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<div class="add">

<b>let </b><font color="Maroon">G</font> be   <a href="graph_1.html#NM1">Graph</a>;<br/><b>let </b><font color="Maroon">vs</font> be    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">c</font> be  <a href="graph_1.html#V8">oriented</a>  <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">G</font>;<br/>





<b>assume </b><a NAME="E1:32"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">vs</font> <a href="graph_4.html#R3" target="_self">is_oriented_vertex_seq_of</a> <font color="Maroon">c</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_1"/>
( <font color="Maroon">c</font> is <a href="graph_4.html#V1" target="_self">Simple</a> or  not <font color="Maroon">c</font> is <a href="graph_4.html#V1" target="_self">Simple</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1"><b>suppose </b></a><a NAME="E1:32_1_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c</font> is <a href="graph_4.html#V1" target="_self">Simple</a>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">fc</font> = <font color="Maroon">c</font> as    <a href="graph_2.html#M1">FinSubsequence</a> of <font color="Maroon">c</font> <b>by </b><i><a class="ref" href="graph_2.html#T28">GRAPH_2:28</a></i>;<br/><b>reconsider </b><font color="Maroon">fvs</font> = <font color="Maroon">vs</font> as    <a href="graph_2.html#M1">FinSubsequence</a> of <font color="Maroon">vs</font> <b>by </b><i><a class="ref" href="graph_2.html#T28">GRAPH_2:28</a></i>;<br/><b>reconsider </b><font color="Maroon">sc</font> = <font color="Maroon">c</font> as  <a href="graph_1.html#V8">oriented</a> <a href="graph_2.html#V3">simple</a>  <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">G</font> <b>by </b><i>, </i>;<br/><b>take </b>
<font color="Maroon">fc</font>
;<br/><b>take </b>
<font color="Maroon">fvs</font>
;<br/><b>take </b>
<font color="Maroon">sc</font>
;<br/><b>take </b>
<font color="Maroon">vs</font>
;<br/><b>thus </b><a NAME="E5:32_1_1"/>
(  <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">fc</font> <a href="hidden.html#R1">=</a> <font color="Maroon">sc</font> &amp;  <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">fvs</font> <a href="hidden.html#R1">=</a> <font color="Maroon">vs</font> )
 <b>by </b><i><a class="ref" href="graph_2.html#T24">GRAPH_2:24</a></i>;<br/><b>thus </b><a NAME="E6:32_1_1"/>
<font color="Maroon">vs</font> <a href="graph_4.html#R3" target="_self">is_oriented_vertex_seq_of</a> <font color="Maroon">sc</font>
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E7:32_1_1"/>
( <font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> 1 &amp; <font color="Maroon">vs</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">vs</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">vs</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">vs</font></span>)</span> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_2"><b>suppose </b></a><a NAME="E1:32_1_2"/><i><font color="Green">E45</font></i>: 
not <font color="Maroon">c</font> is <a href="graph_4.html#V1" target="_self">Simple</a>
 ;<br/><div class="add"><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b> ex <font color="Olive">fc</font> being   <a href="graph_2.html#M1">FinSubsequence</a> of <font color="Maroon">c</font> ex <font color="Olive">fvs</font> being   <a href="graph_2.html#M1">FinSubsequence</a> of <font color="Maroon">vs</font> ex <font color="Olive">c1</font> being <a href="graph_1.html#V8">oriented</a>  <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">G</font> ex <font color="Olive">vs1</font> being   <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G</font> st <br/>(  <a href="finseq_1.html#K15">Seq</a> <font color="Olive">fc</font> <a href="hidden.html#R1">=</a> <font color="Olive">c1</font> &amp;  <a href="finseq_1.html#K15">Seq</a> <font color="Olive">fvs</font> <a href="hidden.html#R1">=</a> <font color="Olive">vs1</font> &amp; <font color="Olive">vs1</font> <a href="graph_4.html#R3" target="_self">is_oriented_vertex_seq_of</a> <font color="Olive">c1</font> &amp; <font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Olive">vs1</font> <a href="funct_1.html#K1">.</a> 1 &amp; <font color="Maroon">vs</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">vs</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">vs1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">vs1</font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">c1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; not <font color="Olive">c1</font> is <a href="graph_4.html#V1" target="_self">Simple</a> );<br/><a NAME="E2:32_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_2_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">fc</font> = <font color="Maroon">c</font> as    <a href="graph_2.html#M1">FinSubsequence</a> of <font color="Maroon">c</font> <b>by </b><i><a class="ref" href="graph_2.html#T28">GRAPH_2:28</a></i>;<br/>
<b>reconsider </b><font color="Maroon">fvs</font> = <font color="Maroon">vs</font> as    <a href="graph_2.html#M1">FinSubsequence</a> of <font color="Maroon">vs</font> <b>by </b><i><a class="ref" href="graph_2.html#T28">GRAPH_2:28</a></i>;<br/>
<b>take </b>
<font color="Maroon">fc</font>
;<br/>

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

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

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

<b>thus </b><a NAME="E3:32_1_2_1"/>
(  <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">fc</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> &amp;  <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">fvs</font> <a href="hidden.html#R1">=</a> <font color="Maroon">vs</font> )
 <b>by </b><i><a class="ref" href="graph_2.html#T24">GRAPH_2:24</a></i>;<br/>

<b>thus </b><a NAME="E4:32_1_2_1"/>
<font color="Maroon">vs</font> <a href="graph_4.html#R3" target="_self">is_oriented_vertex_seq_of</a> <font color="Maroon">c</font>
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E5:32_1_2_1"/>
( <font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> 1 &amp; <font color="Maroon">vs</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">vs</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">vs</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">vs</font></span>)</span> )
 ;<br/>

<b>thus </b><a NAME="E6:32_1_2_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font> &amp; not <font color="Maroon">c</font> is <a href="graph_4.html#V1" target="_self">Simple</a> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><a NAME="E3:32_1_2"/><b>then </b><i><font color="Green">E46</font></i>: 
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 ;<br/><b>consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E5:32_1_2"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>] &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5">NAT_1:sch 5</a>();<br/></i><b>consider </b><font color="Maroon">fc</font> being    <a href="graph_2.html#M1">FinSubsequence</a> of <font color="Maroon">c</font>, <font color="Maroon">fvs</font> being    <a href="graph_2.html#M1">FinSubsequence</a> of <font color="Maroon">vs</font>, <font color="Maroon">c1</font> being  <a href="graph_1.html#V8">oriented</a>  <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">G</font>, <font color="Maroon">vs1</font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E7:32_1_2"/><i><font color="Green">E48</font></i>: 
(  <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">fc</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c1</font> &amp;  <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">fvs</font> <a href="hidden.html#R1">=</a> <font color="Maroon">vs1</font> &amp; <font color="Maroon">vs1</font> <a href="graph_4.html#R3" target="_self">is_oriented_vertex_seq_of</a> <font color="Maroon">c1</font> &amp; <font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">vs1</font> <a href="funct_1.html#K1">.</a> 1 &amp; <font color="Maroon">vs</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">vs</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">vs1</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">vs1</font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp; not <font color="Maroon">c1</font> is <a href="graph_4.html#V1" target="_self">Simple</a> )
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">fc1</font> being    <a href="graph_2.html#M1">FinSubsequence</a> of <font color="Maroon">c1</font>, <font color="Maroon">fvs1</font> being    <a href="graph_2.html#M1">FinSubsequence</a> of <font color="Maroon">vs1</font>, <font color="Maroon">c2</font> being  <a href="graph_1.html#V8">oriented</a>  <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">G</font>, <font color="Maroon">vs2</font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E9:32_1_2"/><i><font color="Green">E49</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c1</font> &amp; <font color="Maroon">vs2</font> <a href="graph_4.html#R3" target="_self">is_oriented_vertex_seq_of</a> <font color="Maroon">c2</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs1</font> &amp; <font color="Maroon">vs1</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">vs2</font> <a href="funct_1.html#K1">.</a> 1 &amp; <font color="Maroon">vs1</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">vs1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">vs2</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">vs2</font></span>)</span> &amp;  <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">fc1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c2</font> &amp;  <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">fvs1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">vs2</font> )
 <b>by </b><i>, </i>;<br/><b>reconsider </b><font color="Maroon">fc'</font> = <font color="Maroon">fc</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p4">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">fc</font></span>)</span></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">fc1</font></span>)</span></span>)</span></span>)</span> as    <a href="graph_2.html#M1">FinSubsequence</a> of <font color="Maroon">c</font> <b>by </b><i><a class="ref" href="graph_2.html#T29">GRAPH_2:29</a></i>;<br/><a NAME="E11:32_1_2"/><i><font color="Green">E52</font></i>: 
 <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">fc'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c2</font>
 <b>by </b><i>, , <a class="ref" href="graph_2.html#T30">GRAPH_2:30</a></i>;<br/><b>reconsider </b><font color="Maroon">fvs'</font> = <font color="Maroon">fvs</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p4">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">fvs</font></span>)</span></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">fvs1</font></span>)</span></span>)</span></span>)</span> as    <a href="graph_2.html#M1">FinSubsequence</a> of <font color="Maroon">vs</font> <b>by </b><i><a class="ref" href="graph_2.html#T29">GRAPH_2:29</a></i>;<br/><a NAME="E13:32_1_2"/><i><font color="Green">E53</font></i>: 
 <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">fvs'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">vs2</font>
 <b>by </b><i>, , <a class="ref" href="graph_2.html#T30">GRAPH_2:30</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:32_1_2_2"/>
<font color="Maroon">c2</font> is  <a href="graph_1.html#V8">oriented</a> <a href="graph_4.html#V1" target="_self">Simple</a>  <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">G</font>
 ;<br/><b>then reconsider </b><font color="Maroon">sc</font> = <font color="Maroon">c2</font> as  <a href="graph_1.html#V8">oriented</a> <a href="graph_2.html#V3">simple</a>  <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">G</font> <b>by </b><i/>;<br/><b>take </b><font color="Maroon">fc'</font> = <font color="Maroon">fc'</font>;<br/><b>take </b><font color="Maroon">sc</font> = <font color="Maroon">sc</font>;<br/><a NAME="E3:32_1_2_2"/>
 <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">fc'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">sc</font>
 <b>by </b><i>, , <a class="ref" href="graph_2.html#T30">GRAPH_2:30</a></i>;<br/><b>hence </b><a NAME="E4:32_1_2_2"/>
 ex <font color="Olive">fc</font> being   <a href="graph_2.html#M1">FinSubsequence</a> of <font color="Maroon">c</font> ex <font color="Olive">fvs</font> being   <a href="graph_2.html#M1">FinSubsequence</a> of <font color="Maroon">vs</font> ex <font color="Olive">sc</font> being <a href="graph_1.html#V8">oriented</a> <a href="graph_2.html#V3">simple</a>  <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">G</font> ex <font color="Olive">vs1</font> being   <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G</font> st <br/>(  <a href="finseq_1.html#K15">Seq</a> <font color="Olive">fc</font> <a href="hidden.html#R1">=</a> <font color="Olive">sc</font> &amp;  <a href="finseq_1.html#K15">Seq</a> <font color="Olive">fvs</font> <a href="hidden.html#R1">=</a> <font color="Olive">vs1</font> &amp; <font color="Olive">vs1</font> <a href="graph_4.html#R3" target="_self">is_oriented_vertex_seq_of</a> <font color="Olive">sc</font> &amp; <font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Olive">vs1</font> <a href="funct_1.html#K1">.</a> 1 &amp; <font color="Maroon">vs</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">vs</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">vs1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">vs1</font></span>)</span> )
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E15:32_1_2"/>
 ex <font color="Olive">fc</font> being   <a href="graph_2.html#M1">FinSubsequence</a> of <font color="Maroon">c</font> ex <font color="Olive">fvs</font> being   <a href="graph_2.html#M1">FinSubsequence</a> of <font color="Maroon">vs</font> ex <font color="Olive">sc</font> being <a href="graph_1.html#V8">oriented</a> <a href="graph_2.html#V3">simple</a>  <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">G</font> ex <font color="Olive">vs1</font> being   <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G</font> st <br/>(  <a href="finseq_1.html#K15">Seq</a> <font color="Olive">fc</font> <a href="hidden.html#R1">=</a> <font color="Olive">sc</font> &amp;  <a href="finseq_1.html#K15">Seq</a> <font color="Olive">fvs</font> <a href="hidden.html#R1">=</a> <font color="Olive">vs1</font> &amp; <font color="Olive">vs1</font> <a href="graph_4.html#R3" target="_self">is_oriented_vertex_seq_of</a> <font color="Olive">sc</font> &amp; <font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Olive">vs1</font> <a href="funct_1.html#K1">.</a> 1 &amp; <font color="Maroon">vs</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">vs</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">vs1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">vs1</font></span>)</span> )
 <b>by </b><i>, , , , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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