<?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">X</font> be  <a href="relat_1.html#V2">V2</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">v</font> be   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>







<b>assume </b><a NAME="E1:5"/><i><font color="Green">E41</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 ;<br/>

<b>set </b><font color="Maroon">G</font> =  <a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">S</font>;<br/>
<a NAME="E2:5"/><i><font color="Green">E42</font></i>: 
 <a href="msafree.html#K7">FreeSort</a> <font color="Maroon">X</font>,<font color="Maroon">v</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font> <a href="msaterm.html#K1">-Terms</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="msaterm.html#T12">MSATERM:12</a></i>;<br/>
<a NAME="E3:5"/>
 <a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#G3">MSAlgebra</a>(# <span class="p1">(<span class="default"><a href="msafree.html#K8">FreeSort</a> <font color="Maroon">X</font></span>)</span>,<span class="p1">(<span class="default"><a href="msafree.html#K10">FreeOper</a> <font color="Maroon">X</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="msafree.html#D16">MSAFREE:def 16</a></i>;<br/>
<a NAME="E4:5"/><b>then </b><i><font color="Green">E43</font></i>: 
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="msafree.html#K7">FreeSort</a> <font color="Maroon">X</font>,<font color="Maroon">v</font>
 
<b>by </b><i><a class="ref" href="msafree.html#D13">MSAFREE:def 13</a></i>;<br/>
<b>thus </b><a NAME="E5:5"/>
(  ex <font color="Olive">t</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> st  <a href="msafree2.html#K9">depth</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> implies  ex <font color="Olive">c</font> being <a href="graph_1.html#V8">directed</a>  <a href="graph_1.html#M1">Chain</a> of  <a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">S</font> st <br/>(  <a href="finseq_1.html#K3">len</a> <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp; <span class="p1">(<span class="default"><a href="graph_2.html#K7">vertex-seq</a> <font color="Olive">c</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">c</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon">t</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font><b> such that </b><a NAME="E1:5_1"/><i><font color="Green">E55</font></i>: 
 <a href="msafree2.html#K9">depth</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 ;<br/>

<a NAME="E2:5_1"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="msafree.html#K7">FreeSort</a> <font color="Maroon">X</font>,<font color="Maroon">v</font>
 
<b>by </b><i/>;<br/>
<b>then 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">X</font> <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">dt</font> being  <a href="finset_1.html#V1">finite</a> <a href="trees_2.html#NM4">DecoratedTree</a>, <font color="Maroon">tr</font> being  <a href="finset_1.html#V1">finite</a> <a href="trees_1.html#NM1">Tree</a><b> such that </b><br/><a NAME="E5:5_1"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">dt</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> &amp; <font color="Maroon">tr</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">dt</font> &amp;  <a href="msafree2.html#K9">depth</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a>  <a href="trees_1.html#K6">height</a> <font color="Maroon">tr</font> )
 <b>by </b><i><a class="ref" href="msafree2.html#D14">MSAFREE2:def 14</a></i>;<br/>
<b>consider </b><font color="Maroon">p</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E7:5_1"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">tr</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="trees_1.html#K6">height</a> <font color="Maroon">tr</font> )
 <b>by </b><i><a class="ref" href="trees_1.html#D15">TREES_1:def 15</a></i>;<br/>
<b>set </b><font color="Maroon">D</font> = the <a href="graph_1.html#U2">Edges</a> of <span class="p1">(<span class="default"><a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">S</font></span>)</span> <a href="finseq_1.html#K13">*</a> ;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>,   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">t1</font>, <font color="Olive">t2</font> being  <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">S</font>,<font color="Maroon">X</font> st <br/>( <font color="Olive">t1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="trees_2.html#K5">|</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> &amp; <font color="Olive">t2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="trees_2.html#K5">|</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> &amp;  ex <font color="Olive">o</font> being  <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font> ex <font color="Olive">rs1</font> being  <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font> ex <font color="Olive">Ck</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="graph_1.html#U2">Edges</a> of <span class="p1">(<span class="default"><a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">S</font></span>)</span> <a href="finseq_1.html#K13">*</a>  st <br/>( <font color="Olive">Ck</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> &amp; <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">o</font>,<font color="Olive">rs1</font></span><a href="tarski.html#K4">]</a></span></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <font color="Olive">Ck</font> &amp; <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="hidden.html#R1">=</a> <font color="Olive">t1</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">rs1</font> <a href="hidden.html#R1">=</a>  <a href="msaterm.html#K7">the_sort_of</a> <font color="Olive">t2</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">o</font>,<font color="Olive">rs1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2">Edges</a> of <span class="p1">(<span class="default"><a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">S</font></span>)</span> ) );<br/>
<b>reconsider </b><font color="Maroon">n1</font> = <font color="Maroon">n</font> as    <a href="subset_1.html#M2">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="E9:5_1"/><i><font color="Green">E61</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n1</font> holds <br/>for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="graph_1.html#U2">Edges</a> of <span class="p1">(<span class="default"><a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">S</font></span>)</span> <a href="finseq_1.html#K13">*</a>   ex <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="graph_1.html#U2">Edges</a> of <span class="p1">(<span class="default"><a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">S</font></span>)</span> <a href="finseq_1.html#K13">*</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>,<font color="Olive">x</font>,<font color="Olive">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:5_1_1"/><i><font color="Green">E62</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n1</font> )
 ;<br/>

<a NAME="E2:5_1_1"/><b>then </b><i><font color="Green">E63</font></i>: 
( <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> )
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="graph_1.html#U2">Edges</a> of <span class="p1">(<span class="default"><a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">S</font></span>)</span> <a href="finseq_1.html#K13">*</a> ;<br/>

<b>reconsider </b><font color="Maroon">pk</font> = <font color="Maroon">p</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font>, <font color="Maroon">pk1</font> = <font color="Maroon">p</font> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as   <a href="trees_4.html#NM1">Node</a> of <font color="Maroon">t</font> <b>by </b><i>, , <a class="ref" href="msscyc_1.html#T19">MSSCYC_1:19</a></i>;<br/>
<b>reconsider </b><font color="Maroon">t1</font> = <font color="Maroon">t'</font> <a href="trees_2.html#K5">|</a> <font color="Maroon">pk</font>, <font color="Maroon">t2</font> = <font color="Maroon">t'</font> <a href="trees_2.html#K5">|</a> <font color="Maroon">pk1</font> as   <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">S</font>,<font color="Maroon">X</font> <b>by </b><i><a class="ref" href="msaterm.html#T29">MSATERM:29</a></i>;<br/>
<b>set </b><font color="Maroon">pk'</font> = <font color="Maroon">p</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">k</font>;<br/>
<a NAME="E5:5_1_1"/><i><font color="Green">E64</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k</font>
 
<b>by </b><i>, , , , <a class="ref" href="rfinseq.html#D2">RFINSEQ:def 2</a></i>;<br/>
<a NAME="E6:5_1_1"/><i><font color="Green">E84</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<a NAME="E7:5_1_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i>, , <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<a NAME="E8:5_1_1"/><b>then </b><i><font color="Green">E85</font></i>: 
not <font color="Maroon">p</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">k</font> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E9:5_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">pk</font> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">k</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rfinseq.html#T21">RFINSEQ:21</a></i>;<br/>
<a NAME="E10:5_1_1"/><b>then </b><i><font color="Green">E86</font></i>: 
<font color="Maroon">p</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">tr</font> <a href="trees_1.html#K4">|</a> <font color="Maroon">pk</font>
 
<b>by </b><i>, , <a class="ref" href="trees_1.html#D9">TREES_1:def 9</a></i>;<br/>
<a NAME="E11:5_1_1"/><b>then </b><i><font color="Green">E87</font></i>: 
<font color="Maroon">p</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">t1</font>
 
<b>by </b><i>, <a class="ref" href="trees_2.html#D11">TREES_2:def 11</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:5_1_1_1"/>
<font color="Maroon">t1</font> is <a href="trees_9.html#V1">root</a>
 ;<br/><a NAME="E2:5_1_1_1"/><b>then </b><i><font color="Green">E88</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">t1</font> <a href="hidden.html#R1">=</a>  <a href="trees_1.html#K2">elementary_tree</a> 0
 <b>by </b><i><a class="ref" href="trees_9.html#D1">TREES_9:def 1</a></i>;<br/><a NAME="E3:5_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">k</font></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="relat_1.html#K1">dom</a> <font color="Maroon">t1</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="trees_1.html#D15">TREES_1:def 15</a></i>;<br/><b>hence </b><a NAME="E4:5_1_1_1"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="trees_1.html#T79">TREES_1:79</a></i>;<br/></div><b>end;</b></div>
<b>then consider </b><font color="Maroon">o</font> being   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E14:5_1_1"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">t1</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="msscyc_1.html#T20">MSSCYC_1:20</a></i>;<br/>
<b>consider </b><font color="Maroon">a</font> being    <a href="msaterm.html#M1">ArgumentSeq</a> of  <a href="msaterm.html#K2">Sym</a> <font color="Maroon">o</font>,<font color="Maroon">X</font><b> such that </b><br/><a NAME="E16:5_1_1"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">t1</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">a</font>
 <b>by </b><i>, <a class="ref" href="msaterm.html#T10">MSATERM:10</a></i>;<br/>
<b>consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">T</font> being   <a href="trees_2.html#NM4">DecoratedTree</a>, <font color="Maroon">q</font> being   <a href="trees_4.html#NM1">Node</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E18:5_1_1"/><i><font color="Green">E95</font></i>: 
( <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font> &amp; <font color="Maroon">T</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">p</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">i</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font> )
 <b>by </b><i>, , , <a class="ref" href="trees_4.html#T11">TREES_4:11</a></i>;<br/>
<b>set </b><font color="Maroon">args</font> =  <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font>;<br/>
<a NAME="E19:5_1_1"/><i><font color="Green">E96</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">a</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/>
<a NAME="E20:5_1_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font> )
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E21:5_1_1"/><b>then </b><i><font color="Green">E97</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</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="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">rs1</font> = <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> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font> <b>by </b><i><a class="ref" href="dtconstr.html#T2">DTCONSTR:2</a></i>;<br/>
<b>set </b><font color="Maroon">e1</font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,<font color="Maroon">rs1</font></span><a href="tarski.html#K4">]</a></span>;<br/>
<a NAME="E23:5_1_1"/><i><font color="Green">E99</font></i>: 
the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font>
 
<b>by </b><i><a class="ref" href="msualg_1.html#D6">MSUALG_1:def 6</a></i>;<br/>
<a NAME="E24:5_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,<font color="Maroon">rs1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="msscyc_2.html#K1" target="_self">InducedEdges</a> <font color="Maroon">S</font>
 
<b>by </b><i>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">E'</font> = the <a href="graph_1.html#U2">Edges</a> of <span class="p1">(<span class="default"><a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">S</font></span>)</span> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">E'</font> <b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<b>reconsider </b><font color="Maroon">e1'</font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,<font color="Maroon">rs1</font></span><a href="tarski.html#K4">]</a></span> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">E'</font> <b>by </b><i>, , </i>;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">e1'</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">x'</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="graph_1.html#U2">Edges</a> of <span class="p1">(<span class="default"><a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">S</font></span>)</span> <a href="finseq_1.html#K13">*</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<b>take </b>
<font color="Maroon">y</font>
;<br/>

<b>take </b>
<font color="Maroon">t1</font>
;<br/>

<b>take </b>
<font color="Maroon">t2</font>
;<br/>

<b>thus </b><a NAME="E29:5_1_1"/>
( <font color="Maroon">t1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="trees_2.html#K5">|</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> &amp; <font color="Maroon">t2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="trees_2.html#K5">|</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 ;<br/>

<b>take </b>
<font color="Maroon">o</font>
;<br/>

<b>take </b>
<font color="Maroon">rs1</font>
;<br/>

<b>take </b>
<font color="Maroon">x</font>
;<br/>

<b>thus </b><a NAME="E30:5_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,<font color="Maroon">rs1</font></span><a href="tarski.html#K4">]</a></span></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">x</font> )
 ;<br/>

<b>thus </b><a NAME="E31:5_1_1"/>
<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="hidden.html#R1">=</a> <font color="Maroon">t1</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i/>;<br/>

<a NAME="E32:5_1_1"/><i><font color="Green">E103</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">k</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_5.html#T6">FINSEQ_5:6</a></i>;<br/>
<i><font color="Green">E104</font></i>: <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K17">|</a> 1 = 
<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
<b>by </b><i>, <a class="ref" href="finseq_5.html#T23">FINSEQ_5:23</a></i>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">k</font></span>)</span> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span>
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/>
<b>reconsider </b><font color="Maroon">pk'</font> = <font color="Maroon">p</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">k</font> as   <a href="trees_4.html#NM1">Node</a> of <font color="Maroon">t1</font> <b>by </b><i>, , <a class="ref" href="trees_2.html#D11">TREES_2:def 11</a></i>;<br/>
<b>reconsider </b><font color="Maroon">p1</font> = <font color="Maroon">pk'</font> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span> as   <a href="trees_4.html#NM1">Node</a> of <font color="Maroon">t1</font> <b>by </b><i><a class="ref" href="msscyc_1.html#T19">MSSCYC_1:19</a></i>;<br/>
<b>reconsider </b><font color="Maroon">t2'</font> = <font color="Maroon">t1</font> <a href="msaterm.html#K8">|</a> <font color="Maroon">p1</font> as   <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">S</font>,<font color="Maroon">X</font> ;<br/>
<a NAME="E37:5_1_1"/>
<font color="Maroon">pk'</font> <a href="finseq_1.html#K17">|</a> 1 <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">i</font></span><a href="finseq_1.html#K12">*&gt;</a></span>
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#T58">FINSEQ_1:58</a></i>;<br/>
<a NAME="E38:5_1_1"/><b>then </b><i><font color="Green">E107</font></i>: 
<font color="Maroon">t2'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i>, , <a class="ref" href="trees_4.html#D4">TREES_4:def 4</a></i>;<br/>
<a NAME="E39:5_1_1"/><i><font color="Green">E108</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">pk1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, , , , <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E40:5_1_1"/><i><font color="Green">E109</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E41:5_1_1"/><b>then </b><i><font color="Green">E110</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">pk1</font>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E42:5_1_1"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font>
 
<b>by </b><i>, , , , , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E43:5_1_1"/><i><font color="Green">E112</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T7">FINSEQ_1:7</a></i>;<br/>
<i><font color="Green">E113</font></i>: <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font> = 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D15">FINSEQ_1:def 15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D15">FINSEQ_1:def 15</a></i>
<br/>.= 
<font color="Maroon">p</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T82">FUNCT_1:82</a></i>
<br/>.= 
<font color="Maroon">pk</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D15">FINSEQ_1:def 15</a></i>
;<br/>
<font color="Maroon">p1</font> = 
<span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span>
<b>by </b><i>, , , , , , <a class="ref" href="rfinseq.html#D2">RFINSEQ:def 2</a></i>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K17">|</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
<b>by </b><i>, <a class="ref" href="finseq_4.html#T85">FINSEQ_4:85</a></i>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">pk1</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span>
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/>
<a NAME="E46:5_1_1"/><b>then </b>
<font color="Maroon">pk1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">pk</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p1</font>
 
<b>by </b><i>, , <a class="ref" href="rfinseq.html#T20">RFINSEQ:20</a></i>;<br/>
<a NAME="E47:5_1_1"/><b>then </b>
<font color="Maroon">t2'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t2</font>
 
<b>by </b><i><a class="ref" href="trees_9.html#T3">TREES_9:3</a></i>;<br/>
<b>hence </b><a NAME="E48:5_1_1"/>
<font color="Maroon">rs1</font> <a href="hidden.html#R1">=</a>  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">t2</font>
 <b>by </b><i>, , , <a class="ref" href="msaterm.html#T23">MSATERM:23</a></i>;<br/>

<b>thus </b><a NAME="E49:5_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,<font color="Maroon">rs1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2">Edges</a> of <span class="p1">(<span class="default"><a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">S</font></span>)</span>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:5_1"/>
not <font color="Maroon">t</font> is <a href="trees_9.html#V1">root</a>
 
<b>by </b><i>, , , <a class="ref" href="trees_1.html#T79">TREES_1:79</a>, <a class="ref" href="trees_9.html#D1">TREES_9:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">o</font> being   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E12:5_1"/><i><font color="Green">E114</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">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="msscyc_1.html#T20">MSSCYC_1:20</a></i>;<br/>
<b>consider </b><font color="Maroon">a</font> being    <a href="msaterm.html#M1">ArgumentSeq</a> of  <a href="msaterm.html#K2">Sym</a> <font color="Maroon">o</font>,<font color="Maroon">X</font><b> such that </b><br/><a NAME="E14:5_1"/><i><font color="Green">E115</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">a</font>
 <b>by </b><i>, <a class="ref" href="msaterm.html#T10">MSATERM:10</a></i>;<br/>
<a NAME="E15:5_1"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a>, , , , </i>;<br/>
<b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">T</font> being   <a href="trees_2.html#NM4">DecoratedTree</a>, <font color="Maroon">q</font> being   <a href="trees_4.html#NM1">Node</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E17:5_1"/><i><font color="Green">E117</font></i>: 
( <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font> &amp; <font color="Maroon">T</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">i</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font> )
 <b>by </b><i>, , , <a class="ref" href="trees_4.html#T11">TREES_4:11</a></i>;<br/>
<b>set </b><font color="Maroon">args</font> =  <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font>;<br/>
<a NAME="E18:5_1"/><i><font color="Green">E118</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">a</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/>
<a NAME="E19:5_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font> )
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E20:5_1"/><b>then </b><i><font color="Green">E119</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</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="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">rs1</font> = <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> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font> <b>by </b><i><a class="ref" href="dtconstr.html#T2">DTCONSTR:2</a></i>;<br/>
<b>set </b><font color="Maroon">e1</font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,<font color="Maroon">rs1</font></span><a href="tarski.html#K4">]</a></span>;<br/>
<a NAME="E22:5_1"/><i><font color="Green">E120</font></i>: 
the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font>
 
<b>by </b><i><a class="ref" href="msualg_1.html#D6">MSUALG_1:def 6</a></i>;<br/>
<a NAME="E23:5_1"/><b>then </b><i><font color="Green">E121</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,<font color="Maroon">rs1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="msscyc_2.html#K1" target="_self">InducedEdges</a> <font color="Maroon">S</font>
 
<b>by </b><i>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">E'</font> = the <a href="graph_1.html#U2">Edges</a> of <span class="p1">(<span class="default"><a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">S</font></span>)</span> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<b>reconsider </b><font color="Maroon">e1'</font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,<font color="Maroon">rs1</font></span><a href="tarski.html#K4">]</a></span> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">E'</font> <b>by </b><i>, , </i>;<br/>
<b>reconsider </b><font color="Maroon">C1'</font> = <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">e1'</font></span><a href="finseq_1.html#K12">*&gt;</a></span> as    <a href="subset_1.html#M1">Element</a> of the <a href="graph_1.html#U2">Edges</a> of <span class="p1">(<span class="default"><a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">S</font></span>)</span> <a href="finseq_1.html#K13">*</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<b>consider </b><font color="Maroon">C</font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U2">Edges</a> of <span class="p1">(<span class="default"><a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">S</font></span>)</span> <a href="finseq_1.html#K13">*</a> <b> such that </b><br/><a NAME="E28:5_1"/><i><font color="Green">E124</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n1</font> &amp; ( <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">C1'</font> or <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> 0 ) &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n1</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>,<font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font>,<font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>] ) )
 <b>from </b><i><a class="ref" href="recdef_1.html#S6">RECDEF_1:sch 6</a>();<br/></i>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b>( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> implies  ex <font color="Olive">Ck</font> being <a href="graph_1.html#V8">directed</a>  <a href="graph_1.html#M1">Chain</a> of  <a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">S</font> ex <font color="Olive">t1</font> being  <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">S</font>,<font color="Maroon">X</font> st <br/>( <font color="Olive">Ck</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">Ck</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">t1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="trees_2.html#K5">|</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="graph_2.html#K7">vertex-seq</a> <font color="Olive">Ck</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">Ck</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> &amp; <span class="p1">(<span class="default"><a href="graph_2.html#K7">vertex-seq</a> <font color="Olive">Ck</font></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="msaterm.html#K7">the_sort_of</a> <font color="Olive">t1</font> ) );<br/>
<a NAME="E29:5_1"/><i><font color="Green">E126</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
;<br/>
<a NAME="E30:5_1"/><i><font color="Green">E127</font></i>: 
for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">i</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">i</font> <a href="xcmplx_0.html#K2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:5_1_2"/><i><font color="Green">E128</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> implies  ex <font color="Olive">Ck</font> being <a href="graph_1.html#V8">directed</a>  <a href="graph_1.html#M1">Chain</a> of  <a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">S</font> ex <font color="Olive">t1</font> being  <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">S</font>,<font color="Maroon">X</font> st <br/>( <font color="Olive">Ck</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">Ck</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp; <font color="Olive">t1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="trees_2.html#K5">|</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="graph_2.html#K7">vertex-seq</a> <font color="Olive">Ck</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">Ck</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> &amp; <span class="p1">(<span class="default"><a href="graph_2.html#K7">vertex-seq</a> <font color="Olive">Ck</font></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="msaterm.html#K7">the_sort_of</a> <font color="Olive">t1</font> ) )
 ;<br/>

<b>assume </b><a NAME="E2:5_1_2"/><i><font color="Green">E129</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1 &amp; <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> )
 ;<br/>

<a NAME="E3:5_1_2"/><i><font color="Green">E130</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E4:5_1_2"/><i><font color="Green">E131</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:5_1_2_1"/>
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_1_2_1_1"><b>suppose </b></a><a NAME="E1:5_1_2_1_1"/><i><font color="Green">E132</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">C1</font> = <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,<font color="Maroon">rs1</font></span><a href="tarski.html#K4">]</a></span></span><a href="finseq_1.html#K9">*&gt;</a></span> as  <a href="graph_1.html#V8">directed</a>  <a href="graph_1.html#M1">Chain</a> of  <a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">S</font> <b>by </b><i>, <a class="ref" href="msscyc_1.html#T5">MSSCYC_1:5</a></i>;<br/><a NAME="E3:5_1_2_1_1"/><i><font color="Green">E134</font></i>: 
 <a href="graph_2.html#K7">vertex-seq</a> <font color="Maroon">C1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><span class="p2">(<span class="default">the <a href="graph_1.html#U3">Source</a> of <span class="p3">(<span class="default"><a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,<font color="Maroon">rs1</font></span><a href="tarski.html#K4">]</a></span></span>)</span>,<span class="p2">(<span class="default">the <a href="graph_1.html#U4">Target</a> of <span class="p3">(<span class="default"><a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,<font color="Maroon">rs1</font></span><a href="tarski.html#K4">]</a></span></span>)</span></span><a href="finseq_1.html#K10">*&gt;</a></span>
 <b>by </b><i>, <a class="ref" href="msscyc_1.html#T7">MSSCYC_1:7</a></i>;<br/><i><font color="Green">E135</font></i>: <span class="p1">(<span class="default"><a href="graph_2.html#K7">vertex-seq</a> <font color="Maroon">C1</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">C1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><a href="graph_2.html#K7">vertex-seq</a> <font color="Maroon">C1</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>
<br/>.= 
the <a href="graph_1.html#U4">Target</a> of <span class="p1">(<span class="default"><a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,<font color="Maroon">rs1</font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
<br/>.= 
the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font> <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>,<font color="Maroon">rs1</font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> </span>)</span>
<b>by </b><i>, </i>
<br/>.= 
the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o</font>
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
 <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font>
<b>by </b><i><a class="ref" href="msualg_1.html#D7">MSUALG_1:def 7</a></i>
<br/>.= 
 <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">t'</font>
<b>by </b><i>, <a class="ref" href="msaterm.html#T17">MSATERM:17</a></i>
<br/>.= 
<font color="Maroon">v</font>
<b>by </b><i>, <a class="ref" href="msaterm.html#D5">MSATERM:def 5</a></i>
;<br/><b>reconsider </b><font color="Maroon">p1</font> = <font color="Maroon">p</font> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span> as   <a href="trees_4.html#NM1">Node</a> of <font color="Maroon">t</font> <b>by </b><i>, , <a class="ref" href="msscyc_1.html#T19">MSSCYC_1:19</a></i>;<br/><b>reconsider </b><font color="Maroon">t2</font> = <font color="Maroon">t'</font> <a href="trees_2.html#K5">|</a> <font color="Maroon">p1</font> as   <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">S</font>,<font color="Maroon">X</font> <b>by </b><i><a class="ref" href="msaterm.html#T29">MSATERM:29</a></i>;<br/><a NAME="E7:5_1_2_1_1"/><i><font color="Green">E136</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="finseq_5.html#T6">FINSEQ_5:6</a></i>;<br/><b>reconsider </b><font color="Maroon">p'</font> = <font color="Maroon">p</font> as   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="numbers.html#K5">NAT</a> , <a href="numbers.html#K5">NAT</a>  ;<br/><font color="Maroon">p</font> <a href="finseq_1.html#K17">|</a> 1 = 
<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p'</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
<b>by </b><i>, <a class="ref" href="finseq_5.html#T23">FINSEQ_5:23</a></i>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span>
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">i</font></span><a href="finseq_1.html#K12">*&gt;</a></span>
<b>by </b><i>, <a class="ref" href="finseq_1.html#T58">FINSEQ_1:58</a></i>
;<br/><a NAME="E10:5_1_2_1_1"/><b>then </b><i><font color="Green">E138</font></i>: 
<font color="Maroon">t2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, , <a class="ref" href="trees_4.html#D4">TREES_4:def 4</a></i>;<br/><i><font color="Green">E139</font></i>: <span class="p1">(<span class="default"><a href="graph_2.html#K7">vertex-seq</a> <font color="Maroon">C1</font></span>)</span> <a href="funct_1.html#K1">.</a> 1 = 
the <a href="graph_1.html#U3">Source</a> of <span class="p1">(<span class="default"><a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,<font color="Maroon">rs1</font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,<font color="Maroon">rs1</font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">rs1</font>
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
 <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">t2</font>
<b>by </b><i>, , , <a class="ref" href="msaterm.html#T23">MSATERM:23</a></i>
;<br/><b>take </b>
<font color="Maroon">C1</font>
;<br/><b>take </b>
<font color="Maroon">t2</font>
;<br/><b>thus </b><a NAME="E12:5_1_2_1_1"/>
<font color="Maroon">C1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
 <b>by </b><i>, , </i>;<br/><b>thus </b><a NAME="E13:5_1_2_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">C1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/><b>thus </b><a NAME="E14:5_1_2_1_1"/>
<font color="Maroon">t2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="trees_2.html#K5">|</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E15:5_1_2_1_1"/>
<span class="p1">(<span class="default"><a href="graph_2.html#K7">vertex-seq</a> <font color="Maroon">C1</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">C1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font>
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E16:5_1_2_1_1"/>
<span class="p1">(<span class="default"><a href="graph_2.html#K7">vertex-seq</a> <font color="Maroon">C1</font></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">t2</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_1_2_1_2"><b>suppose </b></a><a NAME="E1:5_1_2_1_2"/><i><font color="Green">E140</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:5_1_2_1_2"/><b>then </b><i><font color="Green">E141</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/><b>consider </b><font color="Maroon">Ck</font> being  <a href="graph_1.html#V8">directed</a>  <a href="graph_1.html#M1">Chain</a> of  <a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">S</font>, <font color="Maroon">t1'</font> being   <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">S</font>,<font color="Maroon">X</font><b> such that </b><br/><a NAME="E4:5_1_2_1_2"/><i><font color="Green">E143</font></i>: 
( <font color="Maroon">Ck</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">Ck</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">t1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="trees_2.html#K5">|</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="graph_2.html#K7">vertex-seq</a> <font color="Maroon">Ck</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">Ck</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> &amp; <span class="p1">(<span class="default"><a href="graph_2.html#K7">vertex-seq</a> <font color="Maroon">Ck</font></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">t1'</font> )
 <b>by </b><i>, , , , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/><a NAME="E5:5_1_2_1_2"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><b>then consider </b><font color="Maroon">t1</font> being   <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">S</font>,<font color="Maroon">X</font>, <font color="Maroon">t2</font> being   <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">S</font>,<font color="Maroon">X</font><b> such that </b><br/><a NAME="E7:5_1_2_1_2"/><i><font color="Green">E144</font></i>: 
( <font color="Maroon">t1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="trees_2.html#K5">|</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> &amp; <font color="Maroon">t2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="trees_2.html#K5">|</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> )
 <b>and </b><br/><a NAME="E8:5_1_2_1_2"/><i><font color="Green">E145</font></i>: 
 ex <font color="Olive">o</font> being  <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font> ex <font color="Olive">rs1</font> being  <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font> ex <font color="Olive">Ck</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="graph_1.html#U2">Edges</a> of <span class="p1">(<span class="default"><a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">S</font></span>)</span> <a href="finseq_1.html#K13">*</a>  st <br/>( <font color="Olive">Ck</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> &amp; <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">o</font>,<font color="Olive">rs1</font></span><a href="tarski.html#K4">]</a></span></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <font color="Olive">Ck</font> &amp; <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="hidden.html#R1">=</a> <font color="Maroon">t1</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">rs1</font> <a href="hidden.html#R1">=</a>  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">t2</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">o</font>,<font color="Olive">rs1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2">Edges</a> of <span class="p1">(<span class="default"><a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">S</font></span>)</span> )
 <b>by </b><i>, , </i>;<br/><b>consider </b><font color="Maroon">o</font> being   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font>, <font color="Maroon">rs1</font> being   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font>, <font color="Maroon">Ck'</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="graph_1.html#U2">Edges</a> of <span class="p1">(<span class="default"><a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">S</font></span>)</span> <a href="finseq_1.html#K13">*</a> <b> such that </b><br/><a NAME="E10:5_1_2_1_2"/><i><font color="Green">E147</font></i>: 
( <font color="Maroon">Ck'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> &amp; <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,<font color="Maroon">rs1</font></span><a href="tarski.html#K4">]</a></span></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">Ck'</font> &amp; <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="hidden.html#R1">=</a> <font color="Maroon">t1</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">rs1</font> <a href="hidden.html#R1">=</a>  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">t2</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,<font color="Maroon">rs1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2">Edges</a> of <span class="p1">(<span class="default"><a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">S</font></span>)</span> )
 <b>by </b><i/>;<br/><b>set </b><font color="Maroon">e1</font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,<font color="Maroon">rs1</font></span><a href="tarski.html#K4">]</a></span>;<br/><b>reconsider </b><font color="Maroon">C1</font> = <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,<font color="Maroon">rs1</font></span><a href="tarski.html#K4">]</a></span></span><a href="finseq_1.html#K9">*&gt;</a></span> as  <a href="graph_1.html#V8">directed</a>  <a href="graph_1.html#M1">Chain</a> of  <a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">S</font> <b>by </b><i>, <a class="ref" href="msscyc_1.html#T5">MSSCYC_1:5</a></i>;<br/><a NAME="E12:5_1_2_1_2"/><i><font color="Green">E148</font></i>: 
not <font color="Maroon">Ck</font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/><a NAME="E13:5_1_2_1_2"/><i><font color="Green">E149</font></i>: 
not  <a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">S</font> is <a href="msscyc_1.html#V2">empty</a>
 <b>by </b><i>, <a class="ref" href="msscyc_1.html#D3">MSSCYC_1:def 3</a></i>;<br/><a NAME="E14:5_1_2_1_2"/><i><font color="Green">E150</font></i>: 
 <a href="graph_2.html#K7">vertex-seq</a> <font color="Maroon">C1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><span class="p2">(<span class="default">the <a href="graph_1.html#U3">Source</a> of <span class="p3">(<span class="default"><a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,<font color="Maroon">rs1</font></span><a href="tarski.html#K4">]</a></span></span>)</span>,<span class="p2">(<span class="default">the <a href="graph_1.html#U4">Target</a> of <span class="p3">(<span class="default"><a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,<font color="Maroon">rs1</font></span><a href="tarski.html#K4">]</a></span></span>)</span></span><a href="finseq_1.html#K10">*&gt;</a></span>
 <b>by </b><i>, <a class="ref" href="msscyc_1.html#T7">MSSCYC_1:7</a></i>;<br/><i><font color="Green">E151</font></i>: <span class="p1">(<span class="default"><a href="graph_2.html#K7">vertex-seq</a> <font color="Maroon">C1</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">C1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><a href="graph_2.html#K7">vertex-seq</a> <font color="Maroon">C1</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>
<br/>.= 
the <a href="graph_1.html#U4">Target</a> of <span class="p1">(<span class="default"><a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,<font color="Maroon">rs1</font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
<br/>.= 
the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font> <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>,<font color="Maroon">rs1</font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> </span>)</span>
<b>by </b><i>, </i>
<br/>.= 
the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o</font>
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
 <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font>
<b>by </b><i><a class="ref" href="msualg_1.html#D7">MSUALG_1:def 7</a></i>
<br/>.= 
 <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">t1</font>
<b>by </b><i>, <a class="ref" href="msaterm.html#T17">MSATERM:17</a></i>
;<br/><b>then reconsider </b><font color="Maroon">d</font> = <font color="Maroon">C1</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">Ck</font> as  <a href="graph_1.html#V8">directed</a>  <a href="graph_1.html#M1">Chain</a> of  <a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">S</font> <b>by </b><i>, , , , <a class="ref" href="msscyc_1.html#T15">MSSCYC_1:15</a></i>;<br/><a NAME="E17:5_1_2_1_2"/><i><font color="Green">E153</font></i>: 
not <font color="Maroon">d</font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i>, , , , <a class="ref" href="msscyc_2.html#D1" target="_self">Def1</a>, <a class="ref" href="msscyc_1.html#T15">MSSCYC_1:15</a></i>;<br/><b>take </b>
<font color="Maroon">d</font>
;<br/><b>take </b>
<font color="Maroon">t2</font>
;<br/><b>thus </b><a NAME="E18:5_1_2_1_2"/>
<font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
 <b>by </b><i>, </i>;<br/><b>thus </b> <a href="finseq_1.html#K3">len</a> <font color="Maroon">d</font> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">C1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k</font>
<b>by </b><i>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>
;<br/><b>thus </b><a NAME="E20:5_1_2_1_2"/>
<font color="Maroon">t2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="trees_2.html#K5">|</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E21:5_1_2_1_2"/>
<span class="p1">(<span class="default"><a href="graph_2.html#K7">vertex-seq</a> <font color="Maroon">d</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">d</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font>
 <b>by </b><i>, , , , <a class="ref" href="msscyc_1.html#T16">MSSCYC_1:16</a></i>;<br/><span class="p1">(<span class="default"><a href="graph_2.html#K7">vertex-seq</a> <font color="Maroon">C1</font></span>)</span> <a href="funct_1.html#K1">.</a> 1 = 
the <a href="graph_1.html#U3">Source</a> of <span class="p1">(<span class="default"><a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,<font color="Maroon">rs1</font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,<font color="Maroon">rs1</font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i>, </i>
<br/>.= 
 <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">t2</font>
<b>by </b><i>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
;<br/><b>hence </b><a NAME="E23:5_1_2_1_2"/>
<span class="p1">(<span class="default"><a href="graph_2.html#K7">vertex-seq</a> <font color="Maroon">d</font></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">t2</font>
 <b>by </b><i>, , , <a class="ref" href="msscyc_1.html#T16">MSSCYC_1:16</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E31:5_1"/>
for <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2">NAT_1:sch 2</a>(, );<br/></i>
<b>then consider </b><font color="Maroon">c</font> being  <a href="graph_1.html#V8">directed</a>  <a href="graph_1.html#M1">Chain</a> of  <a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">S</font>, <font color="Maroon">t1</font> being   <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">S</font>,<font color="Maroon">X</font><b> such that </b><br/><a NAME="E33:5_1"/><i><font color="Green">E154</font></i>: 
( <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">t1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="trees_2.html#K5">|</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="graph_2.html#K7">vertex-seq</a> <font color="Maroon">c</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> &amp; <span class="p1">(<span class="default"><a href="graph_2.html#K7">vertex-seq</a> <font color="Maroon">c</font></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">t1</font> )
 <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E34:5_1"/>
 ex <font color="Olive">c</font> being <a href="graph_1.html#V8">directed</a>  <a href="graph_1.html#M1">Chain</a> of  <a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">S</font> st <br/>(  <a href="finseq_1.html#K3">len</a> <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp; <span class="p1">(<span class="default"><a href="graph_2.html#K7">vertex-seq</a> <font color="Olive">c</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">c</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>

<b>given </b><font color="Maroon">c</font> being  <a href="graph_1.html#V8">directed</a>  <a href="graph_1.html#M1">Chain</a> of  <a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">S</font><b> such that </b><a NAME="E6:5"/><i><font color="Green">E155</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp; <span class="p1">(<span class="default"><a href="graph_2.html#K7">vertex-seq</a> <font color="Maroon">c</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> )
 ;<br/>

<b>set </b><font color="Maroon">cS</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>;<br/>
<b>set </b><font color="Maroon">EG</font> = the <a href="graph_1.html#U2">Edges</a> of <span class="p1">(<span class="default"><a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">S</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">D</font> = <font color="Maroon">S</font> <a href="msaterm.html#K1">-Terms</a> <font color="Maroon">X</font>;<br/>
<b>set </b><font color="Maroon">SG</font> = the <a href="graph_1.html#U3">Source</a> of <span class="p1">(<span class="default"><a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">S</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">TG</font> = the <a href="graph_1.html#U4">Target</a> of <span class="p1">(<span class="default"><a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">S</font></span>)</span>;<br/>
<a NAME="E7:5"/><i><font color="Green">E160</font></i>: 
<font color="Maroon">c</font> is    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U2">Edges</a> of <span class="p1">(<span class="default"><a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">S</font></span>)</span>
 
<b>by </b><i><a class="ref" href="msscyc_1.html#D1">MSSCYC_1:def 1</a></i>;<br/>
<a NAME="E8:5"/><i><font color="Green">E161</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c</font>
 
<b>by </b><i>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E9:5"/><b>then </b><i><font color="Green">E162</font></i>: 
<font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a>  <a href="msscyc_2.html#K1" target="_self">InducedEdges</a> <font color="Maroon">S</font>
 
<b>by </b><i>, <a class="ref" href="dtconstr.html#T2">DTCONSTR:2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">EG</font> = the <a href="graph_1.html#U2">Edges</a> of <span class="p1">(<span class="default"><a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">S</font></span>)</span> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i>, , <a class="ref" href="dtconstr.html#T2">DTCONSTR:2</a></i>;<br/>
<a NAME="E11:5"/><i><font color="Green">E163</font></i>: 
not  <a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">S</font> is <a href="msscyc_1.html#V2">empty</a>
 
<b>by </b><i>, <a class="ref" href="msscyc_1.html#D3">MSSCYC_1:def 3</a></i>;<br/>
<a NAME="E12:5"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i>, </i>;<br/>
<a NAME="E13:5"/><b>then </b><i><font color="Green">E164</font></i>: 
not <font color="Maroon">c</font> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</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>  = <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E14:5"/><i><font color="Green">E165</font></i>: 
for <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">e</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Olive">e</font>) <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="pboole.html#D16">PBOOLE:def 16</a></i>;<br/>
<b>consider </b><font color="Maroon">cX</font> being    <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E16:5"/><i><font color="Green">E167</font></i>: 
for <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">e</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> holds <br/><font color="Maroon">cX</font> <a href="funct_1.html#K1">.</a> <font color="Olive">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">e</font>)
 <b>from </b><i><a class="ref" href="pboole.html#S1">PBOOLE:sch 1</a>();<br/></i>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>,   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">o</font> being  <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font> ex <font color="Olive">rs1</font> being  <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font> ex <font color="Olive">Ck</font>, <font color="Olive">Ck1</font> being  <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">S</font>,<font color="Maroon">X</font> ex <font color="Olive">a</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">X</font> st <br/>( <font color="Olive">Ck</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> &amp; <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">Ck1</font> &amp; <font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">o</font>,<font color="Olive">rs1</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Olive">Ck1</font> <a href="hidden.html#R1">=</a> <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">a</font> &amp; ( for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Olive">a</font> holds <br/> ex <font color="Olive">t</font> being  <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">S</font>,<font color="Maroon">X</font> st <br/>( <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp;  <a href="msaterm.html#K7">the_sort_of</a> <font color="Olive">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="Olive">o</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; (  <a href="msaterm.html#K7">the_sort_of</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Olive">rs1</font> &amp;  <a href="msaterm.html#K7">the_sort_of</a> <font color="Olive">Ck</font> <a href="hidden.html#R1">=</a> <font color="Olive">rs1</font> implies <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Olive">Ck</font> ) &amp; ( (  <a href="msaterm.html#K7">the_sort_of</a> <font color="Olive">t</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">rs1</font> or  <a href="msaterm.html#K7">the_sort_of</a> <font color="Olive">Ck</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">rs1</font> ) implies <font color="Olive">t</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"><span class="p2">(<span class="default"><font color="Maroon">cX</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="msaterm.html#K7">the_sort_of</a> <font color="Olive">t</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="msaterm.html#K7">the_sort_of</a> <font color="Olive">t</font></span>)</span></span><a href="tarski.html#K4">]</a></span> ) ) ) );<br/>
<b>reconsider </b><font color="Maroon">n1</font> = <font color="Maroon">n</font> as    <a href="subset_1.html#M2">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="E18:5"/><i><font color="Green">E169</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n1</font> holds <br/>for <font color="Olive">x</font> being   <a href="trees_9.html#M1">Element</a> of <font color="Maroon">S</font> <a href="msaterm.html#K1">-Terms</a> <font color="Maroon">X</font>  ex <font color="Olive">y</font> being   <a href="trees_9.html#M1">Element</a> of <font color="Maroon">S</font> <a href="msaterm.html#K1">-Terms</a> <font color="Maroon">X</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>,<font color="Olive">x</font>,<font color="Olive">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:5_2"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n1</font> )
 ;<br/>

<a NAME="E2:5_2"/><b>then </b><i><font color="Green">E170</font></i>: 
( <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> )
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>let </b><font color="Maroon">x</font> be    <a href="trees_9.html#M1">Element</a> of <font color="Maroon">S</font> <a href="msaterm.html#K1">-Terms</a> <font color="Maroon">X</font>;<br/>

<a NAME="E3:5_2"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E4:5_2"/><b>then </b>
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c</font>
 
<b>by </b><i>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">ck1</font> = <font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">EG</font> <b>by </b><i>, <a class="ref" href="dtconstr.html#T2">DTCONSTR:2</a></i>;<br/>
<b>consider </b><font color="Maroon">o</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">rs1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:5_2"/><i><font color="Green">E172</font></i>: 
( <font color="Maroon">ck1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,<font color="Maroon">rs1</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font> &amp; <font color="Maroon">rs1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> )
 <b>and </b><br/><a NAME="E8:5_2"/>
 ex <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> ex <font color="Olive">args</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="finseq_1.html#K13">*</a>  st <br/>( the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Olive">args</font> &amp; <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Olive">args</font> &amp; <font color="Olive">args</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">rs1</font> )
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">o</font> = <font color="Maroon">o</font> as   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">rs1</font> = <font color="Maroon">rs1</font> as   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font> <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">DA</font> =  <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>;<br/>
<b>set </b><font color="Maroon">ar</font> =  <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>( ( <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">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">rs1</font> &amp;  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">rs1</font> implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> ) &amp; ( ( <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">a<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">rs1</font> or  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">rs1</font> ) implies <font color="Maroon">a<sub>2</sub></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"><span class="p2">(<span class="default"><font color="Maroon">cX</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<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">a<sub>1</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<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">a<sub>1</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> ) );<br/>
<a NAME="E11:5_2"/><i><font color="Green">E175</font></i>: 
for <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">e</font> <a href="hidden.html#R2">in</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> holds <br/> ex <font color="Olive">u</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> <a href="msaterm.html#K1">-Terms</a> <font color="Maroon">X</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">e</font>,<font color="Olive">u</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_2_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:5_2_1"/><i><font color="Green">E177</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</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>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:5_2_1_1"/>
( ( <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">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">rs1</font> &amp;  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">rs1</font> ) or <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">e</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">rs1</font> or  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">rs1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_2_1_1_1"><b>suppose </b></a><a NAME="E1:5_2_1_1_1"/><i><font color="Green">E178</font></i>: 
( <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">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">rs1</font> &amp;  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">rs1</font> )
 ;<br/><div class="add"><b>take </b>
<font color="Maroon">x</font>
;<br/><b>thus </b><a NAME="E2:5_2_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> <a href="msaterm.html#K1">-Terms</a> <font color="Maroon">X</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">e</font>,<font color="Maroon">x</font>] )
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_2_1_1_2"><b>suppose </b></a><a NAME="E1:5_2_1_1_2"/><i><font color="Green">E179</font></i>: 
( <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">e</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">rs1</font> or  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">rs1</font> )
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">s</font> = <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">e</font> as   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font> <b>by </b><i>, <a class="ref" href="dtconstr.html#T2">DTCONSTR:2</a></i>;<br/><b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">cX</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<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">e</font></span>)</span> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <b>by </b><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">x</font>,<font color="Maroon">s</font></span><a href="tarski.html#K4">]</a></span> as   <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">S</font>,<font color="Maroon">X</font> <b>by </b><i><a class="ref" href="msaterm.html#T4">MSATERM:4</a></i>;<br/><b>take </b>
<font color="Maroon">t</font>
;<br/><b>thus </b><a NAME="E5:5_2_1_1_2"/>
( <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> <a href="msaterm.html#K1">-Terms</a> <font color="Maroon">X</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">e</font>,<font color="Maroon">t</font>] )
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>consider </b><font color="Maroon">a</font> being   <a href="funct_2.html#NM1">Function</a> of  <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>,<font color="Maroon">S</font> <a href="msaterm.html#K1">-Terms</a> <font color="Maroon">X</font><b> such that </b><br/><a NAME="E13:5_2"/><i><font color="Green">E181</font></i>: 
for <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">e</font> <a href="hidden.html#R2">in</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> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">e</font>,<font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <font color="Olive">e</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1">FUNCT_2:sch 1</a>();<br/></i>
<a NAME="E14:5_2"/>
 <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> <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> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font> as    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">S</font> <a href="msaterm.html#K1">-Terms</a> <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="finseq_2.html#T28">FINSEQ_2:28</a></i>;<br/>
<a NAME="E16:5_2"/><i><font color="Green">E182</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">a</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="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_2_2"><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="E1:5_2_2"/><i><font color="Green">E183</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">a</font>
 ;<br/><b>then reconsider </b><font color="Maroon">t</font> = <font color="Maroon">a</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">X</font> <b>by </b><i><a class="ref" href="dtconstr.html#T2">DTCONSTR:2</a></i>;<br/><b>take </b><font color="Maroon">t</font> = <font color="Maroon">t</font>;<br/><b>thus </b><a NAME="E3:5_2_2"/>
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:5_2_2_1"/>
( ( <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> <a href="hidden.html#R1">=</a> <font color="Maroon">rs1</font> &amp;  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">rs1</font> ) or <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> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">rs1</font> or  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">rs1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_2_2_1_1"><b>suppose </b></a><a NAME="E1:5_2_2_1_1"/>
( <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> <a href="hidden.html#R1">=</a> <font color="Maroon">rs1</font> &amp;  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">rs1</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:5_2_2_1_1"/>
 <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>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_2_2_1_2"><b>suppose </b></a><a NAME="E1:5_2_2_1_2"/>
( <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> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">rs1</font> or  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">rs1</font> )
 ;<br/><div class="add"><a NAME="E2:5_2_2_1_2"/><b>then </b><i><font color="Green">E184</font></i>: 
<font color="Maroon">t</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"><span class="p2">(<span class="default"><font color="Maroon">cX</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<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></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<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></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, , </i>;<br/><b>reconsider </b><font color="Maroon">s</font> = <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> as   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font> <b>by </b><i>, , <a class="ref" href="dtconstr.html#T2">DTCONSTR:2</a></i>;<br/><a NAME="E4:5_2_2_1_2"/>
<font color="Maroon">cX</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<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></span>)</span> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E5:5_2_2_1_2"/>
 <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#T14">MSATERM:14</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font> as    <a href="msaterm.html#M1">ArgumentSeq</a> of  <a href="msaterm.html#K2">Sym</a> <font color="Maroon">o</font>,<font color="Maroon">X</font> <b>by </b><i>, <a class="ref" href="msaterm.html#T24">MSATERM:24</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y</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">a</font> as   <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">S</font>,<font color="Maroon">X</font> <b>by </b><i><a class="ref" href="msaterm.html#T1">MSATERM:1</a></i>;<br/>
<b>take </b>
<font color="Maroon">y</font>
;<br/>

<b>take </b>
<font color="Maroon">o</font>
;<br/>

<b>take </b>
<font color="Maroon">rs1</font>
;<br/>

<b>take </b>
<font color="Maroon">x</font>
;<br/>

<b>take </b>
<font color="Maroon">y</font>
;<br/>

<b>take </b>
<font color="Maroon">a</font>
;<br/>

<b>thus </b><a NAME="E20:5_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 ;<br/>

<b>thus </b><a NAME="E21:5_2"/>
<font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</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>,<font color="Maroon">rs1</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E22:5_2"/>
<font color="Maroon">y</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">a</font>
 ;<br/>

<b>let </b><font color="Maroon">i</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E23:5_2"/><i><font color="Green">E185</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">a</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">t</font> = <font color="Maroon">a</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">X</font> <b>by </b><i><a class="ref" href="dtconstr.html#T2">DTCONSTR:2</a></i>;<br/>
<b>take </b>
<font color="Maroon">t</font>
;<br/>

<b>thus </b><a NAME="E25:5_2"/>
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 ;<br/>

<b>thus </b><a NAME="E26:5_2"/>
 <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/>

<b>hence </b><a NAME="E27:5_2"/>
( (  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">rs1</font> &amp;  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">rs1</font> implies <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> ) &amp; ( (  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">t</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">rs1</font> or  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">rs1</font> ) implies <font color="Maroon">t</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"><span class="p2">(<span class="default"><font color="Maroon">cX</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">t</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">t</font></span>)</span></span><a href="tarski.html#K4">]</a></span> ) )
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">o</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">rs1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E20:5"/><i><font color="Green">E186</font></i>: 
( <font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,<font color="Maroon">rs1</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font> &amp; <font color="Maroon">rs1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> &amp;  ex <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> ex <font color="Olive">args</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="finseq_1.html#K13">*</a>  st <br/>( the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Olive">args</font> &amp; <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Olive">args</font> &amp; <font color="Olive">args</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">rs1</font> ) )
 <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">o</font> = <font color="Maroon">o</font> as   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">rs1</font> = <font color="Maroon">rs1</font> as   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font> <b>by </b><i/>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  =  <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">cX</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<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">a<sub>1</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<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">a<sub>1</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>;<br/>
<b>consider </b><font color="Maroon">a</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E24:5"/><i><font color="Green">E187</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</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> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span></span>)</span> holds <br/><font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">k</font>) ) )
 <b>from </b><i><a class="ref" href="finseq_1.html#S2">FINSEQ_1:sch 2</a>();<br/></i>
<a NAME="E25:5"/><i><font color="Green">E188</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">a</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">a</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E26:5"/>
for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">a</font> holds <br/> ex <font color="Olive">t</font> being  <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">S</font>,<font color="Maroon">X</font> st <br/>( <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp;  <a href="msaterm.html#K7">the_sort_of</a> <font color="Olive">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="Olive">i</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_3"><b>proof </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="E1:5_3"/><i><font color="Green">E189</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">a</font>
 ;<br/>

<b>set </b><font color="Maroon">s</font> = <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>;<br/>
<b>set </b><font color="Maroon">x</font> = <font color="Maroon">cX</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<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></span>)</span>;<br/>
<a NAME="E2:5_3"/>
 <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> <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> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">s</font> = <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> as   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font> <b>by </b><i>, , , <a class="ref" href="dtconstr.html#T2">DTCONSTR:2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">cX</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<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></span>)</span> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <b>by </b><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">x</font>,<font color="Maroon">s</font></span><a href="tarski.html#K4">]</a></span> as   <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">S</font>,<font color="Maroon">X</font> <b>by </b><i><a class="ref" href="msaterm.html#T4">MSATERM:4</a></i>;<br/>
<b>take </b>
<font color="Maroon">t</font>
;<br/>

<b>thus </b><a NAME="E6:5_3"/>
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, , </i>;<br/>

<b>thus </b><a NAME="E7:5_3"/>
 <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#T14">MSATERM:14</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font> as    <a href="msaterm.html#M1">ArgumentSeq</a> of  <a href="msaterm.html#K2">Sym</a> <font color="Maroon">o</font>,<font color="Maroon">X</font> <b>by </b><i>, <a class="ref" href="msaterm.html#T24">MSATERM:24</a></i>;<br/>
<b>set </b><font color="Maroon">C1</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">a</font>;<br/>
<b>reconsider </b><font color="Maroon">C1</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">a</font> as   <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">S</font>,<font color="Maroon">X</font> <b>by </b><i><a class="ref" href="msaterm.html#T1">MSATERM:1</a></i>;<br/>
<b>consider </b><font color="Maroon">C</font> being    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">S</font> <a href="msaterm.html#K1">-Terms</a> <font color="Maroon">X</font><b> such that </b><br/><a NAME="E30:5"/><i><font color="Green">E190</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n1</font> &amp; ( <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">C1</font> or <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> 0 ) &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n1</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>,<font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font>,<font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>] ) )
 <b>from </b><i><a class="ref" href="recdef_1.html#S6">RECDEF_1:sch 6</a>();<br/></i>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b>( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> implies  ex <font color="Olive">C0</font> being  <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">S</font>,<font color="Maroon">X</font> ex <font color="Olive">o</font> being  <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font> st <br/>( <font color="Olive">C0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">o</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K1">`1</a>  &amp;  <a href="msaterm.html#K7">the_sort_of</a> <font color="Olive">C0</font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Olive">o</font> &amp;  <a href="trees_1.html#K6">height</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Olive">C0</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> ) );<br/>
<a NAME="E31:5"/><i><font color="Green">E192</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
;<br/>
<a NAME="E32:5"/><i><font color="Green">E193</font></i>: 
for <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font> <a href="xcmplx_0.html#K2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:5_4"/><i><font color="Green">E194</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> implies  ex <font color="Olive">Ck</font> being  <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">S</font>,<font color="Maroon">X</font> ex <font color="Olive">o</font> being  <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font> st <br/>( <font color="Olive">Ck</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> &amp; <font color="Olive">o</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  &amp;  <a href="msaterm.html#K7">the_sort_of</a> <font color="Olive">Ck</font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Olive">o</font> &amp;  <a href="trees_1.html#K6">height</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Olive">Ck</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> ) )
 ;<br/>

<b>assume </b><a NAME="E2:5_4"/><i><font color="Green">E195</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1 &amp; <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> )
 ;<br/>

<a NAME="E3:5_4"/><i><font color="Green">E196</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E4:5_4"/><b>then </b><i><font color="Green">E197</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E5:5_4"/><i><font color="Green">E198</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:5_4_1"/>
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_4_1_1"><b>suppose </b></a><a NAME="E1:5_4_1_1"/><i><font color="Green">E199</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>take </b>
<font color="Maroon">C1</font>
;<br/><b>take </b>
<font color="Maroon">o</font>
;<br/><b>thus </b><a NAME="E2:5_4_1_1"/>
<font color="Maroon">C1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
 <b>by </b><i>, , </i>;<br/><b>thus </b><a NAME="E3:5_4_1_1"/>
<font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K1">`1</a> 
 <b>by </b><i>, , <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/><a NAME="E4:5_4_1_1"/>
<font color="Maroon">C1</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/><b>hence </b><a NAME="E5:5_4_1_1"/>
 <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">C1</font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font>
 <b>by </b><i><a class="ref" href="msaterm.html#T17">MSATERM:17</a></i>;<br/><a NAME="E6:5_4_1_1"/><i><font color="Green">E200</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">C1</font> <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">a</font></span>)</span>
 <b>by </b><i><a class="ref" href="trees_4.html#T10">TREES_4:10</a></i>;<br/><a NAME="E7:5_4_1_1"/><i><font color="Green">E201</font></i>: 
 <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">a</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="trees_3.html#T39">TREES_3:39</a></i>;<br/><b>consider </b><font color="Maroon">i</font> being   <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">args</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="finseq_1.html#K13">*</a> <b> such that </b><br/><a NAME="E9:5_4_1_1"/><i><font color="Green">E202</font></i>: 
( the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">args</font> &amp; <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">args</font> &amp; <font color="Maroon">args</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">rs1</font> )
 <b>by </b><i/>;<br/><a NAME="E10:5_4_1_1"/><i><font color="Green">E203</font></i>: 
<font color="Maroon">args</font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font>
 <b>by </b><i>, <a class="ref" href="msualg_1.html#D6">MSUALG_1:def 6</a></i>;<br/><a NAME="E11:5_4_1_1"/><i><font color="Green">E204</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">args</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">args</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon">t</font> = <font color="Maroon">a</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">X</font> <b>by </b><i>, , , , <a class="ref" href="msaterm.html#T22">MSATERM:22</a></i>;<br/><b>reconsider </b><font color="Maroon">w</font> =  <a href="trees_4.html#K11">doms</a> <font color="Maroon">a</font> as  <a href="trees_3.html#V5">FinTree-yielding</a> <a href="finseq_1.html#NM1">FinSequence</a> ;<br/><b>reconsider </b><font color="Maroon">dt</font> =  <a href="relat_1.html#K1">dom</a> <font color="Maroon">t</font> as  <a href="finset_1.html#V1">finite</a> <a href="trees_1.html#NM1">Tree</a> ;<br/><a NAME="E15:5_4_1_1"/>
<span class="p1">(<span class="default"><a href="trees_4.html#K11">doms</a> <font color="Maroon">a</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="funct_6.html#T31">FUNCT_6:31</a></i>;<br/><a NAME="E16:5_4_1_1"/><b>then </b><i><font color="Green">E206</font></i>: 
 <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> <font color="Maroon">w</font>
 <b>by </b><i>, , , , , , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E17:5_4_1_1"/><i><font color="Green">E207</font></i>: 
<font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</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"><span class="p2">(<span class="default"><font color="Maroon">cX</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<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></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<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></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, , , </i>;<br/><a NAME="E18:5_4_1_1"/><b>then </b><i><font color="Green">E208</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a>  <a href="trees_1.html#K2">elementary_tree</a> 0
 <b>by </b><i><a class="ref" href="trees_4.html#T3">TREES_4:3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_4_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">t'</font> be  <a href="finset_1.html#V1">finite</a> <a href="trees_1.html#NM1">Tree</a>;<br/><b>assume </b><a NAME="E1:5_4_1_1_1"/>
<font color="Maroon">t'</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">w</font>
 ;<br/><b>then consider </b><font color="Maroon">j</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:5_4_1_1_1"/><i><font color="Green">E210</font></i>: 
( <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">w</font> &amp; <font color="Maroon">w</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t'</font> )
 <b>by </b><i><a class="ref" href="finseq_2.html#T11">FINSEQ_2:11</a></i>;<br/><a NAME="E4:5_4_1_1_1"/><i><font color="Green">E211</font></i>: 
<font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</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"><span class="p2">(<span class="default"><font color="Maroon">cX</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<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">j</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<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">j</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, , , </i>;<br/><b>reconsider </b><font color="Maroon">t''</font> = <font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> as   <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">S</font>,<font color="Maroon">X</font> <b>by </b><i>, , <a class="ref" href="msaterm.html#T22">MSATERM:22</a></i>;<br/><a NAME="E6:5_4_1_1_1"/><i><font color="Green">E213</font></i>: 
<font color="Maroon">w</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</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="funct_6.html#T31">FUNCT_6:31</a></i>;<br/><a NAME="E7:5_4_1_1_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">t''</font> <a href="hidden.html#R1">=</a>  <a href="trees_1.html#K2">elementary_tree</a> 0
 <b>by </b><i>, <a class="ref" href="trees_4.html#T3">TREES_4:3</a></i>;<br/><b>hence </b><a NAME="E8:5_4_1_1_1"/>
 <a href="trees_1.html#K6">height</a> <font color="Maroon">t'</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="trees_1.html#K6">height</a> <font color="Maroon">dt</font>
 <b>by </b><i>, , , <a class="ref" href="trees_4.html#T3">TREES_4:3</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E20:5_4_1_1"/>
 <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">C1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i>, , , <a class="ref" href="msscyc_2.html#D1" target="_self">Def1</a>, <a class="ref" href="trees_1.html#T79">TREES_1:79</a>, <a class="ref" href="trees_3.html#T82">TREES_3:82</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_4_1_2"><b>suppose </b></a><a NAME="E1:5_4_1_2"/><i><font color="Green">E214</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:5_4_1_2"/><b>then </b><i><font color="Green">E215</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/><b>consider </b><font color="Maroon">Ck</font> being   <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">S</font>,<font color="Maroon">X</font>, <font color="Maroon">o</font> being   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E4:5_4_1_2"/><i><font color="Green">E216</font></i>: 
( <font color="Maroon">Ck</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  &amp;  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">Ck</font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font> &amp;  <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">Ck</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> )
 <b>by </b><i>, , , , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/><b>reconsider </b><font color="Maroon">kk</font> = <font color="Maroon">k</font> as    <a href="subset_1.html#M2">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/><b>consider </b><font color="Maroon">o1</font> being   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font>, <font color="Maroon">rs1</font> being   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font>, <font color="Maroon">Ck'</font> being   <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">S</font>,<font color="Maroon">X</font>, <font color="Maroon">Ck1</font> being   <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">S</font>,<font color="Maroon">X</font>, <font color="Maroon">a</font> being    <a href="msaterm.html#M1">ArgumentSeq</a> of  <a href="msaterm.html#K2">Sym</a> <font color="Maroon">o1</font>,<font color="Maroon">X</font><b> such that </b><br/><a NAME="E7:5_4_1_2"/><i><font color="Green">E219</font></i>: 
( <font color="Maroon">Ck'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> &amp; <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Ck1</font> &amp; <font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">rs1</font></span><a href="tarski.html#K4">]</a></span> )
 <b>and </b><br/><a NAME="E8:5_4_1_2"/><i><font color="Green">E220</font></i>: 
<font color="Maroon">Ck1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o1</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">a</font>
 <b>and </b><br/><a NAME="E9:5_4_1_2"/><i><font color="Green">E221</font></i>: 
for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">a</font> holds <br/> ex <font color="Olive">t</font> being  <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">S</font>,<font color="Maroon">X</font> st <br/>( <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp;  <a href="msaterm.html#K7">the_sort_of</a> <font color="Olive">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">o1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; (  <a href="msaterm.html#K7">the_sort_of</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">rs1</font> &amp;  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">Ck'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">rs1</font> implies <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Ck'</font> ) &amp; ( (  <a href="msaterm.html#K7">the_sort_of</a> <font color="Olive">t</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">rs1</font> or  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">Ck'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">rs1</font> ) implies <font color="Olive">t</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"><span class="p2">(<span class="default"><font color="Maroon">cX</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="msaterm.html#K7">the_sort_of</a> <font color="Olive">t</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="msaterm.html#K7">the_sort_of</a> <font color="Olive">t</font></span>)</span></span><a href="tarski.html#K4">]</a></span> ) )
 <b>by </b><i>, , </i>;<br/><a NAME="E10:5_4_1_2"/><i><font color="Green">E222</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o1</font></span>)</span>
 <b>by </b><i><a class="ref" href="msaterm.html#T22">MSATERM:22</a></i>;<br/><a NAME="E11:5_4_1_2"/><i><font color="Green">E223</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">Ck1</font> <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">a</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="trees_4.html#T10">TREES_4:10</a></i>;<br/><a NAME="E12:5_4_1_2"/><i><font color="Green">E224</font></i>: 
 <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">a</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="trees_3.html#T39">TREES_3:39</a></i>;<br/><a NAME="E13:5_4_1_2"/><i><font color="Green">E225</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">a</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">a</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>set </b><font color="Maroon">ck1</font> = <font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><a NAME="E14:5_4_1_2"/><i><font color="Green">E226</font></i>: 
<font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c</font>
 <b>by </b><i>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E15:5_4_1_2"/><b>then </b>
<font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">EG</font>
 <b>by </b><i>, <a class="ref" href="dtconstr.html#T2">DTCONSTR:2</a></i>;<br/><b>then consider </b><font color="Maroon">o'</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">rs1'</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E17:5_4_1_2"/><i><font color="Green">E229</font></i>: 
( <font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">kk</font> <a href="nat_1.html#K1">+</a> 1</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>,<font color="Maroon">rs1'</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">o'</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font> &amp; <font color="Maroon">rs1'</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> &amp;  ex <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> ex <font color="Olive">args</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="finseq_1.html#K13">*</a>  st <br/>( the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o'</font> <a href="hidden.html#R1">=</a> <font color="Olive">args</font> &amp; <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Olive">args</font> &amp; <font color="Olive">args</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">rs1'</font> ) )
 <b>by </b><i/>;<br/><a NAME="E18:5_4_1_2"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c</font>
 <b>by </b><i>, , , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then reconsider </b><font color="Maroon">ck</font> = <font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">EG</font> <b>by </b><i>, <a class="ref" href="dtconstr.html#T2">DTCONSTR:2</a></i>;<br/><b>reconsider </b><font color="Maroon">ck1</font> = <font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">EG</font> <b>by </b><i>, , <a class="ref" href="dtconstr.html#T2">DTCONSTR:2</a></i>;<br/><i><font color="Green">E231</font></i>:  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">Ck</font> = 
the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">ck</font> <a href="mcart_1.html#K1">`1</a> </span>)</span>
<b>by </b><i>, <a class="ref" href="msualg_1.html#D7">MSUALG_1:def 7</a></i>
<br/>.= 
the <a href="graph_1.html#U4">Target</a> of <span class="p1">(<span class="default"><a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">ck</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="graph_2.html#K7">vertex-seq</a> <font color="Maroon">c</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, , , , , <a class="ref" href="msscyc_1.html#T11">MSSCYC_1:11</a></i>
<br/>.= 
the <a href="graph_1.html#U3">Source</a> of <span class="p1">(<span class="default"><a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">ck1</font>
<b>by </b><i>, , , , <a class="ref" href="msscyc_1.html#T11">MSSCYC_1:11</a></i>
<br/>.= 
<font color="Maroon">ck1</font> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i/>
<br/>.= 
<font color="Maroon">rs1</font>
<b>by </b><i>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
;<br/><a NAME="E22:5_4_1_2"/><i><font color="Green">E232</font></i>: 
( <font color="Maroon">o1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o'</font> &amp; <font color="Maroon">rs1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">rs1'</font> )
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>then consider </b><font color="Maroon">i</font> being   <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">args</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="finseq_1.html#K13">*</a> <b> such that </b><br/><a NAME="E24:5_4_1_2"/><i><font color="Green">E233</font></i>: 
( the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">args</font> &amp; <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">args</font> &amp; <font color="Maroon">args</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">rs1</font> )
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">o'</font> = <font color="Maroon">o'</font> as   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font> <b>by </b><i/>;<br/><a NAME="E26:5_4_1_2"/><i><font color="Green">E234</font></i>: 
<font color="Maroon">args</font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o'</font>
 <b>by </b><i><a class="txt" href="msscyc_2.html#E4:5"><i><font color="Green">E43</font></i></a>, <a class="ref" href="msualg_1.html#D6">MSUALG_1:def 6</a></i>;<br/><a NAME="E27:5_4_1_2"/><i><font color="Green">E235</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">args</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">args</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">t</font> being   <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">S</font>,<font color="Maroon">X</font><b> such that </b><br/><a NAME="E29:5_4_1_2"/><i><font color="Green">E236</font></i>: 
( <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</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">o1</font></span>)</span> <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> <font color="Maroon">rs1</font> &amp;  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">Ck'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">rs1</font> implies <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Ck'</font> ) &amp; ( (  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">t</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">rs1</font> or  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">Ck'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">rs1</font> ) implies <font color="Maroon">t</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"><span class="p2">(<span class="default"><font color="Maroon">cX</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">t</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">t</font></span>)</span></span><a href="tarski.html#K4">]</a></span> ) )
 <b>by </b><i>, , , <a class="txt" href="msscyc_2.html#E2:5"><i><font color="Green">E42</font></i></a>, <a class="txt" href="msscyc_2.html#E4:5"><i><font color="Green">E43</font></i></a>, </i>;<br/><b>reconsider </b><font color="Maroon">w</font> =  <a href="trees_4.html#K11">doms</a> <font color="Maroon">a</font> as  <a href="trees_3.html#V5">FinTree-yielding</a> <a href="finseq_1.html#NM1">FinSequence</a> ;<br/><b>reconsider </b><font color="Maroon">dt</font> =  <a href="relat_1.html#K1">dom</a> <font color="Maroon">t</font> as  <a href="finset_1.html#V1">finite</a> <a href="trees_1.html#NM1">Tree</a> ;<br/><a NAME="E32:5_4_1_2"/>
<span class="p1">(<span class="default"><a href="trees_4.html#K11">doms</a> <font color="Maroon">a</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="txt" href="msscyc_2.html#E2:5"><i><font color="Green">E42</font></i></a>, <a class="txt" href="msscyc_2.html#E4:5"><i><font color="Green">E43</font></i></a>, , , <a class="ref" href="msscyc_2.html#T1" target="_self">Th1</a>, <a class="ref" href="funct_6.html#T31">FUNCT_6:31</a></i>;<br/><a NAME="E33:5_4_1_2"/><b>then </b><i><font color="Green">E237</font></i>: 
 <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> <font color="Maroon">w</font>
 <b>by </b><i>, <a class="ref" href="msscyc_2.html#D1" target="_self">Def1</a>, , <a class="txt" href="msscyc_2.html#E2:5"><i><font color="Green">E42</font></i></a>, <a class="txt" href="msscyc_2.html#E4:5"><i><font color="Green">E43</font></i></a>, , , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><div><i><font color="Green">E238</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="5_4_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">t'</font> be  <a href="finset_1.html#V1">finite</a> <a href="trees_1.html#NM1">Tree</a>;<br/><b>assume </b><a NAME="E1:5_4_1_2_2"/>
<font color="Maroon">t'</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">w</font>
 ;<br/><b>then consider </b><font color="Maroon">j</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:5_4_1_2_2"/><i><font color="Green">E239</font></i>: 
( <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">w</font> &amp; <font color="Maroon">w</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t'</font> )
 <b>by </b><i><a class="ref" href="finseq_2.html#T11">FINSEQ_2:11</a></i>;<br/><b>consider </b><font color="Maroon">t''</font> being   <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">S</font>,<font color="Maroon">X</font><b> such that </b><br/><a NAME="E5:5_4_1_2_2"/><i><font color="Green">E240</font></i>: 
( <font color="Maroon">t''</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</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">o1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> &amp; (  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">t''</font> <a href="hidden.html#R1">=</a> <font color="Maroon">rs1</font> &amp;  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">Ck'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">rs1</font> implies <font color="Maroon">t''</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Ck'</font> ) &amp; ( (  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">t''</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">rs1</font> or  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">Ck'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">rs1</font> ) implies <font color="Maroon">t''</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"><span class="p2">(<span class="default"><font color="Maroon">cX</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">t''</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">t''</font></span>)</span></span><a href="tarski.html#K4">]</a></span> ) )
 <b>by </b><i>, <a class="ref" href="msscyc_2.html#D1" target="_self">Def1</a>, <a class="ref" href="msscyc_2.html#D2" target="_self">Def2</a></i>;<br/><a NAME="E6:5_4_1_2_2"/><i><font color="Green">E241</font></i>: 
<font color="Maroon">w</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</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="msscyc_2.html#D1" target="_self">Def1</a>, <a class="ref" href="msscyc_2.html#D2" target="_self">Def2</a>, , <a class="ref" href="funct_6.html#T31">FUNCT_6:31</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_4_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:5_4_1_2_2_1"/>
( (  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">t''</font> <a href="hidden.html#R1">=</a> <font color="Maroon">rs1</font> &amp;  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">Ck'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">rs1</font> ) or  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">t''</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">rs1</font> or  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">Ck'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">rs1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_4_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:5_4_1_2_2_1_1"/>
(  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">t''</font> <a href="hidden.html#R1">=</a> <font color="Maroon">rs1</font> &amp;  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">Ck'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">rs1</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:5_4_1_2_2_1_1"/>
 <a href="trees_1.html#K6">height</a> <font color="Maroon">t'</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="trees_1.html#K6">height</a> <font color="Maroon">dt</font>
 <b>by </b><i><a class="ref" href="msscyc_2.html#D1" target="_self">Def1</a>, <a class="txt" href="msscyc_2.html#E2:5"><i><font color="Green">E42</font></i></a>, <a class="txt" href="msscyc_2.html#E4:5"><i><font color="Green">E43</font></i></a>, <a class="ref" href="msscyc_2.html#T1" target="_self">Th1</a>, <a class="ref" href="msscyc_2.html#D2" target="_self">Def2</a>, , <a class="ref" href="funct_6.html#T31">FUNCT_6:31</a>, <a class="ref" href="msualg_1.html#D6">MSUALG_1:def 6</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_4_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:5_4_1_2_2_1_2"/>
(  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">t''</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">rs1</font> or  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">Ck'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">rs1</font> )
 ;<br/><div class="add"><a NAME="E2:5_4_1_2_2_1_2"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">t''</font> <a href="hidden.html#R1">=</a>  <a href="trees_1.html#K2">elementary_tree</a> 0
 <b>by </b><i>, <a class="ref" href="trees_4.html#T3">TREES_4:3</a></i>;<br/><b>hence </b><a NAME="E3:5_4_1_2_2_1_2"/>
 <a href="trees_1.html#K6">height</a> <font color="Maroon">t'</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="trees_1.html#K6">height</a> <font color="Maroon">dt</font>
 <b>by </b><i><a class="ref" href="msscyc_2.html#D2" target="_self">Def2</a>, , <a class="ref" href="trees_1.html#T79">TREES_1:79</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>take </b>
<font color="Maroon">Ck1</font>
;<br/><b>take </b>
<font color="Maroon">o1</font>
;<br/><b>thus </b><a NAME="E35:5_4_1_2"/>
<font color="Maroon">Ck1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E36:5_4_1_2"/>
<font color="Maroon">o1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K1">`1</a> 
 <b>by </b><i>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/><a NAME="E37:5_4_1_2"/>
<font color="Maroon">Ck1</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">o1</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/><b>hence </b><a NAME="E38:5_4_1_2"/>
 <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">Ck1</font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o1</font>
 <b>by </b><i><a class="ref" href="msaterm.html#T17">MSATERM:17</a></i>;<br/><b>thus </b><a NAME="E39:5_4_1_2"/>
 <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">Ck1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i>, , , <a class="txt" href="msscyc_2.html#E1:5"><i><font color="Green">E41</font></i></a>, <a class="txt" href="msscyc_2.html#E2:5"><i><font color="Green">E42</font></i></a>, <a class="txt" href="msscyc_2.html#E4:5"><i><font color="Green">E43</font></i></a>, <a class="ref" href="msscyc_2.html#T1" target="_self">Th1</a>, , , <a class="ref" href="msualg_1.html#D6">MSUALG_1:def 6</a>, <a class="ref" href="trees_3.html#T82">TREES_3:82</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E33:5"/>
for <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2">NAT_1:sch 2</a>(, );<br/></i>
<b>then consider </b><font color="Maroon">Cn</font> being   <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">S</font>,<font color="Maroon">X</font>, <font color="Maroon">o</font> being   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E35:5"/><i><font color="Green">E243</font></i>: 
( <font color="Maroon">Cn</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  &amp;  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">Cn</font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font> &amp;  <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">Cn</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 <b>by </b><i/>;<br/>
<a NAME="E36:5"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i>, </i>;<br/>
<a NAME="E37:5"/><b>then </b><i><font color="Green">E244</font></i>: 
not <font color="Maroon">c</font> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E38:5"/><i><font color="Green">E245</font></i>: 
not  <a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">S</font> is <a href="msscyc_1.html#V2">empty</a>
 
<b>by </b><i>, <a class="ref" href="msscyc_1.html#D3">MSSCYC_1:def 3</a></i>;<br/>
<b>set </b><font color="Maroon">cn</font> = <font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font></span>)</span>;<br/>
<a NAME="E39:5"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c</font>
 
<b>by </b><i>, , <a class="ref" href="finseq_5.html#T6">FINSEQ_5:6</a></i>;<br/>
<a NAME="E40:5"/><b>then </b><i><font color="Green">E247</font></i>: 
<font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="msscyc_2.html#K1" target="_self">InducedEdges</a> <font color="Maroon">S</font>
 
<b>by </b><i>, , <a class="ref" href="dtconstr.html#T2">DTCONSTR:2</a></i>;<br/>
<span class="p1">(<span class="default"><a href="graph_2.html#K7">vertex-seq</a> <font color="Maroon">c</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
the <a href="graph_1.html#U4">Target</a> of <span class="p1">(<span class="default"><a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="msscyc_1.html#T14">MSSCYC_1:14</a></i>
<br/>.= 
the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span>
<b>by </b><i>, </i>
<br/>.= 
 <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font>
<b>by </b><i>, , <a class="ref" href="msualg_1.html#D7">MSUALG_1:def 7</a></i>
;<br/>
<b>then reconsider </b><font color="Maroon">Cn</font> = <font color="Maroon">Cn</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> <b>by </b><i>, , , <a class="ref" href="msaterm.html#D5">MSATERM:def 5</a></i>;<br/>
<b>take </b>
<font color="Maroon">Cn</font>
;<br/>

<b>thus </b><a NAME="E43:5"/>
 <a href="msafree2.html#K9">depth</a> <font color="Maroon">Cn</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="msafree2.html#D14">MSAFREE2:def 14</a></i>;<br/>


</div>
