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<b>let </b><font color="Maroon" title="c1">S</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> non <a href="msualg_1.html#V2" title="MSUALG_1:attr.2">void</a>  <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a> ;<br/><b>let </b><font color="Maroon" title="c2">V</font> be  <a href="relat_1.html#V2" title="RELAT_1:attr.2">V5</a>  <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>;<br/><b>let </b><font color="Maroon" title="c3">X</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="msaterm.html#K1" title="MSATERM:func.1">-Terms</a> <font color="Maroon" title="c2">V</font></span>)</span>;<br/>





<b>set </b><font color="Maroon" title="c4">C</font> = <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c1">S</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>;<br/>
<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon" title="c5">v</font> be   <a href="msafree2.html#NM1" title="MSAFREE2:NM.1">Vertex</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="circtrm1.html#K1" title="CIRCTRM1:func.1">-CircuitStr</a> </span>)</span>;<br/><b>consider </b><font color="Maroon" title="c6">t</font> being    <a href="trees_3.html#M3" title="TREES_3:mode.3">Element</a> of <font color="Maroon" title="c3">X</font>, <font color="Maroon" title="c7">p</font> being   <a href="trees_4.html#NM1" title="TREES_4:NM.1">Node</a> of <font color="Maroon" title="c6">t</font><b> such that </b><br/><a NAME="E2:7_1"/><i><font color="Green" title="E5">A1</font></i>: 
<font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">t</font> <a href="trees_2.html#K5" title="TREES_2:func.5">|</a> <font color="Maroon" title="c7">p</font>
 <b>by </b><i><a class="ref" href="trees_9.html#T20" title="TREES_9:th.20">TREES_9:20</a></i>;<br/><a NAME="E3:7_1"/>
<font color="Maroon" title="c6">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 ;<br/><b>hence </b><a NAME="E4:7_1"/>
<font color="Maroon" title="c5">v</font> is   <a href="msaterm.html#NM1" title="MSATERM:NM.1">Term</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">V</font>
 <b>by </b><i><a class="txt" href="circtrm1.html#E2:7_1"><i><font color="Green" title="E5">A1</font></i></a>, <a class="ref" href="msaterm.html#T29" title="MSATERM:th.29">MSATERM:29</a></i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon" title="c5">s</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E2:7"/>
<font color="Maroon" title="c5">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="circtrm1.html#K1" title="CIRCTRM1:func.1">-CircuitStr</a> </span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c6">t</font> being    <a href="trees_3.html#M3" title="TREES_3:mode.3">Element</a> of <font color="Maroon" title="c3">X</font>, <font color="Maroon" title="c7">p</font> being   <a href="trees_4.html#NM1" title="TREES_4:NM.1">Node</a> of <font color="Maroon" title="c6">t</font><b> such that </b><br/><a NAME="E4:7"/><i><font color="Green" title="E5">A2</font></i>: 
( <font color="Maroon" title="c5">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">t</font> <a href="trees_2.html#K5" title="TREES_2:func.5">|</a> <font color="Maroon" title="c7">p</font> &amp; (  not <font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_1.html#K3" title="TREES_1:func.3">Leaves</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c6">t</font></span>)</span> or <font color="Maroon" title="c6">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c1">S</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> ) )
 <b>by </b><i><a class="ref" href="trees_9.html#T25" title="TREES_9:th.25">TREES_9:25</a></i>;<br/>
<a NAME="E5:7"/>
<font color="Maroon" title="c6">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c8">s</font> = <font color="Maroon" title="c5">s</font> as   <a href="msaterm.html#NM1" title="MSATERM:NM.1">Term</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">V</font> <b>by </b><i><a class="txt" href="circtrm1.html#E4:7"><i><font color="Green" title="E5">A2</font></i></a>, <a class="ref" href="msaterm.html#T29" title="MSATERM:th.29">MSATERM:29</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">e</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  as   <a href="trees_4.html#NM1" title="TREES_4:NM.1">Node</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="trees_2.html#K5" title="TREES_2:func.5">|</a> <font color="Maroon" title="c7">p</font></span>)</span> <b>by </b><i><a class="ref" href="trees_1.html#T47" title="TREES_1:th.47">TREES_1:47</a></i>;<br/>
<a NAME="E8:7"/>
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="trees_2.html#K5" title="TREES_2:func.5">|</a> <font color="Maroon" title="c7">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c6">t</font></span>)</span> <a href="trees_1.html#K4" title="TREES_1:func.4">|</a> <font color="Maroon" title="c7">p</font> &amp; <font color="Maroon" title="c7">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c9">e</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T47" title="FINSEQ_1:th.47">FINSEQ_1:47</a>, <a class="ref" href="trees_2.html#D11" title="TREES_2:def.11">TREES_2:def 11</a></i>;<br/>
<a NAME="E9:7"/><b>then </b>
( <font color="Maroon" title="c6">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e</font> &amp; ( <font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_1.html#K3" title="TREES_1:func.3">Leaves</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c6">t</font></span>)</span> implies <font color="Maroon" title="c8">s</font> is <a href="trees_9.html#V1" title="TREES_9:attr.1">root</a> ) &amp; ( <font color="Maroon" title="c8">s</font> is <a href="trees_9.html#V1" title="TREES_9:attr.1">root</a> implies <font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_1.html#K3" title="TREES_1:func.3">Leaves</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c6">t</font></span>)</span> ) )
 
<b>by </b><i><a class="txt" href="circtrm1.html#E4:7"><i><font color="Green" title="E5">A2</font></i></a>, <a class="ref" href="trees_2.html#D11" title="TREES_2:def.11">TREES_2:def 11</a>, <a class="ref" href="trees_9.html#T6" title="TREES_9:th.6">TREES_9:6</a></i>;<br/>
<b>hence </b><a NAME="E10:7"/>
<font color="Maroon" title="c5">s</font> is    <a href="msaterm.html#M2" title="MSATERM:mode.2">CompoundTerm</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">V</font>
 <b>by </b><i><a class="txt" href="circtrm1.html#E4:7"><i><font color="Green" title="E5">A2</font></i></a>, <a class="ref" href="msaterm.html#T28" title="MSATERM:th.28">MSATERM:28</a>, <a class="ref" href="msaterm.html#D6" title="MSATERM:def.6">MSATERM:def 6</a></i>;<br/>


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