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

<b>set </b><font color="Maroon">IT</font> = <span class="p1">{<span class="default"> <font color="Olive">c</font> where <font color="Olive">c</font> is    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">G</font> <a href="graph_3.html#K9" target="_self">-CycleSet</a>  : (  <a href="relat_1.html#K2">rng</a> <font color="Olive">c</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> &amp; not <font color="Olive">c</font> is <a href="xboole_0.html#V1">empty</a> &amp;  ex <font color="Olive">vs</font> being   <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G</font> st <br/>( <font color="Olive">vs</font> <a href="graph_2.html#R1">is_vertex_seq_of</a> <font color="Olive">c</font> &amp; <font color="Olive">vs</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> ) ) </span>}</span> ;<br/>
<a NAME="E1:78_1_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">X</font> <a href="graph_3.html#K12" target="_self">-PathSet</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">c</font> where <font color="Olive">c</font> is    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">X</font> <a href="finseq_2.html#K3">*</a>  : ( <font color="Olive">c</font> is   <a href="graph_1.html#NM4">Path</a> of <font color="Maroon">G</font> &amp; not <font color="Olive">c</font> is <a href="xboole_0.html#V1">empty</a> &amp;  ex <font color="Olive">vs</font> being   <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G</font> st <br/>( <font color="Olive">vs</font> <a href="graph_2.html#R1">is_vertex_seq_of</a> <font color="Olive">c</font> &amp; <font color="Olive">vs</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> ) ) </span>}</span> 
 
<b>by </b><i>, <a class="ref" href="graph_3.html#T7" target="_self">Th7</a></i>;<br/>
<b>set </b><font color="Maroon">XL</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">p</font></span>)</span> where <font color="Olive">p</font> is    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">X</font> <a href="graph_3.html#K12" target="_self">-PathSet</a> <font color="Maroon">v</font> : verum </span>}</span> ;<br/>
<b>consider </b><font color="Maroon">p</font> being    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">X</font> <a href="graph_3.html#K12" target="_self">-PathSet</a> <font color="Maroon">v</font>;<br/>
<a NAME="E3:78_1_1"/><i><font color="Green">E37</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">p</font></span>)</span> where <font color="Olive">p</font> is    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">X</font> <a href="graph_3.html#K12" target="_self">-PathSet</a> <font color="Maroon">v</font> : verum </span>}</span> 
 
;<br/>
<b>reconsider </b><font color="Maroon">E</font> = the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="graph_1.html#D8">GRAPH_1:def 8</a></i>;<br/>
<a NAME="E5:78_1_1"/><i><font color="Green">E46</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">p</font></span>)</span> where <font color="Olive">p</font> is    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">X</font> <a href="graph_3.html#K12" target="_self">-PathSet</a> <font color="Maroon">v</font> : verum </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">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">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:78_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">p</font></span>)</span> where <font color="Olive">p</font> is    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">X</font> <a href="graph_3.html#K12" target="_self">-PathSet</a> <font color="Maroon">v</font> : verum </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">p</font> being    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">X</font> <a href="graph_3.html#K12" target="_self">-PathSet</a> <font color="Maroon">v</font><b> such that </b><br/><a NAME="E3:78_1_1_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font>
 ;<br/>
<a NAME="E4:78_1_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="graph_3.html#K12" target="_self">-PathSet</a> <font color="Maroon">v</font>
 
;<br/>
<b>then consider </b><font color="Maroon">c</font> being    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">X</font> <a href="finseq_2.html#K3">*</a> <b> such that </b><br/><a NAME="E6:78_1_1_1"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> &amp; <font color="Maroon">c</font> is   <a href="graph_1.html#NM4">Path</a> of <font color="Maroon">G</font> &amp; not <font color="Maroon">c</font> is <a href="xboole_0.html#V1">empty</a> &amp;  ex <font color="Olive">vs</font> being   <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G</font> st <br/>( <font color="Olive">vs</font> <a href="graph_2.html#R1">is_vertex_seq_of</a> <font color="Maroon">c</font> &amp; <font color="Olive">vs</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> ) )
 <b>by </b><i/>;<br/>
<a NAME="E7:78_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E8:78_1_1_1"/><b>then </b>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font>
 
;<br/>
<a NAME="E9:78_1_1_1"/><b>then </b><i><font color="Green">E72</font></i>: 
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E10:78_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">E</font>
 
<b>by </b><i>, <a class="ref" href="graph_3.html#T8" target="_self">Th8</a></i>;<br/>
<b>hence </b><a NAME="E11:78_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">E</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="graph_3.html#E9:78_1_1_1"><i><font color="Green">E72</font></i></a>, <a class="ref" href="finseq_1.html#T3">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">len</a> <font color="Olive">p</font></span>)</span> where <font color="Olive">p</font> is    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">X</font> <a href="graph_3.html#K12" target="_self">-PathSet</a> <font color="Maroon">v</font> : verum </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">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">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:78_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">p</font></span>)</span> where <font color="Olive">p</font> is    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">X</font> <a href="graph_3.html#K12" target="_self">-PathSet</a> <font color="Maroon">v</font> : verum </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">p</font> being    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">X</font> <a href="graph_3.html#K12" target="_self">-PathSet</a> <font color="Maroon">v</font><b> such that </b><br/><a NAME="E3:78_1_1_2"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font>
 ;<br/>
<b>thus </b><a NAME="E4:78_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">XL</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">p</font></span>)</span> where <font color="Olive">p</font> is    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">X</font> <a href="graph_3.html#K12" target="_self">-PathSet</a> <font color="Maroon">v</font> : verum </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, <a class="ref" href="graph_3.html#T9" target="_self">Th9</a>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/>
<b>set </b><font color="Maroon">maxl</font> =  <a href="pre_circ.html#K1">max</a> <font color="Maroon">XL</font>;<br/>
<a NAME="E8:78_1_1"/>
 <a href="pre_circ.html#K1">max</a> <font color="Maroon">XL</font> <a href="hidden.html#R2">in</a> <font color="Maroon">XL</font>
 
<b>by </b><i><a class="ref" href="pre_circ.html#D1">PRE_CIRC:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">p</font> being    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">X</font> <a href="graph_3.html#K12" target="_self">-PathSet</a> <font color="Maroon">v</font><b> such that </b><br/><a NAME="E10:78_1_1"/><i><font color="Green">E75</font></i>: 
 <a href="pre_circ.html#K1">max</a> <font color="Maroon">XL</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font>
 ;<br/>
<a NAME="E11:78_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="graph_3.html#K12" target="_self">-PathSet</a> <font color="Maroon">v</font>
 
;<br/>
<b>then consider </b><font color="Maroon">c</font> being    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">X</font> <a href="finseq_2.html#K3">*</a> <b> such that </b><br/><a NAME="E13:78_1_1"/><i><font color="Green">E76</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> &amp; <font color="Maroon">c</font> is   <a href="graph_1.html#NM4">Path</a> of <font color="Maroon">G</font> &amp; not <font color="Maroon">c</font> is <a href="xboole_0.html#V1">empty</a> &amp;  ex <font color="Olive">vs</font> being   <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G</font> st <br/>( <font color="Olive">vs</font> <a href="graph_2.html#R1">is_vertex_seq_of</a> <font color="Maroon">c</font> &amp; <font color="Olive">vs</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> ) )
 <b>by </b><i/>;<br/>
<a NAME="E14:78_1_1"/><i><font color="Green">E77</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">p</font> as   <a href="graph_1.html#NM4">Path</a> of <font color="Maroon">G</font> <b>by </b><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">E79</font></i>: 
not <font color="Maroon">p</font> is <a href="msscyc_1.html#V1">cyclic</a>
 ;<br/><b>consider </b><font color="Maroon">vs</font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E3:78_1_1_3"/><i><font color="Green">E80</font></i>: 
( <font color="Maroon">vs</font> <a href="graph_2.html#R1">is_vertex_seq_of</a> <font color="Maroon">p</font> &amp; <font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> )
 <b>by </b><i/>;<br/><a NAME="E4:78_1_1_3"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="graph_2.html#D7">GRAPH_2:def 7</a></i>;<br/><a NAME="E5:78_1_1_3"/><b>then </b><i><font color="Green">E81</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E6:78_1_1_3"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">vs</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then reconsider </b><font color="Maroon">v1</font> = <font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font></span>)</span> as   <a href="graph_3.html#NM2">Vertex</a> of <font color="Maroon">G</font> <b>by </b><i><a class="ref" href="finseq_2.html#T13">FINSEQ_2:13</a></i>;<br/><a NAME="E8:78_1_1_3"/><i><font color="Green">E82</font></i>: 
not  <a href="graph_3.html#K7" target="_self">Degree</a> <font color="Maroon">v1</font>,<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font></span>)</span> is <a href="abian.html#V1">even</a>
 <b>by </b><i>, , </i>;<br/><a NAME="E9:78_1_1_3"/><i><font color="Green">E89</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/><a NAME="E10:78_1_1_3"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="txt" href="graph_3.html#E13:78_1_1"><i><font color="Green">E76</font></i></a></i>;<br/><a NAME="E11:78_1_1_3"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="txt" href="graph_3.html#E14:78_1_1"><i><font color="Green">E77</font></i></a>, <a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/><b>then consider </b><font color="Maroon">xx</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E13:78_1_1_3"/><i><font color="Green">E90</font></i>: 
( <font color="Maroon">xx</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; not <font color="Maroon">xx</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font> )
 <b>by </b><i><a class="ref" href="rlsub_2.html#T77">RLSUB_2:77</a></i>;<br/><b>reconsider </b><font color="Maroon">Xp</font> = <font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font></span>)</span> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>set </b><font color="Maroon">S</font> = the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font>;<br/><b>set </b><font color="Maroon">T</font> = the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font>;<br/><a NAME="E15:78_1_1_3"/><i><font color="Green">E91</font></i>: 
 <a href="graph_3.html#K7" target="_self">Degree</a> <font color="Maroon">v1</font>,<font color="Maroon">Xp</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="graph_3.html#K7" target="_self">Degree</a> <font color="Maroon">v1</font>,<font color="Maroon">X</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="graph_3.html#K7" target="_self">Degree</a> <font color="Maroon">v1</font>,<span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="graph_3.html#E14:78_1_1"><i><font color="Green">E77</font></i></a>, <a class="ref" href="graph_3.html#T1" target="_self">Th1</a></i>;<br/><a NAME="E16:78_1_1_3"/>
 <a href="graph_3.html#K7" target="_self">Degree</a> <font color="Maroon">v1</font>,<font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="graph_3.html#K7" target="_self">Degree</a> <font color="Maroon">v1</font>,<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="graph_3.html#E13:78_1_1"><i><font color="Green">E76</font></i></a></i>;<br/><a NAME="E17:78_1_1_3"/><b>then </b><i><font color="Green">E93</font></i>: 
 <a href="graph_3.html#K7" target="_self">Degree</a> <font color="Maroon">v1</font>,<font color="Maroon">Xp</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="graph_3.html#E1:78_1_1_3"><i><font color="Green">E79</font></i></a></i>;<br/><b>set </b><font color="Maroon">e</font> =  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="graph_3.html#K4" target="_self">Edges_At</a> <font color="Maroon">v1</font>,<font color="Maroon">Xp</font></span>)</span>;<br/><a NAME="E18:78_1_1_3"/><i><font color="Green">E94</font></i>: 
not  <a href="graph_3.html#K4" target="_self">Edges_At</a> <font color="Maroon">v1</font>,<font color="Maroon">Xp</font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="txt" href="graph_3.html#E3:78_1_1_3"><i><font color="Green">E80</font></i></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">choose</a> <span class="p1">(<span class="default"><a href="graph_3.html#K4" target="_self">Edges_At</a> <font color="Maroon">v1</font>,<font color="Maroon">Xp</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="graph_3.html#K2" target="_self">Edges_In</a> <font color="Maroon">v1</font>,<font color="Maroon">Xp</font> or  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="graph_3.html#K4" target="_self">Edges_At</a> <font color="Maroon">v1</font>,<font color="Maroon">Xp</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="graph_3.html#K3" target="_self">Edges_Out</a> <font color="Maroon">v1</font>,<font color="Maroon">Xp</font> )
 </span><b>by </b><i><a class="txt" href="graph_3.html#E5:78_1_1_3"><i><font color="Green">E81</font></i></a>, <a class="ref" href="xboole_0.html#D2">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">choose</a> <span class="p1">(<span class="default"><a href="graph_3.html#K4" target="_self">Edges_At</a> <font color="Maroon">v1</font>,<font color="Maroon">Xp</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="graph_3.html#K2" target="_self">Edges_In</a> <font color="Maroon">v1</font>,<font color="Maroon">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">choose</a> <span class="p1">(<span class="default"><a href="graph_3.html#K4" target="_self">Edges_At</a> <font color="Maroon">v1</font>,<font color="Maroon">Xp</font></span>)</span> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> &amp;  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="graph_3.html#K4" target="_self">Edges_At</a> <font color="Maroon">v1</font>,<font color="Maroon">Xp</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">Xp</font> &amp; the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="graph_3.html#K4" target="_self">Edges_At</a> <font color="Maroon">v1</font>,<font color="Maroon">Xp</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:78_1_1_3_1_1_1"/>
 ex <font color="Olive">e'</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Xp</font> st <br/>(  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="graph_3.html#K4" target="_self">Edges_At</a> <font color="Maroon">v1</font>,<font color="Maroon">Xp</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">e'</font> &amp; <font color="Olive">e'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Xp</font> &amp; <font color="Olive">e'</font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> &amp; ( <font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Olive">e'</font> or <font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Olive">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">choose</a> <span class="p1">(<span class="default"><a href="graph_3.html#K4" target="_self">Edges_At</a> <font color="Maroon">v1</font>,<font color="Maroon">Xp</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="graph_3.html#K3" target="_self">Edges_Out</a> <font color="Maroon">v1</font>,<font color="Maroon">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">choose</a> <span class="p1">(<span class="default"><a href="graph_3.html#K4" target="_self">Edges_At</a> <font color="Maroon">v1</font>,<font color="Maroon">Xp</font></span>)</span> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> &amp;  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="graph_3.html#K4" target="_self">Edges_At</a> <font color="Maroon">v1</font>,<font color="Maroon">Xp</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">Xp</font> &amp; the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="graph_3.html#K4" target="_self">Edges_At</a> <font color="Maroon">v1</font>,<font color="Maroon">Xp</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:78_1_1_3_1_1_2"/>
 ex <font color="Olive">e'</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Xp</font> st <br/>(  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="graph_3.html#K4" target="_self">Edges_At</a> <font color="Maroon">v1</font>,<font color="Maroon">Xp</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">e'</font> &amp; <font color="Olive">e'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Xp</font> &amp; <font color="Olive">e'</font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> &amp; ( <font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Olive">e'</font> or <font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Olive">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">e'</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Xp</font><b> such that </b><br/><a NAME="E21:78_1_1_3"/><i><font color="Green">E95</font></i>: 
(  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="graph_3.html#K4" target="_self">Edges_At</a> <font color="Maroon">v1</font>,<font color="Maroon">Xp</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">e'</font> &amp; <font color="Maroon">e'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Xp</font> &amp; <font color="Maroon">e'</font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> &amp; ( <font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e'</font> or <font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e'</font> ) )
 ;<br/><b>reconsider </b><font color="Maroon">ep</font> = <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="graph_3.html#K4" target="_self">Edges_At</a> <font color="Maroon">v1</font>,<font color="Maroon">Xp</font></span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span> as   <a href="graph_1.html#NM4">Path</a> of <font color="Maroon">G</font> <b>by </b><i><a class="txt" href="graph_3.html#E8:78_1_1_3"><i><font color="Green">E82</font></i></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">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E1:78_1_1_3_2"/><i><font color="Green">E123</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">ep</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font> )
 ;<br/><a NAME="E2:78_1_1_3_2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">ep</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="graph_3.html#K4" target="_self">Edges_At</a> <font color="Maroon">v1</font>,<font color="Maroon">Xp</font></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T55">FINSEQ_1:55</a></i>;<br/><a NAME="E3:78_1_1_3_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Xp</font>
 <b>by </b><i><a class="txt" href="graph_3.html#E8:78_1_1_3"><i><font color="Green">E82</font></i></a>, <a class="ref" href="graph_3.html#T12" target="_self">Th12</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E4:78_1_1_3_2"/>
contradiction
 <b>by </b><i><a class="ref" href="graph_3.html#T12" target="_self">Th12</a>, <a class="ref" href="xboole_0.html#D4">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">E125</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">ep</font> <a href="xboole_0.html#R1">misses</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><b>reconsider </b><font color="Maroon">t</font> = the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="graph_3.html#K4" target="_self">Edges_At</a> <font color="Maroon">v1</font>,<font color="Maroon">Xp</font></span>)</span></span>)</span>, <font color="Maroon">s</font> = the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="graph_3.html#K4" target="_self">Edges_At</a> <font color="Maroon">v1</font>,<font color="Maroon">Xp</font></span>)</span></span>)</span> as   <a href="graph_3.html#NM2">Vertex</a> of <font color="Maroon">G</font> <b>by </b><i><a class="txt" href="graph_3.html#E8:78_1_1_3"><i><font color="Green">E82</font></i></a>, <a class="ref" href="funct_2.html#T7">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">v1</font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="graph_3.html#K4" target="_self">Edges_At</a> <font color="Maroon">v1</font>,<font color="Maroon">Xp</font></span>)</span></span>)</span> or <font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="graph_3.html#K4" target="_self">Edges_At</a> <font color="Maroon">v1</font>,<font color="Maroon">Xp</font></span>)</span></span>)</span> )
 </span><b>by </b><i><a class="txt" href="graph_3.html#E8:78_1_1_3"><i><font color="Green">E82</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">E126</font></i>: 
<font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="graph_3.html#K4" target="_self">Edges_At</a> <font color="Maroon">v1</font>,<font color="Maroon">Xp</font></span>)</span></span>)</span>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">ts</font> = <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">t</font>,<font color="Maroon">s</font></span><a href="finseq_4.html#K2">*&gt;</a></span> as    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G</font> ;<br/><a NAME="E3:78_1_1_3_3_1"/><i><font color="Green">E127</font></i>: 
<font color="Maroon">ts</font> <a href="graph_2.html#R1">is_vertex_seq_of</a> <font color="Maroon">ep</font>
 <b>by </b><i/>;<br/><a NAME="E4:78_1_1_3_3_1"/><i><font color="Green">E128</font></i>: 
<font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">ts</font> <a href="funct_1.html#K1">.</a> 1
 <b>by </b><i><a class="ref" href="graph_3.html#T13" target="_self">Th13</a>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/><b>then reconsider </b><font color="Maroon">p1</font> = <font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">ep</font> as   <a href="graph_1.html#NM4">Path</a> of <font color="Maroon">G</font> <b>by </b><i>, <a class="ref" href="graph_3.html#T12" target="_self">Th12</a>, , </i>;<br/><i><font color="Green">E131</font></i>:  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">ep</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>
;<br/><b>reconsider </b><font color="Maroon">vs1</font> = <font color="Maroon">vs</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">ts</font> as    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G</font> ;<br/><a NAME="E8:78_1_1_3_3_1"/><i><font color="Green">E132</font></i>: 
<font color="Maroon">vs1</font> <a href="graph_2.html#R1">is_vertex_seq_of</a> <font color="Maroon">p1</font>
 <b>by </b><i>, , , <a class="ref" href="graph_2.html#T47">GRAPH_2:47</a></i>;<br/><b>reconsider </b><font color="Maroon">X'</font> = <font color="Maroon">X</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="txt" href="graph_3.html#E8:78_1_1_3"><i><font color="Green">E82</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">e'</font> =  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="graph_3.html#K4" target="_self">Edges_At</a> <font color="Maroon">v1</font>,<font color="Maroon">Xp</font></span>)</span> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X'</font> <b>by </b><i><a class="txt" href="graph_3.html#E8:78_1_1_3"><i><font color="Green">E82</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">ep</font> = <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">e'</font></span><a href="finseq_1.html#K12">*&gt;</a></span> as    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">X</font> ;<br/><b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">p</font> as    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">X</font> <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">xp1</font> = <font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">ep</font> as    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">X</font> <a href="finseq_2.html#K3">*</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/><a NAME="E14:78_1_1_3_3_1"/><i><font color="Green">E133</font></i>: 
not <font color="Maroon">xp1</font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/><a NAME="E15:78_1_1_3_3_1"/>
<font color="Maroon">vs1</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">v</font>
 <b>by </b><i>, <a class="txt" href="graph_3.html#E10:78_1_1"><i><font color="Green">E75</font></i></a>, <a class="ref" href="graph_2.html#T14">GRAPH_2:14</a></i>;<br/><a NAME="E16:78_1_1_3_3_1"/><b>then </b>
<font color="Maroon">xp1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="graph_3.html#K12" target="_self">-PathSet</a> <font color="Maroon">v</font>
 <b>by </b><i>, , <a class="txt" href="graph_3.html#E9:78_1_1_3"><i><font color="Green">E89</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon">xp1</font> = <font color="Maroon">xp1</font> as    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">X</font> <a href="graph_3.html#K12" target="_self">-PathSet</a> <font color="Maroon">v</font> ;<br/><a NAME="E18:78_1_1_3_3_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">xp1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">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">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> 0
 <b>by </b><i>, , <a class="ref" href="pre_circ.html#D1">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">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">E134</font></i>: 
<font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="graph_3.html#K4" target="_self">Edges_At</a> <font color="Maroon">v1</font>,<font color="Maroon">Xp</font></span>)</span></span>)</span>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">ts</font> = <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">t</font></span><a href="finseq_4.html#K2">*&gt;</a></span> as    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G</font> ;<br/><a NAME="E3:78_1_1_3_3_2"/><i><font color="Green">E135</font></i>: 
<font color="Maroon">ts</font> <a href="graph_2.html#R1">is_vertex_seq_of</a> <font color="Maroon">ep</font>
 <b>by </b><i><a class="ref" href="msscyc_1.html#T4">MSSCYC_1:4</a></i>;<br/><a NAME="E4:78_1_1_3_3_2"/><i><font color="Green">E136</font></i>: 
<font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">ts</font> <a href="funct_1.html#K1">.</a> 1
 <b>by </b><i><a class="ref" href="graph_3.html#T13" target="_self">Th13</a>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/><b>then reconsider </b><font color="Maroon">p1</font> = <font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">ep</font> as   <a href="graph_1.html#NM4">Path</a> of <font color="Maroon">G</font> <b>by </b><i>, <a class="ref" href="graph_3.html#T12" target="_self">Th12</a>, , </i>;<br/><i><font color="Green">E138</font></i>:  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">ep</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>
;<br/><b>reconsider </b><font color="Maroon">vs1</font> = <font color="Maroon">vs</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">ts</font> as    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G</font> ;<br/><a NAME="E8:78_1_1_3_3_2"/><i><font color="Green">E139</font></i>: 
<font color="Maroon">vs1</font> <a href="graph_2.html#R1">is_vertex_seq_of</a> <font color="Maroon">p1</font>
 <b>by </b><i>, , , <a class="ref" href="graph_2.html#T47">GRAPH_2:47</a></i>;<br/><b>reconsider </b><font color="Maroon">X'</font> = <font color="Maroon">X</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="txt" href="graph_3.html#E8:78_1_1_3"><i><font color="Green">E82</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">e'</font> =  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="graph_3.html#K4" target="_self">Edges_At</a> <font color="Maroon">v1</font>,<font color="Maroon">Xp</font></span>)</span> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X'</font> <b>by </b><i><a class="txt" href="graph_3.html#E8:78_1_1_3"><i><font color="Green">E82</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">ep</font> = <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">e'</font></span><a href="finseq_1.html#K12">*&gt;</a></span> as    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">X</font> ;<br/><b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">p</font> as    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">X</font> <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">xp1</font> = <font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">ep</font> as    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">X</font> <a href="finseq_2.html#K3">*</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/><a NAME="E14:78_1_1_3_3_2"/><i><font color="Green">E140</font></i>: 
not <font color="Maroon">xp1</font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/><a NAME="E15:78_1_1_3_3_2"/>
<font color="Maroon">vs1</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">v</font>
 <b>by </b><i>, <a class="txt" href="graph_3.html#E10:78_1_1"><i><font color="Green">E75</font></i></a>, <a class="ref" href="graph_2.html#T14">GRAPH_2:14</a></i>;<br/><a NAME="E16:78_1_1_3_3_2"/><b>then </b>
<font color="Maroon">xp1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="graph_3.html#K12" target="_self">-PathSet</a> <font color="Maroon">v</font>
 <b>by </b><i>, , <a class="txt" href="graph_3.html#E9:78_1_1_3"><i><font color="Green">E89</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon">xp1</font> = <font color="Maroon">xp1</font> as    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">X</font> <a href="graph_3.html#K12" target="_self">-PathSet</a> <font color="Maroon">v</font> ;<br/><a NAME="E18:78_1_1_3_3_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">xp1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">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">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> 0
 <b>by </b><i>, , <a class="ref" href="pre_circ.html#D1">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">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">c</font> = <font color="Maroon">c</font> as    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">G</font> <a href="graph_3.html#K9" target="_self">-CycleSet</a>  <b>by </b><i>, </i>;<br/>
<a NAME="E18:78_1_1"/><i><font color="Green">E141</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">c</font> where <font color="Olive">c</font> is    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">G</font> <a href="graph_3.html#K9" target="_self">-CycleSet</a>  : (  <a href="relat_1.html#K2">rng</a> <font color="Olive">c</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> &amp; not <font color="Olive">c</font> is <a href="xboole_0.html#V1">empty</a> &amp;  ex <font color="Olive">vs</font> being   <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G</font> st <br/>( <font color="Olive">vs</font> <a href="graph_2.html#R1">is_vertex_seq_of</a> <font color="Olive">c</font> &amp; <font color="Olive">vs</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> ) ) </span>}</span> 
 
<b>by </b><i>, </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">c</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:78_1_1_4"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">c</font> where <font color="Olive">c</font> is    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">G</font> <a href="graph_3.html#K9" target="_self">-CycleSet</a>  : (  <a href="relat_1.html#K2">rng</a> <font color="Olive">c</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> &amp; not <font color="Olive">c</font> is <a href="xboole_0.html#V1">empty</a> &amp;  ex <font color="Olive">vs</font> being   <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G</font> st <br/>( <font color="Olive">vs</font> <a href="graph_2.html#R1">is_vertex_seq_of</a> <font color="Olive">c</font> &amp; <font color="Olive">vs</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> ) ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">c'</font> being    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">G</font> <a href="graph_3.html#K9" target="_self">-CycleSet</a> <b> such that </b><br/><a NAME="E3:78_1_1_4"/><i><font color="Green">E142</font></i>: 
( <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c'</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> &amp; not <font color="Maroon">c'</font> is <a href="xboole_0.html#V1">empty</a> &amp;  ex <font color="Olive">vs</font> being   <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G</font> st <br/>( <font color="Olive">vs</font> <a href="graph_2.html#R1">is_vertex_seq_of</a> <font color="Maroon">c'</font> &amp; <font color="Olive">vs</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> ) )
 ;<br/><b>thus </b><a NAME="E4:78_1_1_4"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="graph_3.html#K9" target="_self">-CycleSet</a> 
 <b>by </b><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">c</font> where <font color="Olive">c</font> is    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">G</font> <a href="graph_3.html#K9" target="_self">-CycleSet</a>  : (  <a href="relat_1.html#K2">rng</a> <font color="Olive">c</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> &amp; not <font color="Olive">c</font> is <a href="xboole_0.html#V1">empty</a> &amp;  ex <font color="Olive">vs</font> being   <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G</font> st <br/>( <font color="Olive">vs</font> <a href="graph_2.html#R1">is_vertex_seq_of</a> <font color="Olive">c</font> &amp; <font color="Olive">vs</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> ) ) </span>}</span>  is  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="graph_3.html#K9" target="_self">-CycleSet</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


</div>
