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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">t1</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>, <font color="Maroon">t2</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/>
<a NAME="E1:14"/><i><font color="Green">E36</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2"><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="p3"><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> <a href="trees_9.html#K13">-ImmediateSubtrees</a> <font color="Maroon">t1</font></span>)</span> <a href="hidden.html#R1">=</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> <a href="trees_9.html#K12">-Subtrees</a> <font color="Maroon">t1</font> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p2">(<span class="default"><span class="p3"><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="p4"><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> <a href="trees_9.html#K12">-Subtrees</a> <font color="Maroon">t1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</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> <a href="trees_9.html#K12">-Subtrees</a> <font color="Maroon">t1</font> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2"><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="p3"><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> <a href="trees_9.html#K13">-ImmediateSubtrees</a> <font color="Maroon">t2</font></span>)</span> <a href="hidden.html#R1">=</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> <a href="trees_9.html#K12">-Subtrees</a> <font color="Maroon">t2</font> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p2">(<span class="default"><span class="p3"><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="p4"><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> <a href="trees_9.html#K12">-Subtrees</a> <font color="Maroon">t2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</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> <a href="trees_9.html#K12">-Subtrees</a> <font color="Maroon">t2</font> )
 
<b>by </b><i><a class="ref" href="relat_1.html#T71">RELAT_1:71</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="partfun1.html#D6">PARTFUN1:def 6</a>
<div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:14_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U2">Arity</a> of <span class="p2">(<span class="default"><font color="Maroon">t1</font> <a href="circtrm1.html#K1" target="_self">-CircuitStr</a> </span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U2">Arity</a> of <span class="p2">(<span class="default"><font color="Maroon">t2</font> <a href="circtrm1.html#K1" target="_self">-CircuitStr</a> </span>)</span></span>)</span>
 ;<br/><a NAME="E2:14_1"/><b>then </b><i><font color="Green">E39</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U2">Arity</a> of <span class="p1">(<span class="default"><font color="Maroon">t1</font> <a href="circtrm1.html#K1" target="_self">-CircuitStr</a> </span>)</span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U2">Arity</a> of <span class="p1">(<span class="default"><font color="Maroon">t2</font> <a href="circtrm1.html#K1" target="_self">-CircuitStr</a> </span>)</span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon">u</font> = <font color="Maroon">x</font> as    <a href="trees_3.html#M3">Element</a> of  <a href="trees_9.html#K9">Subtrees</a> <font color="Maroon">t1</font> <b>by </b><i/>;<br/><a NAME="E4:14_1"/>
<span class="p1">(<span class="default"><span class="p2"><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="p3"><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> <a href="trees_9.html#K13">-ImmediateSubtrees</a> <font color="Maroon">t1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="trees_9.html#K9">Subtrees</a> <font color="Maroon">t1</font></span>)</span> <a href="finseq_1.html#K13">*</a> 
 <b>by </b><i>, , <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/><b>then reconsider </b><font color="Maroon">y1</font> = the <a href="msualg_1.html#U2">Arity</a> of <span class="p1">(<span class="default"><font color="Maroon">t1</font> <a href="circtrm1.html#K1" target="_self">-CircuitStr</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="trees_9.html#K9">Subtrees</a> <font color="Maroon">t1</font> <b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/><a NAME="E6:14_1"/>
the <a href="msualg_1.html#U2">Arity</a> of <span class="p1">(<span class="default"><font color="Maroon">t2</font> <a href="circtrm1.html#K1" target="_self">-CircuitStr</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="trees_9.html#K9">Subtrees</a> <font color="Maroon">t2</font></span>)</span> <a href="finseq_1.html#K13">*</a> 
 <b>by </b><i>, , <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/><b>then reconsider </b><font color="Maroon">y2</font> = the <a href="msualg_1.html#U2">Arity</a> of <span class="p1">(<span class="default"><font color="Maroon">t2</font> <a href="circtrm1.html#K1" target="_self">-CircuitStr</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="trees_9.html#K9">Subtrees</a> <font color="Maroon">t2</font> <b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1"><b>now </b></a><div class="add"><div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">t</font> be    <a href="trees_3.html#M3">Element</a> of <font color="Maroon">t1</font>;<br/><a NAME="E1:14_1_1_1"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">t1</font>
 ;<br/><a NAME="E2:14_1_1_1"/><b>then </b>
<font color="Maroon">t</font> is   <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">S</font>,<font color="Maroon">V</font>
 ;<br/><b>hence </b><a NAME="E3:14_1_1_1"/>
<font color="Maroon">t</font> is <a href="finset_1.html#V1">finite</a>
 ;<br/></div>
<b>end;</b></div><div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">t</font> be    <a href="trees_3.html#M3">Element</a> of <font color="Maroon">t2</font>;<br/><a NAME="E1:14_1_1_2"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">t2</font>
 ;<br/><a NAME="E2:14_1_1_2"/><b>then </b>
<font color="Maroon">t</font> is   <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">S</font>,<font color="Maroon">V</font>
 ;<br/><b>hence </b><a NAME="E3:14_1_1_2"/>
<font color="Maroon">t</font> is <a href="finset_1.html#V1">finite</a>
 ;<br/></div>
<b>end;</b></div></div><b>end;</b></div><a NAME="E9:14_1"/><b>then </b>
( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a> </span>)</span> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">y1</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a> </span>)</span> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">y2</font> )
 <b>by </b><i>, , <a class="ref" href="trees_9.html#D13">TREES_9:def 13</a></i>;<br/><b>hence </b><a NAME="E10:14_1"/>
the <a href="msualg_1.html#U2">Arity</a> of <span class="p1">(<span class="default"><font color="Maroon">t1</font> <a href="circtrm1.html#K1" target="_self">-CircuitStr</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U2">Arity</a> of <span class="p1">(<span class="default"><font color="Maroon">t2</font> <a href="circtrm1.html#K1" target="_self">-CircuitStr</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="trees_4.html#T15">TREES_4:15</a></i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="partfun1.html#D6">PARTFUN1:def 6</a><br/>

<b>assume </b><a NAME="E3:14"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U3">ResultSort</a> of <span class="p2">(<span class="default"><font color="Maroon">t1</font> <a href="circtrm1.html#K1" target="_self">-CircuitStr</a> </span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U3">ResultSort</a> of <span class="p2">(<span class="default"><font color="Maroon">t2</font> <a href="circtrm1.html#K1" target="_self">-CircuitStr</a> </span>)</span></span>)</span>
 ;<br/>

<a NAME="E4:14"/><b>then </b><i><font color="Green">E42</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U3">ResultSort</a> of <span class="p1">(<span class="default"><font color="Maroon">t1</font> <a href="circtrm1.html#K1" target="_self">-CircuitStr</a> </span>)</span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U3">ResultSort</a> of <span class="p1">(<span class="default"><font color="Maroon">t2</font> <a href="circtrm1.html#K1" target="_self">-CircuitStr</a> </span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b>the <a href="msualg_1.html#U3">ResultSort</a> of <span class="p1">(<span class="default"><font color="Maroon">t1</font> <a href="circtrm1.html#K1" target="_self">-CircuitStr</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>
<br/>.= 
the <a href="msualg_1.html#U3">ResultSort</a> of <span class="p1">(<span class="default"><font color="Maroon">t2</font> <a href="circtrm1.html#K1" target="_self">-CircuitStr</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>
;<br/><br/>


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