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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">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/>
<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">v</font> be   <a href="msafree2.html#NM1">Vertex</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>consider </b><font color="Maroon">t</font> being    <a href="trees_3.html#M3">Element</a> of <font color="Maroon">X</font>, <font color="Maroon">p</font> being   <a href="trees_4.html#NM1">Node</a> of <font color="Maroon">t</font><b> such that </b><br/><a NAME="E2:7_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="trees_2.html#K5">|</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="trees_9.html#T20">TREES_9:20</a></i>;<br/><a NAME="E3:7_1"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><b>hence </b><a NAME="E4:7_1"/>
<font color="Maroon">v</font> is   <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">S</font>,<font color="Maroon">V</font>
 <b>by </b><i>, <a class="ref" href="msaterm.html#T29">MSATERM:29</a></i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">s</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E2:7"/>
<font color="Maroon">s</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>
 ;<br/>

<b>then consider </b><font color="Maroon">t</font> being    <a href="trees_3.html#M3">Element</a> of <font color="Maroon">X</font>, <font color="Maroon">p</font> being   <a href="trees_4.html#NM1">Node</a> of <font color="Maroon">t</font><b> such that </b><br/><a NAME="E4:7"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="trees_2.html#K5">|</a> <font color="Maroon">p</font> &amp; (  not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K3">Leaves</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">t</font></span>)</span> or <font color="Maroon">t</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> <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="trees_9.html#T25">TREES_9:25</a></i>;<br/>
<a NAME="E5:7"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">s</font> = <font color="Maroon">s</font> as   <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">S</font>,<font color="Maroon">V</font> <b>by </b><i>, <a class="ref" href="msaterm.html#T29">MSATERM:29</a></i>;<br/>
<b>reconsider </b><font color="Maroon">e</font> =  <a href="xboole_0.html#K1">{}</a>  as   <a href="trees_4.html#NM1">Node</a> of <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="trees_2.html#K5">|</a> <font color="Maroon">p</font></span>)</span> <b>by </b><i><a class="ref" href="trees_1.html#T47">TREES_1:47</a></i>;<br/>
<a NAME="E8:7"/>
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="trees_2.html#K5">|</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">t</font></span>)</span> <a href="trees_1.html#K4">|</a> <font color="Maroon">p</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">e</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a>, <a class="ref" href="trees_2.html#D11">TREES_2:def 11</a></i>;<br/>
<a NAME="E9:7"/><b>then </b>
( <font color="Maroon">t</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> &amp; ( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K3">Leaves</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">t</font></span>)</span> implies <font color="Maroon">s</font> is <a href="trees_9.html#V1">root</a> ) &amp; ( <font color="Maroon">s</font> is <a href="trees_9.html#V1">root</a> implies <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K3">Leaves</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">t</font></span>)</span> ) )
 
<b>by </b><i>, <a class="ref" href="trees_2.html#D11">TREES_2:def 11</a>, <a class="ref" href="trees_9.html#T6">TREES_9:6</a></i>;<br/>
<b>hence </b><a NAME="E10:7"/>
<font color="Maroon">s</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#T28">MSATERM:28</a>, <a class="ref" href="msaterm.html#D6">MSATERM:def 6</a></i>;<br/>


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