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<b>let </b><font color="Maroon" title="c1">G</font> be   <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Graph</a>;<br/><b>let </b><font color="Maroon" title="c2">vs</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c1">G</font>;<br/><b>let </b><font color="Maroon" title="c3">c</font> be  <a href="graph_1.html#V8" title="GRAPH_1:attr.8">oriented</a>  <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c1">G</font>;<br/>





<b>assume </b><a NAME="E1:32"/><i><font color="Green" title="E19">A1</font></i>: 
<font color="Maroon" title="c2">vs</font> <a href="graph_4.html#R3" title="GRAPH_4:pred.3">is_oriented_vertex_seq_of</a> <font color="Maroon" title="c3">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" title="c3">c</font> is <a href="graph_4.html#V1" title="GRAPH_4:attr.1">Simple</a> or  not <font color="Maroon" title="c3">c</font> is <a href="graph_4.html#V1" title="GRAPH_4:attr.1">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" title="E20">A2</font></i>: 
<font color="Maroon" title="c3">c</font> is <a href="graph_4.html#V1" title="GRAPH_4:attr.1">Simple</a>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon" title="c4">fc</font> = <font color="Maroon" title="c3">c</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c3">c</font> <b>by </b><i><a class="ref" href="graph_2.html#T28" title="GRAPH_2:th.28">GRAPH_2:28</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c5">fvs</font> = <font color="Maroon" title="c2">vs</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">vs</font> <b>by </b><i><a class="ref" href="graph_2.html#T28" title="GRAPH_2:th.28">GRAPH_2:28</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c6">sc</font> = <font color="Maroon" title="c3">c</font> as  <a href="graph_1.html#V8" title="GRAPH_1:attr.8">oriented</a> <a href="graph_2.html#V3" title="GRAPH_2:attr.3">simple</a>  <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="txt" href="graph_4.html#E1:32_1_1"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="graph_4.html#T21" target="_self" title="GRAPH_4:th.21">Th21</a></i>;<br/><b>take </b>
<font color="Maroon" title="c4">fc</font>
;<br/><b>take </b>
<font color="Maroon" title="c5">fvs</font>
;<br/><b>take </b>
<font color="Maroon" title="c6">sc</font>
;<br/><b>take </b>
<font color="Maroon" title="c2">vs</font>
;<br/><b>thus </b><a NAME="E5:32_1_1"/>
(  <a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Maroon" title="c4">fc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">sc</font> &amp;  <a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Maroon" title="c5">fvs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">vs</font> )
 <b>by </b><i><a class="ref" href="finseq_3.html#T125" title="FINSEQ_3:th.125">FINSEQ_3:125</a></i>;<br/><b>thus </b><a NAME="E6:32_1_1"/>
<font color="Maroon" title="c2">vs</font> <a href="graph_4.html#R3" title="GRAPH_4:pred.3">is_oriented_vertex_seq_of</a> <font color="Maroon" title="c6">sc</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E1:32"><i><font color="Green" title="E19">A1</font></i></a></i>;<br/><b>thus </b><a NAME="E7:32_1_1"/>
( <font color="Maroon" title="c2">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 &amp; <font color="Maroon" title="c2">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">vs</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">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" title="E20">A3</font></i>: 
not <font color="Maroon" title="c3">c</font> is <a href="graph_4.html#V1" title="GRAPH_4:attr.1">Simple</a>
 ;<br/><div class="add"><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b> ex <font color="Olive" title="b1">fc</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c3">c</font> ex <font color="Olive" title="b2">fvs</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">vs</font> ex <font color="Olive" title="b3">c1</font> being <a href="graph_1.html#V8" title="GRAPH_1:attr.8">oriented</a>  <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c1">G</font> ex <font color="Olive" title="b4">vs1</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c1">G</font> st <br/>(  <a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Olive" title="b1">fc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c1</font> &amp;  <a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Olive" title="b2">fvs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">vs1</font> &amp; <font color="Olive" title="b4">vs1</font> <a href="graph_4.html#R3" title="GRAPH_4:pred.3">is_oriented_vertex_seq_of</a> <font color="Olive" title="b3">c1</font> &amp; <font color="Maroon" title="c2">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">vs1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 &amp; <font color="Maroon" title="c2">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">vs</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">vs1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b4">vs1</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; not <font color="Olive" title="b3">c1</font> is <a href="graph_4.html#V1" title="GRAPH_4:attr.1">Simple</a> );<br/><a NAME="E2:32_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">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" title="c4">fc</font> = <font color="Maroon" title="c3">c</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c3">c</font> <b>by </b><i><a class="ref" href="graph_2.html#T28" title="GRAPH_2:th.28">GRAPH_2:28</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">fvs</font> = <font color="Maroon" title="c2">vs</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">vs</font> <b>by </b><i><a class="ref" href="graph_2.html#T28" title="GRAPH_2:th.28">GRAPH_2:28</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c4">fc</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c5">fvs</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c3">c</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c2">vs</font>
;<br/>

<b>thus </b><a NAME="E3:32_1_2_1"/>
(  <a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Maroon" title="c4">fc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font> &amp;  <a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Maroon" title="c5">fvs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">vs</font> )
 <b>by </b><i><a class="ref" href="finseq_3.html#T125" title="FINSEQ_3:th.125">FINSEQ_3:125</a></i>;<br/>

<b>thus </b><a NAME="E4:32_1_2_1"/>
<font color="Maroon" title="c2">vs</font> <a href="graph_4.html#R3" title="GRAPH_4:pred.3">is_oriented_vertex_seq_of</a> <font color="Maroon" title="c3">c</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E1:32"><i><font color="Green" title="E19">A1</font></i></a></i>;<br/>

<b>thus </b><a NAME="E5:32_1_2_1"/>
( <font color="Maroon" title="c2">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 &amp; <font color="Maroon" title="c2">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">vs</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">vs</font></span>)</span> )
 ;<br/>

<b>thus </b><a NAME="E6:32_1_2_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font> &amp; not <font color="Maroon" title="c3">c</font> is <a href="graph_4.html#V1" title="GRAPH_4:attr.1">Simple</a> )
 <b>by </b><i><a class="txt" href="graph_4.html#E1:32_1_2"><i><font color="Green" title="E20">A3</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:32_1_2"/><b>then </b><i><font color="Green" title="E21">A4</font></i>: 
 ex <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 ;<br/><b>consider </b><font color="Maroon" title="c4">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E5:32_1_2"/><i><font color="Green" title="E22">A5</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">k</font>] &amp; ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon" title="c4">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5" title="NAT_1:sch.5">NAT_1:sch 5</a>(<a class="txt" href="graph_4.html#E3:32_1_2"><i><font color="Green" title="E21">A4</font></i></a>);<br/></i><b>consider </b><font color="Maroon" title="c5">fc</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c3">c</font>, <font color="Maroon" title="c6">fvs</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">vs</font>, <font color="Maroon" title="c7">c1</font> being  <a href="graph_1.html#V8" title="GRAPH_1:attr.8">oriented</a>  <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c1">G</font>, <font color="Maroon" title="c8">vs1</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E7:32_1_2"/><i><font color="Green" title="E23">A6</font></i>: 
(  <a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Maroon" title="c5">fc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">c1</font> &amp;  <a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Maroon" title="c6">fvs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">vs1</font> &amp; <font color="Maroon" title="c8">vs1</font> <a href="graph_4.html#R3" title="GRAPH_4:pred.3">is_oriented_vertex_seq_of</a> <font color="Maroon" title="c7">c1</font> &amp; <font color="Maroon" title="c2">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">vs1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 &amp; <font color="Maroon" title="c2">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">vs</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">vs1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">vs1</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">k</font> &amp; not <font color="Maroon" title="c7">c1</font> is <a href="graph_4.html#V1" title="GRAPH_4:attr.1">Simple</a> )
 <b>by </b><i><a class="txt" href="graph_4.html#E5:32_1_2"><i><font color="Green" title="E22">A5</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c9">fc1</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c7">c1</font>, <font color="Maroon" title="c10">fvs1</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c8">vs1</font>, <font color="Maroon" title="c11">c2</font> being  <a href="graph_1.html#V8" title="GRAPH_1:attr.8">oriented</a>  <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c1">G</font>, <font color="Maroon" title="c12">vs2</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E9:32_1_2"/><i><font color="Green" title="E24">A7</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">c2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">c1</font> &amp; <font color="Maroon" title="c12">vs2</font> <a href="graph_4.html#R3" title="GRAPH_4:pred.3">is_oriented_vertex_seq_of</a> <font color="Maroon" title="c11">c2</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">vs2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">vs1</font> &amp; <font color="Maroon" title="c8">vs1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">vs2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 &amp; <font color="Maroon" title="c8">vs1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">vs1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">vs2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">vs2</font></span>)</span> &amp;  <a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Maroon" title="c9">fc1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">c2</font> &amp;  <a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Maroon" title="c10">fvs1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">vs2</font> )
 <b>by </b><i><a class="txt" href="graph_4.html#E7:32_1_2"><i><font color="Green" title="E23">A6</font></i></a>, <a class="ref" href="graph_4.html#T22" target="_self" title="GRAPH_4:th.22">Th22</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c13">fc'</font> = <font color="Maroon" title="c5">fc</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">fc</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c9">fc1</font></span>)</span></span>)</span></span>)</span> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c3">c</font> <b>by </b><i><a class="ref" href="graph_2.html#T29" title="GRAPH_2:th.29">GRAPH_2:29</a></i>;<br/><a NAME="E11:32_1_2"/><i><font color="Green" title="E25">A8</font></i>: 
 <a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Maroon" title="c13">fc'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">c2</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E7:32_1_2"><i><font color="Green" title="E23">A6</font></i></a>, <a class="txt" href="graph_4.html#E9:32_1_2"><i><font color="Green" title="E24">A7</font></i></a>, <a class="ref" href="graph_2.html#T30" title="GRAPH_2:th.30">GRAPH_2:30</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c14">fvs'</font> = <font color="Maroon" title="c6">fvs</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">fvs</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c10">fvs1</font></span>)</span></span>)</span></span>)</span> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">vs</font> <b>by </b><i><a class="ref" href="graph_2.html#T29" title="GRAPH_2:th.29">GRAPH_2:29</a></i>;<br/><a NAME="E13:32_1_2"/><i><font color="Green" title="E26">A9</font></i>: 
 <a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Maroon" title="c14">fvs'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">vs2</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E7:32_1_2"><i><font color="Green" title="E23">A6</font></i></a>, <a class="txt" href="graph_4.html#E9:32_1_2"><i><font color="Green" title="E24">A7</font></i></a>, <a class="ref" href="graph_2.html#T30" title="GRAPH_2:th.30">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" title="c11">c2</font> is  <a href="graph_1.html#V8" title="GRAPH_1:attr.8">oriented</a> <a href="graph_4.html#V1" title="GRAPH_4:attr.1">Simple</a>  <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c1">G</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c15">sc</font> = <font color="Maroon" title="c11">c2</font> as  <a href="graph_1.html#V8" title="GRAPH_1:attr.8">oriented</a> <a href="graph_2.html#V3" title="GRAPH_2:attr.3">simple</a>  <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="ref" href="graph_4.html#T21" target="_self" title="GRAPH_4:th.21">Th21</a></i>;<br/><b>take </b><font color="Maroon" title="c16">fc'</font> = <font color="Maroon" title="c13">fc'</font>;<br/><b>take </b><font color="Maroon" title="c17">sc</font> = <font color="Maroon" title="c15">sc</font>;<br/><a NAME="E3:32_1_2_2"/>
 <a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Maroon" title="c16">fc'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">sc</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E7:32_1_2"><i><font color="Green" title="E23">A6</font></i></a>, <a class="txt" href="graph_4.html#E9:32_1_2"><i><font color="Green" title="E24">A7</font></i></a>, <a class="ref" href="graph_2.html#T30" title="GRAPH_2:th.30">GRAPH_2:30</a></i>;<br/><b>hence </b><a NAME="E4:32_1_2_2"/>
 ex <font color="Olive" title="b1">fc</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c3">c</font> ex <font color="Olive" title="b2">fvs</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">vs</font> ex <font color="Olive" title="b3">sc</font> being <a href="graph_1.html#V8" title="GRAPH_1:attr.8">oriented</a> <a href="graph_2.html#V3" title="GRAPH_2:attr.3">simple</a>  <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c1">G</font> ex <font color="Olive" title="b4">vs1</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c1">G</font> st <br/>(  <a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Olive" title="b1">fc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">sc</font> &amp;  <a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Olive" title="b2">fvs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">vs1</font> &amp; <font color="Olive" title="b4">vs1</font> <a href="graph_4.html#R3" title="GRAPH_4:pred.3">is_oriented_vertex_seq_of</a> <font color="Olive" title="b3">sc</font> &amp; <font color="Maroon" title="c2">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">vs1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 &amp; <font color="Maroon" title="c2">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">vs</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">vs1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b4">vs1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="graph_4.html#E7:32_1_2"><i><font color="Green" title="E23">A6</font></i></a>, <a class="txt" href="graph_4.html#E9:32_1_2"><i><font color="Green" title="E24">A7</font></i></a>, <a class="txt" href="graph_4.html#E13:32_1_2"><i><font color="Green" title="E26">A9</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E15:32_1_2"/>
 ex <font color="Olive" title="b1">fc</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c3">c</font> ex <font color="Olive" title="b2">fvs</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">vs</font> ex <font color="Olive" title="b3">sc</font> being <a href="graph_1.html#V8" title="GRAPH_1:attr.8">oriented</a> <a href="graph_2.html#V3" title="GRAPH_2:attr.3">simple</a>  <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c1">G</font> ex <font color="Olive" title="b4">vs1</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c1">G</font> st <br/>(  <a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Olive" title="b1">fc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">sc</font> &amp;  <a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Olive" title="b2">fvs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">vs1</font> &amp; <font color="Olive" title="b4">vs1</font> <a href="graph_4.html#R3" title="GRAPH_4:pred.3">is_oriented_vertex_seq_of</a> <font color="Olive" title="b3">sc</font> &amp; <font color="Maroon" title="c2">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">vs1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 &amp; <font color="Maroon" title="c2">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">vs</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">vs1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b4">vs1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="graph_4.html#E5:32_1_2"><i><font color="Green" title="E22">A5</font></i></a>, <a class="txt" href="graph_4.html#E7:32_1_2"><i><font color="Green" title="E23">A6</font></i></a>, <a class="txt" href="graph_4.html#E9:32_1_2"><i><font color="Green" title="E24">A7</font></i></a>, <a class="txt" href="graph_4.html#E11:32_1_2"><i><font color="Green" title="E25">A8</font></i></a>, <a class="txt" href="graph_4.html#E13:32_1_2"><i><font color="Green" title="E26">A9</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
