span><a href="tarski.html#K4">]</a></span> or <font color="Maroon">t</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> )
 <b>by </b><i><a class="ref" href="msaterm.html#T2">MSATERM:2</a></i>;<br/><a NAME="E6:20_1"/><b>then </b>
(  ex <font color="Olive">s</font> being  <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font> ex <font color="Olive">v</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <font color="Olive">s</font> st <font color="Maroon">t</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">v</font>,<font color="Olive">s</font></span><a href="tarski.html#K4">]</a></span> or <font color="Maroon">t</font> is    <a href="msaterm.html#M2">CompoundTerm</a> of <font color="Maroon">S</font>,<font color="Maroon">V</font> )
 <b>by </b><i><a class="ref" href="msaterm.html#D6">MSATERM:def 6</a></i>;<br/><b>then consider </b><font color="Maroon">s</font> being   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font>, <font color="Maroon">v</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font><b> such that </b><br/><a NAME="E8:20_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">t</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">v</font>,<font color="Maroon">s</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, </i>;<br/><b>take </b><font color="Maroon">s</font> = <font color="Maroon">s</font>;<br/><b>reconsider </b><font color="Maroon">v</font> = <font color="Maroon">v</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> ;<br/><b>take </b><font color="Maroon">v</font> = <font color="Maroon">v</font>;<br/><b>thus </b><a NAME="E10:20_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">v</font>,<font color="Maroon">s</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, <a class="ref" href="msaterm.html#T5">MSATERM:5</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E4:20"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="trees_9.html#K9">Subtrees</a> <font color="Maroon">X</font>
 ;<br/>

<b>given </b><font color="Maroon">s</font> being   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font>, <font color="Maroon">v</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font><b> such that </b><a NAME="E5:20"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">v</font>,<font color="Maroon">s</font></span><a href="tarski.html#K4">]</a></span>
 ;<br/>

<b>assume </b><a NAME="E6:20"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K2">InputVertices</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="circtrm1.html#K1" target="_self">-CircuitStr</a> </span>)</span>
 ;<br/>

<a NAME="E7:20"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="circtrm1.html#K1" target="_self">-CircuitStr</a> </span>)</span>
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">t</font> = <font color="Maroon">x</font> as    <a href="msaterm.html#M2">CompoundTerm</a> of <font color="Maroon">S</font>,<font color="Maroon">V</font> <b>by </b><i/>;<br/>
<a NAME="E9:20"/>
<font color="Maroon">t</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">v</font>,<font color="Maroon">s</font></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i>, <a class="ref" href="trees_4.html#T3">TREES_4:3</a></i>;<br/>
<a NAME="E10:20"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">v</font>,<font color="Maroon">s</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="msaterm.html#D6">MSATERM:def 6</a></i>;<br/>
<a NAME="E11:20"/><b>then </b>
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T129">ZFMISC_1:129</a></i>;<br/>
<a NAME="E12:20"/><b>then </b>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">s</font>
 
;<br/>
<b>hence </b><a NAME="E13:20"/>
contradiction
 ;<br/>


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