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

<b>set </b><font color="Maroon" title="c4">IT</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">c</font> where <font color="Olive" title="b1">c</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">G</font> <a href="graph_3.html#K9" title="GRAPH_3:func.9">-CycleSet</a>  : (  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">c</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">X</font> &amp; not <font color="Olive" title="b1">c</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp;  ex <font color="Olive" title="b2">vs</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/>( <font color="Olive" title="b2">vs</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Olive" title="b1">c</font> &amp; <font color="Olive" title="b2">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">v</font> ) ) </span>}</span> ;<br/>
<a NAME="E1:78_1_1"/><i><font color="Green" title="E67">A3</font></i>: 
<font color="Maroon" title="c3">X</font> <a href="graph_3.html#K12" title="GRAPH_3:func.12">-PathSet</a> <font color="Maroon" title="c2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">c</font> where <font color="Olive" title="b1">c</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c3">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  : ( <font color="Olive" title="b1">c</font> is   <a href="graph_1.html#NM4" title="GRAPH_1:NM.4">Path</a> of <font color="Maroon" title="c1">G</font> &amp; not <font color="Olive" title="b1">c</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp;  ex <font color="Olive" title="b2">vs</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/>( <font color="Olive" title="b2">vs</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Olive" title="b1">c</font> &amp; <font color="Olive" title="b2">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">v</font> ) ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="graph_3.html#E2:78"><i><font color="Green" title="E66">A2</font></i></a>, <a class="ref" href="graph_3.html#D11" target="_self" title="GRAPH_3:def.11">Def11</a></i>;<br/>
<b>set </b><font color="Maroon" title="c5">XL</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">p</font></span>)</span> where <font color="Olive" title="b1">p</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c3">X</font> <a href="graph_3.html#K12" title="GRAPH_3:func.12">-PathSet</a> <font color="Maroon" title="c2">v</font> : verum </span>}</span> ;<br/>
<b>consider </b><font color="Maroon" title="c6">p</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c3">X</font> <a href="graph_3.html#K12" title="GRAPH_3:func.12">-PathSet</a> <font color="Maroon" title="c2">v</font>;<br/>
<a NAME="E3:78_1_1"/><i><font color="Green" title="E68">A4</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">p</font></span>)</span> where <font color="Olive" title="b1">p</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c3">X</font> <a href="graph_3.html#K12" title="GRAPH_3:func.12">-PathSet</a> <font color="Maroon" title="c2">v</font> : verum </span>}</span> 
 
;<br/>
<b>reconsider </b><font color="Maroon" title="c7">E</font> = the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c1">G</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="ref" href="graph_1.html#D8" title="GRAPH_1:def.8">GRAPH_1:def 8</a></i>;<br/>
<a NAME="E5:78_1_1"/><i><font color="Green" title="E69">A5</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">p</font></span>)</span> where <font color="Olive" title="b1">p</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c3">X</font> <a href="graph_3.html#K12" title="GRAPH_3:func.12">-PathSet</a> <font color="Maroon" title="c2">v</font> : verum </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c7">E</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:78_1_1_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">p</font></span>)</span> where <font color="Olive" title="b1">p</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c3">X</font> <a href="graph_3.html#K12" title="GRAPH_3:func.12">-PathSet</a> <font color="Maroon" title="c2">v</font> : verum </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c9">p</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c3">X</font> <a href="graph_3.html#K12" title="GRAPH_3:func.12">-PathSet</a> <font color="Maroon" title="c2">v</font><b> such that </b><br/><a NAME="E3:78_1_1_1"/><i><font color="Green" title="E70">A6</font></i>: 
<font color="Maroon" title="c8">x</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="c9">p</font>
 ;<br/>
<a NAME="E4:78_1_1_1"/>
<font color="Maroon" title="c9">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> <a href="graph_3.html#K12" title="GRAPH_3:func.12">-PathSet</a> <font color="Maroon" title="c2">v</font>
 
;<br/>
<b>then consider </b><font color="Maroon" title="c10">c</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c3">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> <b> such that </b><br/><a NAME="E6:78_1_1_1"/><i><font color="Green" title="E71">A7</font></i>: 
( <font color="Maroon" title="c9">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">c</font> &amp; <font color="Maroon" title="c10">c</font> is   <a href="graph_1.html#NM4" title="GRAPH_1:NM.4">Path</a> of <font color="Maroon" title="c1">G</font> &amp; not <font color="Maroon" title="c10">c</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp;  ex <font color="Olive" title="b1">vs</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/>( <font color="Olive" title="b1">vs</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Maroon" title="c10">c</font> &amp; <font color="Olive" title="b1">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">v</font> ) )
 <b>by </b><i><a class="txt" href="graph_3.html#E1:78_1_1"><i><font color="Green" title="E67">A3</font></i></a></i>;<br/>
<a NAME="E7:78_1_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="graph_3.html#E6:78_1_1_1"><i><font color="Green" title="E71">A7</font></i></a>, <a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E8:78_1_1_1"/><b>then </b>
0 <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="c9">p</font>
 
;<br/>
<a NAME="E9:78_1_1_1"/><b>then </b><i><font color="Green" title="E72">A8</font></i>: 
0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">p</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E10:78_1_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c7">E</font>
 
<b>by </b><i><a class="txt" href="graph_3.html#E2:78"><i><font color="Green" title="E66">A2</font></i></a>, <a class="ref" href="graph_3.html#T59" target="_self" title="GRAPH_3:th.59">Th59</a></i>;<br/>
<b>hence </b><a NAME="E11:78_1_1_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c7">E</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_3.html#E3:78_1_1_1"><i><font color="Green" title="E70">A6</font></i></a>, <a class="txt" href="graph_3.html#E9:78_1_1_1"><i><font color="Green" title="E72">A8</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:78_1_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">p</font></span>)</span> where <font color="Olive" title="b1">p</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c3">X</font> <a href="graph_3.html#K12" title="GRAPH_3:func.12">-PathSet</a> <font color="Maroon" title="c2">v</font> : verum </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:78_1_1_2"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">p</font></span>)</span> where <font color="Olive" title="b1">p</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c3">X</font> <a href="graph_3.html#K12" title="GRAPH_3:func.12">-PathSet</a> <font color="Maroon" title="c2">v</font> : verum </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c9">p</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c3">X</font> <a href="graph_3.html#K12" title="GRAPH_3:func.12">-PathSet</a> <font color="Maroon" title="c2">v</font><b> such that </b><br/><a NAME="E3:78_1_1_2"/><i><font color="Green" title="E70">A9</font></i>: 
<font color="Maroon" title="c8">x</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="c9">p</font>
 ;<br/>
<b>thus </b><a NAME="E4:78_1_1_2"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="txt" href="graph_3.html#E3:78_1_1_2"><i><font color="Green" title="E70">A9</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c8">XL</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">p</font></span>)</span> where <font color="Olive" title="b1">p</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c3">X</font> <a href="graph_3.html#K12" title="GRAPH_3:func.12">-PathSet</a> <font color="Maroon" title="c2">v</font> : verum </span>}</span>  as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="graph_3.html#E3:78_1_1"><i><font color="Green" title="E68">A4</font></i></a>, <a class="txt" href="graph_3.html#E5:78_1_1"><i><font color="Green" title="E69">A5</font></i></a>, <a class="ref" href="finset_1.html#T13" title="FINSET_1:th.13">FINSET_1:13</a></i>;<br/>
<b>set </b><font color="Maroon" title="c9">maxl</font> =  <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c8">XL</font>;<br/>
<a NAME="E8:78_1_1"/>
 <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c8">XL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">XL</font>
 
<b>by </b><i><a class="ref" href="pre_circ.html#D1" title="PRE_CIRC:def.1">PRE_CIRC:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c10">p</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c3">X</font> <a href="graph_3.html#K12" title="GRAPH_3:func.12">-PathSet</a> <font color="Maroon" title="c2">v</font><b> such that </b><br/><a NAME="E10:78_1_1"/><i><font color="Green" title="E70">A10</font></i>: 
 <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c8">XL</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="c10">p</font>
 ;<br/>
<a NAME="E11:78_1_1"/>
<font color="Maroon" title="c10">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> <a href="graph_3.html#K12" title="GRAPH_3:func.12">-PathSet</a> <font color="Maroon" title="c2">v</font>
 
;<br/>
<b>then consider </b><font color="Maroon" title="c11">c</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c3">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> <b> such that </b><br/><a NAME="E13:78_1_1"/><i><font color="Green" title="E71">A11</font></i>: 
( <font color="Maroon" title="c10">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">c</font> &amp; <font color="Maroon" title="c11">c</font> is   <a href="graph_1.html#NM4" title="GRAPH_1:NM.4">Path</a> of <font color="Maroon" title="c1">G</font> &amp; not <font color="Maroon" title="c11">c</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp;  ex <font color="Olive" title="b1">vs</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/>( <font color="Olive" title="b1">vs</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Maroon" title="c11">c</font> &amp; <font color="Olive" title="b1">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">v</font> ) )
 <b>by </b><i><a class="txt" href="graph_3.html#E1:78_1_1"><i><font color="Green" title="E67">A3</font></i></a></i>;<br/>
<a NAME="E14:78_1_1"/><i><font color="Green" title="E72">A12</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c11">c</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">X</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">p</font> = <font color="Maroon" title="c10">p</font> as   <a href="graph_1.html#NM4" title="GRAPH_1:NM.4">Path</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="txt" href="graph_3.html#E13:78_1_1"><i><font color="Green" title="E71">A11</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_1_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:78_1_1_3"/><i><font color="Green" title="E73">A13</font></i>: 
not <font color="Maroon" title="c12">p</font> is <a href="msscyc_1.html#V1" title="MSSCYC_1:attr.1">cyclic</a>
 ;<br/><b>consider </b><font color="Maroon" title="c13">vs</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="E3:78_1_1_3"/><i><font color="Green" title="E74">A14</font></i>: 
( <font color="Maroon" title="c13">vs</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Maroon" title="c12">p</font> &amp; <font color="Maroon" title="c13">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">v</font> )
 <b>by </b><i><a class="txt" href="graph_3.html#E13:78_1_1"><i><font color="Green" title="E71">A11</font></i></a></i>;<br/><a NAME="E4:78_1_1_3"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">vs</font> <a href="hidden.html#R1" title="HIDDEN:pred.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">p</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="graph_3.html#E3:78_1_1_3"><i><font color="Green" title="E74">A14</font></i></a>, <a class="ref" href="graph_2.html#D7" title="GRAPH_2:def.7">GRAPH_2:def 7</a></i>;<br/><a NAME="E5:78_1_1_3"/><b>then </b><i><font color="Green" title="E75">A15</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">vs</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E6:78_1_1_3"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">vs</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c13">vs</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c14">v1</font> = <font color="Maroon" title="c13">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="c13">vs</font></span>)</span> as   <a href="graph_3.html#NM2" title="GRAPH_3:NM.2">Vertex</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="ref" href="finseq_2.html#T13" title="FINSEQ_2:th.13">FINSEQ_2:13</a></i>;<br/><a NAME="E8:78_1_1_3"/><i><font color="Green" title="E76">A16</font></i>: 
not  <a href="graph_3.html#K7" title="GRAPH_3:func.7">Degree</a> <font color="Maroon" title="c14">v1</font>,<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">p</font></span>)</span> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a>
 <b>by </b><i><a class="txt" href="graph_3.html#E1:78_1_1_3"><i><font color="Green" title="E73">A13</font></i></a>, <a class="txt" href="graph_3.html#E3:78_1_1_3"><i><font color="Green" title="E74">A14</font></i></a>, <a class="ref" href="graph_3.html#T55" target="_self" title="GRAPH_3:th.55">Th55</a></i>;<br/><a NAME="E9:78_1_1_3"/><i><font color="Green" title="E77">A17</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">X</font>
 <b>by </b><i><a class="txt" href="graph_3.html#E13:78_1_1"><i><font color="Green" title="E71">A11</font></i></a>, <a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/><a NAME="E10:78_1_1_3"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">X</font>
 <b>by </b><i><a class="txt" href="graph_3.html#E1:78"><i><font color="Green" title="E65">A1</font></i></a>, <a class="txt" href="graph_3.html#E8:78_1_1_3"><i><font color="Green" title="E76">A16</font></i></a></i>;<br/><a NAME="E11:78_1_1_3"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">p</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c3">X</font>
 <b>by </b><i><a class="txt" href="graph_3.html#E9:78_1_1_3"><i><font color="Green" title="E77">A17</font></i></a>, <a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></i>;<br/><b>then consider </b><font color="Maroon" title="c15">xx</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E13:78_1_1_3"/><i><font color="Green" title="E78">A18</font></i>: 
( <font color="Maroon" title="c15">xx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; not <font color="Maroon" title="c15">xx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">p</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T6" title="XBOOLE_0:th.6">XBOOLE_0:6</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c16">Xp</font> = <font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">p</font></span>)</span> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="txt" href="graph_3.html#E13:78_1_1_3"><i><font color="Green" title="E78">A18</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><b>set </b><font color="Maroon" title="c17">S</font> = the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c1">G</font>;<br/><b>set </b><font color="Maroon" title="c18">T</font> = the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c1">G</font>;<br/><a NAME="E15:78_1_1_3"/><i><font color="Green" title="E79">A19</font></i>: 
 <a href="graph_3.html#K7" title="GRAPH_3:func.7">Degree</a> <font color="Maroon" title="c14">v1</font>,<font color="Maroon" title="c16">Xp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="graph_3.html#K7" title="GRAPH_3:func.7">Degree</a> <font color="Maroon" title="c14">v1</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="graph_3.html#K7" title="GRAPH_3:func.7">Degree</a> <font color="Maroon" title="c14">v1</font>,<span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">p</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="graph_3.html#E9:78_1_1_3"><i><font color="Green" title="E77">A17</font></i></a>, <a class="ref" href="graph_3.html#T34" target="_self" title="GRAPH_3:th.34">Th34</a></i>;<br/><a NAME="E16:78_1_1_3"/>
 <a href="graph_3.html#K7" title="GRAPH_3:func.7">Degree</a> <font color="Maroon" title="c14">v1</font>,<font color="Maroon" title="c3">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="graph_3.html#K7" title="GRAPH_3:func.7">Degree</a> <font color="Maroon" title="c14">v1</font>,<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">p</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_3.html#E1:78"><i><font color="Green" title="E65">A1</font></i></a>, <a class="txt" href="graph_3.html#E8:78_1_1_3"><i><font color="Green" title="E76">A16</font></i></a></i>;<br/><a NAME="E17:78_1_1_3"/><b>then </b><i><font color="Green" title="E80">A20</font></i>: 
 <a href="graph_3.html#K7" title="GRAPH_3:func.7">Degree</a> <font color="Maroon" title="c14">v1</font>,<font color="Maroon" title="c16">Xp</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="graph_3.html#E15:78_1_1_3"><i><font color="Green" title="E79">A19</font></i></a></i>;<br/><b>set </b><font color="Maroon" title="c19">e</font> =  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="graph_3.html#K4" title="GRAPH_3:func.4">Edges_At</a> <font color="Maroon" title="c14">v1</font>,<font color="Maroon" title="c16">Xp</font></span>)</span>;<br/><a NAME="E18:78_1_1_3"/><i><font color="Green" title="E81">A21</font></i>: 
not  <a href="graph_3.html#K4" title="GRAPH_3:func.4">Edges_At</a> <font color="Maroon" title="c14">v1</font>,<font color="Maroon" title="c16">Xp</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>by </b><i><a class="txt" href="graph_3.html#E17:78_1_1_3"><i><font color="Green" title="E80">A20</font></i></a>, <a class="ref" href="graph_3.html#T30" target="_self" title="GRAPH_3:th.30">Th30</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_1_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_1_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:78_1_1_3_1_1"/>
(  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="graph_3.html#K4" title="GRAPH_3:func.4">Edges_At</a> <font color="Maroon" title="c14">v1</font>,<font color="Maroon" title="c16">Xp</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graph_3.html#K2" title="GRAPH_3:func.2">Edges_In</a> <font color="Maroon" title="c14">v1</font>,<font color="Maroon" title="c16">Xp</font> or  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="graph_3.html#K4" title="GRAPH_3:func.4">Edges_At</a> <font color="Maroon" title="c14">v1</font>,<font color="Maroon" title="c16">Xp</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graph_3.html#K3" title="GRAPH_3:func.3">Edges_Out</a> <font color="Maroon" title="c14">v1</font>,<font color="Maroon" title="c16">Xp</font> )
 </span><b>by </b><i><a class="txt" href="graph_3.html#E18:78_1_1_3"><i><font color="Green" title="E81">A21</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_1_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:78_1_1_3_1_1_1"/>
 <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="graph_3.html#K4" title="GRAPH_3:func.4">Edges_At</a> <font color="Maroon" title="c14">v1</font>,<font color="Maroon" title="c16">Xp</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graph_3.html#K2" title="GRAPH_3:func.2">Edges_In</a> <font color="Maroon" title="c14">v1</font>,<font color="Maroon" title="c16">Xp</font>
 ;<br/><div class="add"><a NAME="E2:78_1_1_3_1_1_1"/><b>then </b>
(  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="graph_3.html#K4" title="GRAPH_3:func.4">Edges_At</a> <font color="Maroon" title="c14">v1</font>,<font color="Maroon" title="c16">Xp</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c1">G</font> &amp;  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="graph_3.html#K4" title="GRAPH_3:func.4">Edges_At</a> <font color="Maroon" title="c14">v1</font>,<font color="Maroon" title="c16">Xp</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">Xp</font> &amp; the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c1">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><a href="graph_3.html#K4" title="GRAPH_3:func.4">Edges_At</a> <font color="Maroon" title="c14">v1</font>,<font color="Maroon" title="c16">Xp</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">v1</font> )
 <b>by </b><i><a class="ref" href="graph_3.html#D1" target="_self" title="GRAPH_3:def.1">Def1</a></i>;<br/><b>hence </b><a NAME="E3:78_1_1_3_1_1_1"/>
 ex <font color="Olive" title="b1">e'</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c16">Xp</font> st <br/>(  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="graph_3.html#K4" title="GRAPH_3:func.4">Edges_At</a> <font color="Maroon" title="c14">v1</font>,<font color="Maroon" title="c16">Xp</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e'</font> &amp; <font color="Olive" title="b1">e'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">Xp</font> &amp; <font color="Olive" title="b1">e'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c1">G</font> &amp; ( <font color="Maroon" title="c14">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c1">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">e'</font> or <font color="Maroon" title="c14">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c1">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">e'</font> ) )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_1_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:78_1_1_3_1_1_2"/>
 <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="graph_3.html#K4" title="GRAPH_3:func.4">Edges_At</a> <font color="Maroon" title="c14">v1</font>,<font color="Maroon" title="c16">Xp</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graph_3.html#K3" title="GRAPH_3:func.3">Edges_Out</a> <font color="Maroon" title="c14">v1</font>,<font color="Maroon" title="c16">Xp</font>
 ;<br/><div class="add"><a NAME="E2:78_1_1_3_1_1_2"/><b>then </b>
(  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="graph_3.html#K4" title="GRAPH_3:func.4">Edges_At</a> <font color="Maroon" title="c14">v1</font>,<font color="Maroon" title="c16">Xp</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c1">G</font> &amp;  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="graph_3.html#K4" title="GRAPH_3:func.4">Edges_At</a> <font color="Maroon" title="c14">v1</font>,<font color="Maroon" title="c16">Xp</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">Xp</font> &amp; the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c1">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><a href="graph_3.html#K4" title="GRAPH_3:func.4">Edges_At</a> <font color="Maroon" title="c14">v1</font>,<font color="Maroon" title="c16">Xp</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">v1</font> )
 <b>by </b><i><a class="ref" href="graph_3.html#D2" target="_self" title="GRAPH_3:def.2">Def2</a></i>;<br/><b>hence </b><a NAME="E3:78_1_1_3_1_1_2"/>
 ex <font color="Olive" title="b1">e'</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c16">Xp</font> st <br/>(  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="graph_3.html#K4" title="GRAPH_3:func.4">Edges_At</a> <font color="Maroon" title="c14">v1</font>,<font color="Maroon" title="c16">Xp</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e'</font> &amp; <font color="Olive" title="b1">e'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">Xp</font> &amp; <font color="Olive" title="b1">e'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c1">G</font> &amp; ( <font color="Maroon" title="c14">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c1">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">e'</font> or <font color="Maroon" title="c14">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c1">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">e'</font> ) )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>then consider </b><font color="Maroon" title="c20">e'</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c16">Xp</font><b> such that </b><br/><a NAME="E21:78_1_1_3"/><i><font color="Green" title="E82">A22</font></i>: 
(  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="graph_3.html#K4" title="GRAPH_3:func.4">Edges_At</a> <font color="Maroon" title="c14">v1</font>,<font color="Maroon" title="c16">Xp</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">e'</font> &amp; <font color="Maroon" title="c20">e'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">Xp</font> &amp; <font color="Maroon" title="c20">e'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c1">G</font> &amp; ( <font color="Maroon" title="c14">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c1">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">e'</font> or <font color="Maroon" title="c14">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c1">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">e'</font> ) )
 ;<br/><b>reconsider </b><font color="Maroon" title="c21">ep</font> = <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p3">(<span class="default"><a href="graph_3.html#K4" title="GRAPH_3:func.4">Edges_At</a> <font color="Maroon" title="c14">v1</font>,<font color="Maroon" title="c16">Xp</font></span>)</span></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> as   <a href="graph_1.html#NM4" title="GRAPH_1:NM.4">Path</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="txt" href="graph_3.html#E21:78_1_1_3"><i><font color="Green" title="E82">A22</font></i></a>, <a class="ref" href="graph_3.html#T7" target="_self" title="GRAPH_3:th.7">Th7</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_1_1_3_2"><b>now </b></a><div class="add"><b>given </b><font color="Maroon" title="c22">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><a NAME="E1:78_1_1_3_2"/><i><font color="Green" title="E83">A23</font></i>: 
( <font color="Maroon" title="c22">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c21">ep</font> &amp; <font color="Maroon" title="c22">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">p</font> )
 ;<br/><a NAME="E2:78_1_1_3_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c21">ep</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p3">(<span class="default"><a href="graph_3.html#K4" title="GRAPH_3:func.4">Edges_At</a> <font color="Maroon" title="c14">v1</font>,<font color="Maroon" title="c16">Xp</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T55" title="FINSEQ_1:th.55">FINSEQ_1:55</a></i>;<br/><a NAME="E3:78_1_1_3_2"/><b>then </b>
<font color="Maroon" title="c22">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">Xp</font>
 <b>by </b><i><a class="txt" href="graph_3.html#E21:78_1_1_3"><i><font color="Green" title="E82">A22</font></i></a>, <a class="txt" href="graph_3.html#E1:78_1_1_3_2"><i><font color="Green" title="E83">A23</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E4:78_1_1_3_2"/>
contradiction
 <b>by </b><i><a class="txt" href="graph_3.html#E1:78_1_1_3_2"><i><font color="Green" title="E83">A23</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div><a NAME="E24:78_1_1_3"/><b>then </b><i><font color="Green" title="E83">A24</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c21">ep</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">p</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c22">t</font> = the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c1">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><a href="graph_3.html#K4" title="GRAPH_3:func.4">Edges_At</a> <font color="Maroon" title="c14">v1</font>,<font color="Maroon" title="c16">Xp</font></span>)</span></span>)</span>, <font color="Maroon" title="c23">s</font> = the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c1">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><a href="graph_3.html#K4" title="GRAPH_3:func.4">Edges_At</a> <font color="Maroon" title="c14">v1</font>,<font color="Maroon" title="c16">Xp</font></span>)</span></span>)</span> as   <a href="graph_3.html#NM2" title="GRAPH_3:NM.2">Vertex</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="txt" href="graph_3.html#E21:78_1_1_3"><i><font color="Green" title="E82">A22</font></i></a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_1_1_3_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:78_1_1_3_3"/>
( <font color="Maroon" title="c14">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c1">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><a href="graph_3.html#K4" title="GRAPH_3:func.4">Edges_At</a> <font color="Maroon" title="c14">v1</font>,<font color="Maroon" title="c16">Xp</font></span>)</span></span>)</span> or <font color="Maroon" title="c14">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c1">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><a href="graph_3.html#K4" title="GRAPH_3:func.4">Edges_At</a> <font color="Maroon" title="c14">v1</font>,<font color="Maroon" title="c16">Xp</font></span>)</span></span>)</span> )
 </span><b>by </b><i><a class="txt" href="graph_3.html#E21:78_1_1_3"><i><font color="Green" title="E82">A22</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_1_1_3_3_1"><b>suppose </b></a><a NAME="E1:78_1_1_3_3_1"/><i><font color="Green" title="E84">A25</font></i>: 
<font color="Maroon" title="c14">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c1">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><a href="graph_3.html#K4" title="GRAPH_3:func.4">Edges_At</a> <font color="Maroon" title="c14">v1</font>,<font color="Maroon" title="c16">Xp</font></span>)</span></span>)</span>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon" title="c24">ts</font> = <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c22">t</font>,<font color="Maroon" title="c23">s</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> as    <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/><a NAME="E3:78_1_1_3_3_1"/><i><font color="Green" title="E85">A26</font></i>: 
<font color="Maroon" title="c24">ts</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Maroon" title="c21">ep</font>
 <b>by </b><i><a class="ref" href="graph_3.html#T16" target="_self" title="GRAPH_3:th.16">Th16</a></i>;<br/><a NAME="E4:78_1_1_3_3_1"/><i><font color="Green" title="E86">A27</font></i>: 
<font color="Maroon" title="c13">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="c13">vs</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">ts</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 <b>by </b><i><a class="txt" href="graph_3.html#E1:78_1_1_3_3_1"><i><font color="Green" title="E84">A25</font></i></a>, <a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c25">p1</font> = <font color="Maroon" title="c12">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c21">ep</font> as   <a href="graph_1.html#NM4" title="GRAPH_1:NM.4">Path</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="txt" href="graph_3.html#E3:78_1_1_3"><i><font color="Green" title="E74">A14</font></i></a>, <a class="txt" href="graph_3.html#E24:78_1_1_3"><i><font color="Green" title="E83">A24</font></i></a>, <a class="txt" href="graph_3.html#E3:78_1_1_3_3_1"><i><font color="Green" title="E85">A26</font></i></a>, <a class="ref" href="graph_3.html#T9" target="_self" title="GRAPH_3:th.9">Th9</a></i>;<br/><i><font color="Green" title="E87">A28</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c25">p1</font> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">p</font></span>)</span> <a href="nat_1.html#K1" title="NAT_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="c21">ep</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">p</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#T56" title="FINSEQ_1:th.56">FINSEQ_1:56</a></i>
;<br/><b>reconsider </b><font color="Maroon" title="c26">vs1</font> = <font color="Maroon" title="c13">vs</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c24">ts</font> as    <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/><a NAME="E8:78_1_1_3_3_1"/><i><font color="Green" title="E88">A29</font></i>: 
<font color="Maroon" title="c26">vs1</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Maroon" title="c25">p1</font>
 <b>by </b><i><a class="txt" href="graph_3.html#E3:78_1_1_3"><i><font color="Green" title="E74">A14</font></i></a>, <a class="txt" href="graph_3.html#E3:78_1_1_3_3_1"><i><font color="Green" title="E85">A26</font></i></a>, <a class="txt" href="graph_3.html#E4:78_1_1_3_3_1"><i><font color="Green" title="E86">A27</font></i></a>, <a class="ref" href="graph_2.html#T47" title="GRAPH_2:th.47">GRAPH_2:47</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c27">X'</font> = <font color="Maroon" title="c3">X</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="txt" href="graph_3.html#E21:78_1_1_3"><i><font color="Green" title="E82">A22</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c28">e'</font> =  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="graph_3.html#K4" title="GRAPH_3:func.4">Edges_At</a> <font color="Maroon" title="c14">v1</font>,<font color="Maroon" title="c16">Xp</font></span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c27">X'</font> <b>by </b><i><a class="txt" href="graph_3.html#E21:78_1_1_3"><i><font color="Green" title="E82">A22</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c29">ep</font> = <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c28">e'</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c3">X</font> ;<br/><b>reconsider </b><font color="Maroon" title="c30">p</font> = <font color="Maroon" title="c12">p</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c3">X</font> <b>by </b><i><a class="txt" href="graph_3.html#E13:78_1_1"><i><font color="Green" title="E71">A11</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c31">xp1</font> = <font color="Maroon" title="c30">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">ep</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c3">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></i>;<br/><a NAME="E14:78_1_1_3_3_1"/><i><font color="Green" title="E89">A30</font></i>: 
not <font color="Maroon" title="c31">xp1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>by </b><i><a class="txt" href="graph_3.html#E6:78_1_1_3_3_1"><i><font color="Green" title="E87">A28</font></i></a>, <a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">FINSEQ_1:25</a></i>;<br/><a NAME="E15:78_1_1_3_3_1"/>
<font color="Maroon" title="c26">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="c2">v</font>
 <b>by </b><i><a class="txt" href="graph_3.html#E3:78_1_1_3"><i><font color="Green" title="E74">A14</font></i></a>, <a class="txt" href="graph_3.html#E5:78_1_1_3"><i><font color="Green" title="E75">A15</font></i></a>, <a class="ref" href="graph_2.html#T14" title="GRAPH_2:th.14">GRAPH_2:14</a></i>;<br/><a NAME="E16:78_1_1_3_3_1"/><b>then </b>
<font color="Maroon" title="c31">xp1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> <a href="graph_3.html#K12" title="GRAPH_3:func.12">-PathSet</a> <font color="Maroon" title="c2">v</font>
 <b>by </b><i><a class="txt" href="graph_3.html#E1:78_1_1"><i><font color="Green" title="E67">A3</font></i></a>, <a class="txt" href="graph_3.html#E8:78_1_1_3_3_1"><i><font color="Green" title="E88">A29</font></i></a>, <a class="txt" href="graph_3.html#E14:78_1_1_3_3_1"><i><font color="Green" title="E89">A30</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c32">xp1</font> = <font color="Maroon" title="c31">xp1</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c3">X</font> <a href="graph_3.html#K12" title="GRAPH_3:func.12">-PathSet</a> <font color="Maroon" title="c2">v</font> ;<br/><a NAME="E18:78_1_1_3_3_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c32">xp1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">XL</font>
 ;<br/><a NAME="E19:78_1_1_3_3_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c30">p</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c30">p</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 0
 <b>by </b><i><a class="txt" href="graph_3.html#E10:78_1_1"><i><font color="Green" title="E70">A10</font></i></a>, <a class="txt" href="graph_3.html#E6:78_1_1_3_3_1"><i><font color="Green" title="E87">A28</font></i></a>, <a class="ref" href="pre_circ.html#D1" title="PRE_CIRC:def.1">PRE_CIRC:def 1</a></i>;<br/><b>hence </b><a NAME="E20:78_1_1_3_3_1"/>
contradiction
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_1_1_3_3_2"><b>suppose </b></a><a NAME="E1:78_1_1_3_3_2"/><i><font color="Green" title="E84">A31</font></i>: 
<font color="Maroon" title="c14">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c1">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><a href="graph_3.html#K4" title="GRAPH_3:func.4">Edges_At</a> <font color="Maroon" title="c14">v1</font>,<font color="Maroon" title="c16">Xp</font></span>)</span></span>)</span>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon" title="c24">ts</font> = <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c23">s</font>,<font color="Maroon" title="c22">t</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> as    <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/><a NAME="E3:78_1_1_3_3_2"/><i><font color="Green" title="E85">A32</font></i>: 
<font color="Maroon" title="c24">ts</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Maroon" title="c21">ep</font>
 <b>by </b><i><a class="ref" href="msscyc_1.html#T4" title="MSSCYC_1:th.4">MSSCYC_1:4</a></i>;<br/><a NAME="E4:78_1_1_3_3_2"/><i><font color="Green" title="E86">A33</font></i>: 
<font color="Maroon" title="c13">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="c13">vs</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">ts</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 <b>by </b><i><a class="txt" href="graph_3.html#E1:78_1_1_3_3_2"><i><font color="Green" title="E84">A31</font></i></a>, <a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c25">p1</font> = <font color="Maroon" title="c12">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c21">ep</font> as   <a href="graph_1.html#NM4" title="GRAPH_1:NM.4">Path</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="txt" href="graph_3.html#E3:78_1_1_3"><i><font color="Green" title="E74">A14</font></i></a>, <a class="txt" href="graph_3.html#E24:78_1_1_3"><i><font color="Green" title="E83">A24</font></i></a>, <a class="txt" href="graph_3.html#E3:78_1_1_3_3_2"><i><font color="Green" title="E85">A32</font></i></a>, <a class="ref" href="graph_3.html#T9" target="_self" title="GRAPH_3:th.9">Th9</a></i>;<br/><i><font color="Green" title="E87">A34</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c25">p1</font> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">p</font></span>)</span> <a href="nat_1.html#K1" title="NAT_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="c21">ep</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">p</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#T56" title="FINSEQ_1:th.56">FINSEQ_1:56</a></i>
;<br/><b>reconsider </b><font color="Maroon" title="c26">vs1</font> = <font color="Maroon" title="c13">vs</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c24">ts</font> as    <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/><a NAME="E8:78_1_1_3_3_2"/><i><font color="Green" title="E88">A35</font></i>: 
<font color="Maroon" title="c26">vs1</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Maroon" title="c25">p1</font>
 <b>by </b><i><a class="txt" href="graph_3.html#E3:78_1_1_3"><i><font color="Green" title="E74">A14</font></i></a>, <a class="txt" href="graph_3.html#E3:78_1_1_3_3_2"><i><font color="Green" title="E85">A32</font></i></a>, <a class="txt" href="graph_3.html#E4:78_1_1_3_3_2"><i><font color="Green" title="E86">A33</font></i></a>, <a class="ref" href="graph_2.html#T47" title="GRAPH_2:th.47">GRAPH_2:47</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c27">X'</font> = <font color="Maroon" title="c3">X</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="txt" href="graph_3.html#E21:78_1_1_3"><i><font color="Green" title="E82">A22</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c28">e'</font> =  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="graph_3.html#K4" title="GRAPH_3:func.4">Edges_At</a> <font color="Maroon" title="c14">v1</font>,<font color="Maroon" title="c16">Xp</font></span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c27">X'</font> <b>by </b><i><a class="txt" href="graph_3.html#E21:78_1_1_3"><i><font color="Green" title="E82">A22</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c29">ep</font> = <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c28">e'</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c3">X</font> ;<br/><b>reconsider </b><font color="Maroon" title="c30">p</font> = <font color="Maroon" title="c12">p</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c3">X</font> <b>by </b><i><a class="txt" href="graph_3.html#E13:78_1_1"><i><font color="Green" title="E71">A11</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c31">xp1</font> = <font color="Maroon" title="c30">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">ep</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c3">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></i>;<br/><a NAME="E14:78_1_1_3_3_2"/><i><font color="Green" title="E89">A36</font></i>: 
not <font color="Maroon" title="c31">xp1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>by </b><i><a class="txt" href="graph_3.html#E6:78_1_1_3_3_2"><i><font color="Green" title="E87">A34</font></i></a>, <a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">FINSEQ_1:25</a></i>;<br/><a NAME="E15:78_1_1_3_3_2"/>
<font color="Maroon" title="c26">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="c2">v</font>
 <b>by </b><i><a class="txt" href="graph_3.html#E3:78_1_1_3"><i><font color="Green" title="E74">A14</font></i></a>, <a class="txt" href="graph_3.html#E5:78_1_1_3"><i><font color="Green" title="E75">A15</font></i></a>, <a class="ref" href="graph_2.html#T14" title="GRAPH_2:th.14">GRAPH_2:14</a></i>;<br/><a NAME="E16:78_1_1_3_3_2"/><b>then </b>
<font color="Maroon" title="c31">xp1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> <a href="graph_3.html#K12" title="GRAPH_3:func.12">-PathSet</a> <font color="Maroon" title="c2">v</font>
 <b>by </b><i><a class="txt" href="graph_3.html#E1:78_1_1"><i><font color="Green" title="E67">A3</font></i></a>, <a class="txt" href="graph_3.html#E8:78_1_1_3_3_2"><i><font color="Green" title="E88">A35</font></i></a>, <a class="txt" href="graph_3.html#E14:78_1_1_3_3_2"><i><font color="Green" title="E89">A36</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c32">xp1</font> = <font color="Maroon" title="c31">xp1</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c3">X</font> <a href="graph_3.html#K12" title="GRAPH_3:func.12">-PathSet</a> <font color="Maroon" title="c2">v</font> ;<br/><a NAME="E18:78_1_1_3_3_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c32">xp1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">XL</font>
 ;<br/><a NAME="E19:78_1_1_3_3_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c30">p</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c30">p</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 0
 <b>by </b><i><a class="txt" href="graph_3.html#E10:78_1_1"><i><font color="Green" title="E70">A10</font></i></a>, <a class="txt" href="graph_3.html#E6:78_1_1_3_3_2"><i><font color="Green" title="E87">A34</font></i></a>, <a class="ref" href="pre_circ.html#D1" title="PRE_CIRC:def.1">PRE_CIRC:def 1</a></i>;<br/><b>hence </b><a NAME="E20:78_1_1_3_3_2"/>
contradiction
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</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" title="c13">c</font> = <font color="Maroon" title="c11">c</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">G</font> <a href="graph_3.html#K9" title="GRAPH_3:func.9">-CycleSet</a>  <b>by </b><i><a class="txt" href="graph_3.html#E13:78_1_1"><i><font color="Green" title="E71">A11</font></i></a>, <a class="ref" href="graph_3.html#D8" target="_self" title="GRAPH_3:def.8">Def8</a></i>;<br/>
<a NAME="E18:78_1_1"/><i><font color="Green" title="E73">A37</font></i>: 
<font color="Maroon" title="c13">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">c</font> where <font color="Olive" title="b1">c</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">G</font> <a href="graph_3.html#K9" title="GRAPH_3:func.9">-CycleSet</a>  : (  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">c</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">X</font> &amp; not <font color="Olive" title="b1">c</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp;  ex <font color="Olive" title="b2">vs</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/>( <font color="Olive" title="b2">vs</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Olive" title="b1">c</font> &amp; <font color="Olive" title="b2">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">v</font> ) ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="graph_3.html#E13:78_1_1"><i><font color="Green" title="E71">A11</font></i></a>, <a class="txt" href="graph_3.html#E14:78_1_1"><i><font color="Green" title="E72">A12</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_1_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c14">c</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:78_1_1_4"/>
<font color="Maroon" title="c14">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">c</font> where <font color="Olive" title="b1">c</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">G</font> <a href="graph_3.html#K9" title="GRAPH_3:func.9">-CycleSet</a>  : (  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">c</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">X</font> &amp; not <font color="Olive" title="b1">c</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp;  ex <font color="Olive" title="b2">vs</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/>( <font color="Olive" title="b2">vs</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Olive" title="b1">c</font> &amp; <font color="Olive" title="b2">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">v</font> ) ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon" title="c15">c'</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">G</font> <a href="graph_3.html#K9" title="GRAPH_3:func.9">-CycleSet</a> <b> such that </b><br/><a NAME="E3:78_1_1_4"/><i><font color="Green" title="E74">A38</font></i>: 
( <font color="Maroon" title="c14">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">c'</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">c'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">X</font> &amp; not <font color="Maroon" title="c15">c'</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp;  ex <font color="Olive" title="b1">vs</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/>( <font color="Olive" title="b1">vs</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Maroon" title="c15">c'</font> &amp; <font color="Olive" title="b1">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">v</font> ) )
 ;<br/><b>thus </b><a NAME="E4:78_1_1_4"/>
<font color="Maroon" title="c14">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="graph_3.html#K9" title="GRAPH_3:func.9">-CycleSet</a> 
 <b>by </b><i><a class="txt" href="graph_3.html#E3:78_1_1_4"><i><font color="Green" title="E74">A38</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E20:78_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">c</font> where <font color="Olive" title="b1">c</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">G</font> <a href="graph_3.html#K9" title="GRAPH_3:func.9">-CycleSet</a>  : (  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">c</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">X</font> &amp; not <font color="Olive" title="b1">c</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp;  ex <font color="Olive" title="b2">vs</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/>( <font color="Olive" title="b2">vs</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Olive" title="b1">c</font> &amp; <font color="Olive" title="b2">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">v</font> ) ) </span>}</span>  is  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="graph_3.html#K9" title="GRAPH_3:func.9">-CycleSet</a> </span>)</span>
 <b>by </b><i><a class="txt" href="graph_3.html#E18:78_1_1"><i><font color="Green" title="E73">A37</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>


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