<?xml version="1.0"?>
<div class="add">

<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">A</font> be  <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V4">locally-finite</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">V</font> be    <a href="msaterm.html#M4">Variables</a> of <font color="Maroon">A</font>;<br/><b>let </b><font color="Maroon">X</font> be    <a href="msaterm.html#M3">SetWithCompoundTerm</a> of <font color="Maroon">S</font>,<font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">q</font> be   <a href="circuit1.html#NM3">State</a> of <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="circtrm1.html#K6" target="_self">-Circuit</a> <font color="Maroon">A</font></span>)</span>;<br/><b>let </b><font color="Maroon">f</font> be    <a href="circtrm1.html#M1" target="_self">CompatibleValuation</a> of <font color="Maroon">q</font>;<br/>











<a NAME="E1:40"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">X</font> <a href="circtrm1.html#K1" target="_self">-CircuitStr</a>  <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#G1">ManySortedSign</a>(# <span class="p1">(<span class="default"><a href="trees_9.html#K9">Subtrees</a> <font color="Maroon">X</font></span>)</span>,<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#K12">-Subtrees</a> <font color="Maroon">X</font></span>)</span>,<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">X</font></span>)</span>,<span class="p1">(<span class="default"><a href="funct_3.html#K6">incl</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">X</font></span>)</span></span>)</span> #)
 
;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[ <a href="finset_1.html#V1">finite</a> <a href="trees_2.html#NM4">DecoratedTree</a>] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="trees_9.html#K9">Subtrees</a> <font color="Maroon">X</font> implies (  <a href="facirc_1.html#K9">Following</a> <font color="Maroon">q</font>,<span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="trees_1.html#K6">height</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="facirc_1.html#R1">is_stable_at</a> <font color="Maroon">a<sub>1</sub></font> &amp; <span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">q</font>,<span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><a href="trees_1.html#K6">height</a> <span class="p4">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> <a href="circtrm1.html#K7" target="_self">@</a> <font color="Maroon">f</font>,<font color="Maroon">A</font> ) );<br/>
<a NAME="E2:40"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">s</font> being  <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font><br/> for <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> holds  <font color="Maroon">S<sub>1</sub></font>[ <a href="trees_4.html#K1">root-tree</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>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s</font> be   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">v</font> be    <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>assume </b><a NAME="E1:40_1"/><i><font color="Green">E42</font></i>: 
 <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> <a href="hidden.html#R2">in</a>  <a href="trees_9.html#K9">Subtrees</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E2:40_1"/><b>then </b><i><font color="Green">E65</font></i>: 
 <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> <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>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:40_1"/>
 <a href="facirc_1.html#K9">Following</a> <font color="Maroon">q</font>,<span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="trees_1.html#K6">height</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p4">(<span class="default"><a href="trees_4.html#K1">root-tree</a> <span class="p5"><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></span>)</span></span>)</span></span>)</span></span>)</span> <a href="facirc_1.html#R1">is_stable_at</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="facirc_1.html#T18">FACIRC_1:18</a></i>;<br/>

<b>reconsider </b><font color="Maroon">t</font> =  <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> as   <a href="msaterm.html#NM3">c-Term</a> of <font color="Maroon">A</font>,<font color="Maroon">V</font> <b>by </b><i><a class="ref" href="msaterm.html#T8">MSATERM:8</a></i>;<br/>
<a NAME="E5:40_1"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">t</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#T4">MSATERM:4</a></i>;<br/>
<a NAME="E6:40_1"/>
<font color="Maroon">q</font> <a href="facirc_1.html#R1">is_stable_at</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="facirc_1.html#T18">FACIRC_1:18</a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">q</font>,<span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><a href="trees_1.html#K6">height</a> <span class="p4">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p5">(<span class="default"><a href="trees_4.html#K1">root-tree</a> <span class="p0"><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></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="trees_4.html#K1">root-tree</a> <span class="p2"><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></span>)</span> = 
<font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="trees_4.html#K1">root-tree</a> <span class="p2"><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></span>)</span>
<b>by </b><i><a class="ref" href="facirc_1.html#D9">FACIRC_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font>
<b>by </b><i><a class="ref" href="circtrm1.html#T1" target="_self">Th1</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="msaterm.html#K5">-term</a> <font color="Maroon">A</font></span>)</span> <a href="msaterm.html#K9">@</a> <font color="Maroon">f</font>
<b>by </b><i><a class="ref" href="msaterm.html#T42">MSATERM:42</a></i>
<br/>.= 
<font color="Maroon">t</font> <a href="msaterm.html#K9">@</a> <font color="Maroon">f</font>
<b>by </b><i><a class="ref" href="msaterm.html#D4">MSATERM:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="trees_4.html#K1">root-tree</a> <span class="p2"><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></span>)</span> <a href="circtrm1.html#K7" target="_self">@</a> <font color="Maroon">f</font>,<font color="Maroon">A</font>
<b>by </b><i>, </i>
;<br/><br/>


</div><b>end;</b></div>
<a NAME="E3:40"/><i><font color="Green">E72</font></i>: 
for <font color="Olive">o</font> being  <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font><br/> for <font color="Olive">p</font> being   <a href="msaterm.html#M1">ArgumentSeq</a> of  <a href="msaterm.html#K2">Sym</a> <font color="Olive">o</font>,<font color="Maroon">V</font>  st ( for <font color="Olive">t</font> being  <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">S</font>,<font color="Maroon">V</font>  st <font color="Olive">t</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Olive">p</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">t</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Olive">p</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">o</font> be   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">p</font> be    <a href="msaterm.html#M1">ArgumentSeq</a> of  <a href="msaterm.html#K2">Sym</a> <font color="Maroon">o</font>,<font color="Maroon">V</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:40_2"/><i><font color="Green">E73</font></i>: 
for <font color="Olive">t</font> being  <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">S</font>,<font color="Maroon">V</font>  st <font color="Olive">t</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font> &amp; <font color="Olive">t</font> <a href="hidden.html#R2">in</a>  <a href="trees_9.html#K9">Subtrees</a> <font color="Maroon">X</font> holds <br/>(  <a href="facirc_1.html#K9">Following</a> <font color="Maroon">q</font>,<span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="trees_1.html#K6">height</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Olive">t</font></span>)</span></span>)</span></span>)</span> <a href="facirc_1.html#R1">is_stable_at</a> <font color="Olive">t</font> &amp; <span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">q</font>,<span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><a href="trees_1.html#K6">height</a> <span class="p4">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Olive">t</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Olive">t</font> <a href="circtrm1.html#K7" target="_self">@</a> <font color="Maroon">f</font>,<font color="Maroon">A</font> )
 <b>and </b><br/><a NAME="E2:40_2"/><i><font color="Green">E75</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="trees_9.html#K9">Subtrees</a> <font color="Maroon">X</font>
 ;<br/>

<b>consider </b><font color="Maroon">tt</font> being    <a href="trees_3.html#M3">Element</a> of <font color="Maroon">X</font>, <font color="Maroon">n</font> being   <a href="trees_4.html#NM1">Node</a> of <font color="Maroon">tt</font><b> such that </b><br/><a NAME="E4:40_2"/><i><font color="Green">E100</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">tt</font> <a href="trees_2.html#K5">|</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="trees_9.html#T20">TREES_9:20</a></i>;<br/>
<a NAME="E5:40_2"/>
(  <a href="finseq_1.html#K6">&lt;*&gt;</a> <a href="numbers.html#K5">NAT</a>  <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">tt</font></span>)</span> <a href="trees_1.html#K4">|</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="finseq_1.html#K7">^</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a>, <a class="ref" href="trees_1.html#T47">TREES_1:47</a></i>;<br/>
<b>then </b><font color="Maroon">tt</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon">tt</font> <a href="trees_2.html#K5">|</a> <font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="ref" href="trees_2.html#D11">TREES_2:def 11</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, <a class="ref" href="trees_4.html#D4">TREES_4:def 4</a></i>
;<br/>
<a NAME="E7:40_2"/><b>then </b>
<font color="Maroon">tt</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</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="zfmisc_1.html#T129">ZFMISC_1:129</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">g</font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">p</font> as   <a href="circcomb.html#NM1">Gate</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="trees_9.html#T25">TREES_9:25</a></i>;<br/>
<a NAME="E9:40_2"/><i><font color="Green">E105</font></i>: 
 <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T34">FUNCT_1:34</a></i>;<br/>
<a NAME="E10:40_2"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">g</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">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i><a class="ref" href="trees_4.html#D4">TREES_4:def 4</a></i>;<br/>
<a NAME="E11:40_2"/>
<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g</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> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">g</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E12:40_2"/><b>then </b><i><font color="Green">E107</font></i>: 
 <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="trees_4.html#T15">TREES_4:15</a></i>;<br/>
<a NAME="E13:40_2"/><i><font color="Green">E108</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">g</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="trees_9.html#K9">Subtrees</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E14:40_2"/><i><font color="Green">E109</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="trees_3.html#K13">tree</a> <span class="p1">(<span class="default"><a href="trees_4.html#K11">doms</a> <font color="Maroon">p</font></span>)</span>
 
<b>by </b><i><a class="ref" href="trees_4.html#T10">TREES_4:10</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_2"><b>now </b></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="E1:40_2_2"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">g</font></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon">t</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">X</font> <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">t</font> = <font color="Maroon">t</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="E4:40_2_2"/><i><font color="Green">E124</font></i>: 
(  <a href="facirc_1.html#K9">Following</a> <font color="Maroon">q</font>,<span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="trees_1.html#K6">height</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">t</font></span>)</span></span>)</span></span>)</span> <a href="facirc_1.html#R1">is_stable_at</a> <font color="Maroon">t</font> &amp; <span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">q</font>,<span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><a href="trees_1.html#K6">height</a> <span class="p4">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">t</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="circtrm1.html#K7" target="_self">@</a> <font color="Maroon">f</font>,<font color="Maroon">A</font> )
 <b>by </b><i>, , </i>;<br/><b>consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:40_2_2"/><i><font color="Green">E126</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> &amp; <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> )
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E7:40_2_2"/>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="trees_4.html#K11">doms</a> <font color="Maroon">p</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="trees_4.html#K11">doms</a> <font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">t</font> )
 <b>by </b><i><a class="txt" href="circtrm1.html#E2:40"><i><font color="Green">E39</font></i></a>, <a class="ref" href="funct_6.html#T31">FUNCT_6:31</a></i>;<br/><a NAME="E8:40_2_2"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="trees_4.html#K11">doms</a> <font color="Maroon">p</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E9:40_2_2"/><b>then </b>
 <a href="trees_1.html#K6">height</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">t</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="trees_1.html#K6">height</a> <span class="p1">(<span class="default"><a href="trees_3.html#K13">tree</a> <span class="p2">(<span class="default"><a href="trees_4.html#K11">doms</a> <font color="Maroon">p</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="trees_3.html#T81">TREES_3:81</a></i>;<br/><a NAME="E10:40_2_2"/><b>then </b>
1 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="trees_1.html#K6">height</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">t</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="trees_1.html#K6">height</a> <span class="p1">(<span class="default"><a href="trees_3.html#K13">tree</a> <span class="p2">(<span class="default"><a href="trees_4.html#K11">doms</a> <font color="Maroon">p</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>then consider </b><font color="Maroon">i</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E12:40_2_2"/><i><font color="Green">E127</font></i>: 
 <a href="trees_1.html#K6">height</a> <span class="p1">(<span class="default"><a href="trees_3.html#K13">tree</a> <span class="p2">(<span class="default"><a href="trees_4.html#K11">doms</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="trees_1.html#K6">height</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">t</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><b>reconsider </b><font color="Maroon">i</font> = <font color="Maroon">i</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><a NAME="E14:40_2_2"/>
 <a href="facirc_1.html#K9">Following</a> <font color="Maroon">q</font>,<span class="p1">(<span class="default"><a href="trees_1.html#K6">height</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p3">(<span class="default"><span class="p4"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="facirc_1.html#K9">Following</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">q</font>,<span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><a href="trees_1.html#K6">height</a> <span class="p4">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">t</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon">i</font>
 <b>by </b><i><a class="txt" href="circtrm1.html#E1:40"><i><font color="Green">E36</font></i></a>, , <a class="ref" href="facirc_1.html#T13">FACIRC_1:13</a></i>;<br/><b>hence </b><a NAME="E15:40_2_2"/>
 <a href="facirc_1.html#K9">Following</a> <font color="Maroon">q</font>,<span class="p1">(<span class="default"><a href="trees_1.html#K6">height</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p3">(<span class="default"><span class="p4"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span> <a href="facirc_1.html#R1">is_stable_at</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="facirc_1.html#T17">FACIRC_1:17</a></i>;<br/></div><b>end;</b></div>
<a NAME="E16:40_2"/><b>then </b>
 <a href="circuit2.html#K6">Following</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">q</font>,<span class="p2">(<span class="default"><a href="trees_1.html#K6">height</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p4">(<span class="default"><span class="p5"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="facirc_1.html#R1">is_stable_at</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">p</font>
 
<b>by </b><i>, <a class="ref" href="facirc_1.html#T19">FACIRC_1:19</a></i>;<br/>
<b>hence </b><a NAME="E17:40_2"/>
 <a href="facirc_1.html#K9">Following</a> <font color="Maroon">q</font>,<span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="trees_1.html#K6">height</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p4">(<span class="default"><span class="p5"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="facirc_1.html#R1">is_stable_at</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="facirc_1.html#T12">FACIRC_1:12</a></i>;<br/>

<b>reconsider </b><font color="Maroon">t</font> = <span class="p1">(<span class="default"><a href="msaterm.html#K2">Sym</a> <font color="Maroon">o</font>,<font color="Maroon">V</font></span>)</span> <a href="msaterm.html#K6">-tree</a> <font color="Maroon">p</font> as   <a href="msaterm.html#NM3">c-Term</a> of <font color="Maroon">A</font>,<font color="Maroon">V</font> <b>by </b><i><a class="ref" href="msaterm.html#T27">MSATERM:27</a></i>;<br/>
<a NAME="E19:40_2"/><i><font color="Green">E128</font></i>: 
(  <a href="msaterm.html#K2">Sym</a> <font color="Maroon">o</font>,<span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="pboole.html#K2">\/</a> <font color="Maroon">V</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> &amp;  <a href="msaterm.html#K2">Sym</a> <font color="Maroon">o</font>,<font color="Maroon">V</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> )
 
<b>by </b><i><a class="ref" href="msafree.html#D11">MSAFREE:def 11</a></i>;<br/>
<a NAME="E20:40_2"/><i><font color="Green">E129</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="msafree.html#D11">MSAFREE:def 11</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">q</font>,<span class="p2">(<span class="default"><a href="trees_1.html#K6">height</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">t</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<b>consider </b><font color="Maroon">vp</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E22:40_2"/><i><font color="Green">E131</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">vp</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font>
 <b>and </b><br/><a NAME="E23:40_2"/><i><font color="Green">E132</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> holds <br/><font color="Maroon">vp</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">i</font>)
 <b>from </b><i><a class="ref" href="finseq_1.html#S2">FINSEQ_1:sch 2</a>();<br/></i>
<a NAME="E24:40_2"/><i><font color="Green">E133</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">vp</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> &amp;  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="circtrm1.html#E2:40"><i><font color="Green">E39</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E25:40_2"/><i><font color="Green">E134</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span>
 
<b>by </b><i><a class="ref" href="msaterm.html#T22">MSATERM:22</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:40_2_3"/><i><font color="Green">E135</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font>
 ;<br/><b>then reconsider </b><font color="Maroon">t</font> = <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</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#T22">MSATERM:22</a></i>;<br/><b>reconsider </b><font color="Maroon">t'</font> = <font color="Maroon">t</font> as   <a href="msaterm.html#NM3">c-Term</a> of <font color="Maroon">A</font>,<font color="Maroon">V</font> <b>by </b><i><a class="ref" href="msaterm.html#T27">MSATERM:27</a></i>;<br/><b>take </b><font color="Maroon">t'</font> = <font color="Maroon">t'</font>;<br/><a NAME="E4:40_2_3"/>
 <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a>  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">t'</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E5:40_2_3"/>
( <font color="Maroon">t'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp;  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">t'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> )
 <b>by </b><i>, <a class="ref" href="msaterm.html#T23">MSATERM:23</a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">p'</font> = <font color="Maroon">p</font> as   <a href="msaterm.html#NM4">ArgumentSeq</a> of <font color="Maroon">o</font>,<font color="Maroon">A</font>,<font color="Maroon">V</font> <b>by </b><i>, <a class="ref" href="msaterm.html#T24">MSATERM:24</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:40_2_4"/><i><font color="Green">E137</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font>
 ;<br/><b>let </b><font color="Maroon">t</font> be   <a href="msaterm.html#NM3">c-Term</a> of <font color="Maroon">A</font>,<font color="Maroon">V</font>;<br/><b>assume </b><a NAME="E2:40_2_4"/><i><font color="Green">E138</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 ;<br/><a NAME="E3:40_2_4"/><b>then </b><i><font color="Green">E139</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>then reconsider </b><font color="Maroon">tt</font> = <font color="Maroon">t</font> as    <a href="trees_3.html#M3">Element</a> of  <a href="trees_9.html#K9">Subtrees</a> <font color="Maroon">X</font> <b>by </b><i>, </i>;<br/><b>reconsider </b><font color="Maroon">tt</font> = <font color="Maroon">tt</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="E6:40_2_4"/><i><font color="Green">E140</font></i>: 
(  <a href="facirc_1.html#K9">Following</a> <font color="Maroon">q</font>,<span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="trees_1.html#K6">height</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">t</font></span>)</span></span>)</span></span>)</span> <a href="facirc_1.html#R1">is_stable_at</a> <font color="Maroon">tt</font> &amp; <span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">q</font>,<span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><a href="trees_1.html#K6">height</a> <span class="p4">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">t</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">tt</font> <a href="hidden.html#R1">=</a> <font color="Maroon">tt</font> <a href="circtrm1.html#K7" target="_self">@</a> <font color="Maroon">f</font>,<font color="Maroon">A</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E7:40_2_4"/><i><font color="Green">E141</font></i>: 
<font color="Maroon">vp</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">q</font>,<span class="p2">(<span class="default"><a href="trees_1.html#K6">height</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p4">(<span class="default"><span class="p5"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font>
 <b>by </b><i>, , , , <a class="ref" href="circtrm1.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E8:40_2_4"/>
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="trees_4.html#K11">doms</a> <font color="Maroon">p</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="trees_4.html#K11">doms</a> <font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">t</font> )
 <b>by </b><i>, <a class="ref" href="circtrm1.html#T2" target="_self">Th2</a>, <a class="ref" href="funct_6.html#T31">FUNCT_6:31</a></i>;<br/><a NAME="E9:40_2_4"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="trees_4.html#K11">doms</a> <font color="Maroon">p</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E10:40_2_4"/><b>then </b>
 <a href="trees_1.html#K6">height</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">t</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="trees_1.html#K6">height</a> <span class="p1">(<span class="default"><a href="trees_3.html#K13">tree</a> <span class="p2">(<span class="default"><a href="trees_4.html#K11">doms</a> <font color="Maroon">p</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="trees_3.html#T81">TREES_3:81</a></i>;<br/><a NAME="E11:40_2_4"/><b>then </b>
1 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="trees_1.html#K6">height</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">t</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="trees_1.html#K6">height</a> <span class="p1">(<span class="default"><a href="trees_3.html#K13">tree</a> <span class="p2">(<span class="default"><a href="trees_4.html#K11">doms</a> <font color="Maroon">p</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>then consider </b><font color="Maroon">j</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E13:40_2_4"/><i><font color="Green">E142</font></i>: 
 <a href="trees_1.html#K6">height</a> <span class="p1">(<span class="default"><a href="trees_3.html#K13">tree</a> <span class="p2">(<span class="default"><a href="trees_4.html#K11">doms</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="trees_1.html#K6">height</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">t</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><b>reconsider </b><font color="Maroon">j</font> = <font color="Maroon">j</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><a NAME="E15:40_2_4"/>
 <a href="facirc_1.html#K9">Following</a> <font color="Maroon">q</font>,<span class="p1">(<span class="default"><a href="trees_1.html#K6">height</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p3">(<span class="default"><span class="p4"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="facirc_1.html#K9">Following</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">q</font>,<span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><a href="trees_1.html#K6">height</a> <span class="p4">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">t</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon">j</font>
 <b>by </b><i><a class="txt" href="circtrm1.html#E1:40"><i><font color="Green">E36</font></i></a>, , <a class="ref" href="facirc_1.html#T13">FACIRC_1:13</a></i>;<br/><a NAME="E16:40_2_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">q</font>,<span class="p2">(<span class="default"><a href="trees_1.html#K6">height</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p4">(<span class="default"><span class="p5"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">q</font>,<span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><a href="trees_1.html#K6">height</a> <span class="p4">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">t</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font>
 <b>by </b><i><a class="ref" href="circtrm1.html#T3" target="_self">Th3</a>, <a class="ref" href="facirc_1.html#D9">FACIRC_1:def 9</a></i>;<br/><b>hence </b><a NAME="E17:40_2_4"/>
<font color="Maroon">vp</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="msaterm.html#K9">@</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="circtrm1.html#T3" target="_self">Th3</a>, <a class="ref" href="circtrm1.html#T4" target="_self">Th4</a>, </i>;<br/></div><b>end;</b></div>
<a NAME="E29:40_2"/><b>then </b><i><font color="Green">E143</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msaterm.html#K2">Sym</a> <font color="Maroon">o</font>,<span class="p3">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="pboole.html#K2">\/</a> <font color="Maroon">V</font></span>)</span></span>)</span> <a href="msaterm.html#K6">-tree</a> <font color="Maroon">p'</font></span>)</span> <a href="msaterm.html#K9">@</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">vp</font>
 
<b>by </b><i><a class="txt" href="circtrm1.html#E2:40"><i><font color="Green">E39</font></i></a>, <a class="ref" href="msaterm.html#T43">MSATERM:43</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_5"><b>now </b></a><div class="add"><a NAME="E1:40_2_5"/>
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font> <a href="tarski.html#R1">c=</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;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">q</font>,<span class="p2">(<span class="default"><a href="trees_1.html#K6">height</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">t</font></span>)</span></span>)</span></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> )
 <b>by </b><i>, <a class="ref" href="circuit1.html#T4">CIRCUIT1:4</a>, <a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/><b>hence </b><a NAME="E2:40_2_5"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">q</font>,<span class="p3">(<span class="default"><a href="trees_1.html#K6">height</a> <span class="p4">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">t</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/><b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E3:40_2_5"/><i><font color="Green">E144</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon">i</font> = <font color="Maroon">z</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><a NAME="E5:40_2_5"/>
<font color="Maroon">vp</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">q</font>,<span class="p2">(<span class="default"><a href="trees_1.html#K6">height</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">t</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="circtrm1.html#T2" target="_self">Th2</a></i>;<br/><b>hence </b><a NAME="E6:40_2_5"/>
<font color="Maroon">vp</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">q</font>,<span class="p3">(<span class="default"><a href="trees_1.html#K6">height</a> <span class="p4">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">t</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
 <b>by </b><i>, <a class="ref" href="circtrm1.html#T2" target="_self">Th2</a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/></div><b>end;</b></div>
<a NAME="E31:40_2"/><b>then </b><i><font color="Green">E145</font></i>: 
<font color="Maroon">vp</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">q</font>,<span class="p2">(<span class="default"><a href="trees_1.html#K6">height</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">t</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">g</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<a NAME="E32:40_2"/>
 <a href="msualg_1.html#K5">Den</a> <font color="Maroon">g</font>,<span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="circtrm1.html#K6" target="_self">-Circuit</a> <font color="Maroon">A</font></span>)</span> <a href="funct_2.html#R1">=</a>  <a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">A</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E33:40_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">vp</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <span class="p2">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">q</font>,<span class="p3">(<span class="default"><a href="trees_1.html#K6">height</a> <span class="p4">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">t</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font>
 
<b>by </b><i>, , <a class="ref" href="circtrm1.html#T2" target="_self">Th2</a>, <a class="ref" href="facirc_1.html#T10">FACIRC_1:10</a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">q</font>,<span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><a href="trees_1.html#K6">height</a> <span class="p4">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p5">(<span class="default"><span class="p0"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">p</font></span>)</span> = 
<font color="Maroon">t</font> <a href="msaterm.html#K9">@</a> <font color="Maroon">f</font>
<b>by </b><i>, , <a class="ref" href="facirc_1.html#T12">FACIRC_1:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">p</font></span>)</span> <a href="circtrm1.html#K7" target="_self">@</a> <font color="Maroon">f</font>,<font color="Maroon">A</font>
<b>by </b><i>, </i>
;<br/><br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E4:40"/>
for <font color="Olive">t</font> being  <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">S</font>,<font color="Maroon">V</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">t</font>]
 <b>from </b><i><a class="ref" href="msaterm.html#S1">MSATERM:sch 1</a>(, <a class="ref" href="circtrm1.html#T1" target="_self">Th1</a>);</i><br/>


</div>
