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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="graph_1.html#NM1">Graph</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be  <a href="graph_1.html#V8">oriented</a>  <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<b>assume </b><a NAME="E1:32"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="graph_4.html#R3" target="_self">is_oriented_vertex_seq_of</a> <font color="Maroon">c<sub>3</sub></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<sub>3</sub></font> is <a href="graph_4.html#V1" target="_self">Simple</a> or  not <font color="Maroon">c<sub>3</sub></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">E20</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="graph_4.html#V1" target="_self">Simple</a>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as    <a href="graph_2.html#M1">FinSubsequence</a> of <font color="Maroon">c<sub>3</sub></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">c<sub>5</sub></font> = <font color="Maroon">c<sub>2</sub></font> as    <a href="graph_2.html#M1">FinSubsequence</a> of <font color="Maroon">c<sub>2</sub></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">c<sub>6</sub></font> = <font color="Maroon">c<sub>3</sub></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">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="graph_4.html#E1:32_1_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="graph_4.html#T21" target="_self">Th21</a></i>;<br/><b>take </b>
<font color="Maroon">c<sub>4</sub></font>
;<br/><b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/><b>take </b>
<font color="Maroon">c<sub>6</sub></font>
;<br/><b>take </b>
<font color="Maroon">c<sub>2</sub></font>
;<br/><b>thus </b><a NAME="E5:32_1_1"/>
(  <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp;  <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></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">c<sub>2</sub></font> <a href="graph_4.html#R3" target="_self">is_oriented_vertex_seq_of</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E1:32"><i><font color="Green">E19</font></i></a></i>;<br/><b>thus </b><a NAME="E7:32_1_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> 1 &amp; <font color="Maroon">c<sub>2</sub></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<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></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<sub>2</sub></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">E20</font></i>: 
not <font color="Maroon">c<sub>3</sub></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="nat_1.html#NM1">Nat</a>] means ex <font color="Olive">b<sub>1</sub></font> being   <a href="graph_2.html#M1">FinSubsequence</a> of <font color="Maroon">c<sub>3</sub></font>ex <font color="Olive">b<sub>2</sub></font> being   <a href="graph_2.html#M1">FinSubsequence</a> of <font color="Maroon">c<sub>2</sub></font>ex <font color="Olive">b<sub>3</sub></font> being <a href="graph_1.html#V8">oriented</a>  <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">c<sub>1</sub></font>ex <font color="Olive">b<sub>4</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>(  <a href="finseq_1.html#K15">Seq</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> &amp;  <a href="finseq_1.html#K15">Seq</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>4</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="graph_4.html#R3" target="_self">is_oriented_vertex_seq_of</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>4</sub></font> <a href="funct_1.html#K1">.</a> 1 &amp; <font color="Maroon">c<sub>2</sub></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<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>4</sub></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">b<sub>4</sub></font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; not <font color="Olive">b<sub>3</sub></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<sub>3</sub></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">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as    <a href="graph_2.html#M1">FinSubsequence</a> of <font color="Maroon">c<sub>3</sub></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">c<sub>5</sub></font> = <font color="Maroon">c<sub>2</sub></font> as    <a href="graph_2.html#M1">FinSubsequence</a> of <font color="Maroon">c<sub>2</sub></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">c<sub>4</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>3</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>2</sub></font>
;<br/>

<b>thus </b><a NAME="E3:32_1_2_1"/>
(  <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></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">c<sub>2</sub></font> <a href="graph_4.html#R3" target="_self">is_oriented_vertex_seq_of</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E1:32"><i><font color="Green">E19</font></i></a></i>;<br/>

<b>thus </b><a NAME="E5:32_1_2_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> 1 &amp; <font color="Maroon">c<sub>2</sub></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<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></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<sub>2</sub></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<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> &amp; not <font color="Maroon">c<sub>3</sub></font> is <a href="graph_4.html#V1" target="_self">Simple</a> )
 <b>by </b><i><a class="txt" href="graph_4.html#E1:32_1_2"><i><font color="Green">E20</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:32_1_2"/><b>then </b><i><font color="Green">E21</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 ;<br/><b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E5:32_1_2"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>] &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5">NAT_1:sch 5</a>(<a class="txt" href="graph_4.html#E3:32_1_2"><i><font color="Green">E21</font></i></a>);<br/></i><b>consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="graph_2.html#M1">FinSubsequence</a> of <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>6</sub></font> being    <a href="graph_2.html#M1">FinSubsequence</a> of <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>7</sub></font> being  <a href="graph_1.html#V8">oriented</a>  <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>8</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E7:32_1_2"/><i><font color="Green">E23</font></i>: 
(  <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp;  <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="graph_4.html#R3" target="_self">is_oriented_vertex_seq_of</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> 1 &amp; <font color="Maroon">c<sub>2</sub></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<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></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<sub>8</sub></font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; not <font color="Maroon">c<sub>7</sub></font> is <a href="graph_4.html#V1" target="_self">Simple</a> )
 <b>by </b><i><a class="txt" href="graph_4.html#E5:32_1_2"><i><font color="Green">E22</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="graph_2.html#M1">FinSubsequence</a> of <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>10</sub></font> being    <a href="graph_2.html#M1">FinSubsequence</a> of <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>11</sub></font> being  <a href="graph_1.html#V8">oriented</a>  <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>12</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E9:32_1_2"/><i><font color="Green">E24</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="graph_4.html#R3" target="_self">is_oriented_vertex_seq_of</a> <font color="Maroon">c<sub>11</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> 1 &amp; <font color="Maroon">c<sub>8</sub></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<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></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<sub>12</sub></font></span>)</span> &amp;  <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp;  <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="txt" href="graph_4.html#E7:32_1_2"><i><font color="Green">E23</font></i></a>, <a class="ref" href="graph_4.html#T22" target="_self">Th22</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>5</sub></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">c<sub>5</sub></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">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span> as    <a href="graph_2.html#M1">FinSubsequence</a> of <font color="Maroon">c<sub>3</sub></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">E25</font></i>: 
 <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E7:32_1_2"><i><font color="Green">E23</font></i></a>, <a class="txt" href="graph_4.html#E9:32_1_2"><i><font color="Green">E24</font></i></a>, <a class="ref" href="graph_2.html#T30">GRAPH_2:30</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>6</sub></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">c<sub>6</sub></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">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span> as    <a href="graph_2.html#M1">FinSubsequence</a> of <font color="Maroon">c<sub>2</sub></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">E26</font></i>: 
 <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E7:32_1_2"><i><font color="Green">E23</font></i></a>, <a class="txt" href="graph_4.html#E9:32_1_2"><i><font color="Green">E24</font></i></a>, <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">c<sub>11</sub></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">c<sub>1</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>11</sub></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">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="graph_4.html#T21" target="_self">Th21</a></i>;<br/><b>take </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>13</sub></font>;<br/><b>take </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>15</sub></font>;<br/><a NAME="E3:32_1_2_2"/>
 <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E7:32_1_2"><i><font color="Green">E23</font></i></a>, <a class="txt" href="graph_4.html#E9:32_1_2"><i><font color="Green">E24</font></i></a>, <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">b<sub>1</sub></font> being   <a href="graph_2.html#M1">FinSubsequence</a> of <font color="Maroon">c<sub>3</sub></font>ex <font color="Olive">b<sub>2</sub></font> being   <a href="graph_2.html#M1">FinSubsequence</a> of <font color="Maroon">c<sub>2</sub></font>ex <font color="Olive">b<sub>3</sub></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">c<sub>1</sub></font>ex <font color="Olive">b<sub>4</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>(  <a href="finseq_1.html#K15">Seq</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> &amp;  <a href="finseq_1.html#K15">Seq</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>4</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="graph_4.html#R3" target="_self">is_oriented_vertex_seq_of</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>4</sub></font> <a href="funct_1.html#K1">.</a> 1 &amp; <font color="Maroon">c<sub>2</sub></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<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>4</sub></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">b<sub>4</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="graph_4.html#E7:32_1_2"><i><font color="Green">E23</font></i></a>, <a class="txt" href="graph_4.html#E9:32_1_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="graph_4.html#E13:32_1_2"><i><font color="Green">E26</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E15:32_1_2"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="graph_2.html#M1">FinSubsequence</a> of <font color="Maroon">c<sub>3</sub></font>ex <font color="Olive">b<sub>2</sub></font> being   <a href="graph_2.html#M1">FinSubsequence</a> of <font color="Maroon">c<sub>2</sub></font>ex <font color="Olive">b<sub>3</sub></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">c<sub>1</sub></font>ex <font color="Olive">b<sub>4</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>(  <a href="finseq_1.html#K15">Seq</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> &amp;  <a href="finseq_1.html#K15">Seq</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>4</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="graph_4.html#R3" target="_self">is_oriented_vertex_seq_of</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>4</sub></font> <a href="funct_1.html#K1">.</a> 1 &amp; <font color="Maroon">c<sub>2</sub></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<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>4</sub></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">b<sub>4</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="graph_4.html#E5:32_1_2"><i><font color="Green">E22</font></i></a>, <a class="txt" href="graph_4.html#E7:32_1_2"><i><font color="Green">E23</font></i></a>, <a class="txt" href="graph_4.html#E9:32_1_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="graph_4.html#E11:32_1_2"><i><font color="Green">E25</font></i></a>, <a class="txt" href="graph_4.html#E13:32_1_2"><i><font color="Green">E26</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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