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

<b>let </b><font color="Maroon" title="c1">G</font> be  <a href="graph_1.html#V6" title="GRAPH_1:attr.6">connected</a> <a href="graph_1.html#V7" title="GRAPH_1:attr.7">finite</a> <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Graph</a>;<br/><b>let </b><font color="Maroon" title="c2">c</font> be    <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> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:80"/><i><font color="Green" title="E68">A1</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c1">G</font>
 <b>and </b><br/><a NAME="E2:80"/><i><font color="Green" title="E69">A2</font></i>: 
not <font color="Maroon" title="c2">c</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="graph_3.html#NM2" title="GRAPH_3:NM.2">Vertex</a> of <font color="Maroon" title="c1">G</font>] <b>means </b>( $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="graph_3.html#K1" title="GRAPH_3:func.1">-VSet</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">c</font></span>)</span> &amp;  <a href="graph_1.html#K6" title="GRAPH_1:func.6">Degree</a> $1 <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> $1,<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">c</font></span>)</span> );<br/>
<b>set </b><font color="Maroon" title="c3">X</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">v'</font> where <font color="Olive" title="b1">v'</font> is   <a href="graph_3.html#NM2" title="GRAPH_3:NM.2">Vertex</a> of <font color="Maroon" title="c1">G</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">v'</font>] </span>}</span> ;<br/>
<b>reconsider </b><font color="Maroon" title="c4">cp</font> = <font color="Maroon" title="c2">c</font> as  <a href="msscyc_1.html#V1" title="MSSCYC_1:attr.1">cyclic</a> <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="ref" href="graph_3.html#D8" target="_self" title="GRAPH_3:def.8">Def8</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">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="E5:80"/><i><font color="Green" title="E70">A3</font></i>: 
<font color="Maroon" title="c5">vs</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Maroon" title="c4">cp</font>
 <b>by </b><i><a class="ref" href="graph_2.html#T36" title="GRAPH_2:th.36">GRAPH_2:36</a></i>;<br/>
<a NAME="E6:80"/><i><font color="Green" title="E71">A4</font></i>: 
<font color="Maroon" title="c1">G</font> <a href="graph_3.html#K1" title="GRAPH_3:func.1">-VSet</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">cp</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">vs</font>
 
<b>by </b><i><a class="txt" href="graph_3.html#E2:80"><i><font color="Green" title="E69">A2</font></i></a>, <a class="txt" href="graph_3.html#E5:80"><i><font color="Green" title="E70">A3</font></i></a>, <a class="ref" href="graph_2.html#T34" title="GRAPH_2:th.34">GRAPH_2:34</a></i>;<br/>
<b>set </b><font color="Maroon" title="c6">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/>
<b>set </b><font color="Maroon" title="c7">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="c8">E</font> = the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c1">G</font>;<br/>
<a NAME="E7:80"/><i><font color="Green" title="E72">A5</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">c</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c1">G</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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1"><b>now </b></a><div class="add"><b>consider </b><font color="Maroon" title="c9">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E2:80_1"/><i><font color="Green" title="E73">A6</font></i>: 
( ( <font color="Maroon" title="c9">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="c2">c</font> &amp; not <font color="Maroon" title="c9">x</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> ) or ( <font color="Maroon" title="c9">x</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; not <font color="Maroon" title="c9">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="c2">c</font> ) )
 <b>by </b><i><a class="txt" href="graph_3.html#E1:80"><i><font color="Green" title="E68">A1</font></i></a>, <a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c10">x</font> = <font color="Maroon" title="c9">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="txt" href="graph_3.html#E7:80"><i><font color="Green" title="E72">A5</font></i></a>, <a class="txt" href="graph_3.html#E2:80_1"><i><font color="Green" title="E73">A6</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c11">v</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> <font color="Maroon" title="c10">x</font> 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#E7:80"><i><font color="Green" title="E72">A5</font></i></a>, <a class="txt" href="graph_3.html#E2:80_1"><i><font color="Green" title="E73">A6</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="80_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:80_1_1"/>
( <font color="Maroon" title="c11">v</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="c5">vs</font> or  not <font color="Maroon" title="c11">v</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="c5">vs</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_1_1_1"><b>suppose </b></a><a NAME="E1:80_1_1_1"/><i><font color="Green" title="E74">A7</font></i>: 
<font color="Maroon" title="c11">v</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="c5">vs</font>
 ;<br/><div class="add"><a NAME="E2:80_1_1_1"/>
 <a href="graph_1.html#K6" title="GRAPH_1:func.6">Degree</a> <font color="Maroon" title="c11">v</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="c11">v</font>,<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">c</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_3.html#E7:80"><i><font color="Green" title="E72">A5</font></i></a>, <a class="txt" href="graph_3.html#E2:80_1"><i><font color="Green" title="E73">A6</font></i></a>, <a class="ref" href="graph_3.html#T31" target="_self" title="GRAPH_3:th.31">Th31</a></i>;<br/><b>hence </b><a NAME="E3:80_1_1_1"/>
 ex <font color="Olive" title="b1">v</font> being  <a href="graph_3.html#NM2" title="GRAPH_3:NM.2">Vertex</a> of <font color="Maroon" title="c1">G</font> st <br/>( <font color="Olive" title="b1">v</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="c5">vs</font> &amp;  <a href="graph_1.html#K6" title="GRAPH_1:func.6">Degree</a> <font color="Olive" title="b1">v</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="Olive" title="b1">v</font>,<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">c</font></span>)</span> )
 <b>by </b><i><a class="txt" href="graph_3.html#E1:80_1_1_1"><i><font color="Green" title="E74">A7</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_1_1_2"><b>suppose </b></a><a NAME="E1:80_1_1_2"/><i><font color="Green" title="E74">A8</font></i>: 
not <font color="Maroon" title="c11">v</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="c5">vs</font>
 ;<br/><div class="add"><a NAME="E2:80_1_1_2"/>
not  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">c</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#E2:80"><i><font color="Green" title="E69">A2</font></i></a>, <a class="ref" href="relat_1.html#T64" title="RELAT_1:th.64">RELAT_1:64</a></i>;<br/><b>then consider </b><font color="Maroon" title="c12">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:80_1_1_2"/><i><font color="Green" title="E75">A9</font></i>: 
<font color="Maroon" title="c12">e</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="c2">c</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/><a NAME="E5:80_1_1_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">c</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="graph_3.html#E7:80"><i><font color="Green" title="E72">A5</font></i></a>, <a class="txt" href="graph_3.html#E4:80_1_1_2"><i><font color="Green" title="E75">A9</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/><b>then consider </b><font color="Maroon" title="c13">v'</font> being   <a href="graph_3.html#NM2" title="GRAPH_3:NM.2">Vertex</a> of <font color="Maroon" title="c1">G</font>, <font color="Maroon" title="c14">e</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E7:80_1_1_2"/><i><font color="Green" title="E76">A10</font></i>: 
( <font color="Maroon" title="c13">v'</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="c5">vs</font> &amp; not <font color="Maroon" title="c14">e</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="c2">c</font> &amp; ( <font color="Maroon" title="c13">v'</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="c14">e</font> or <font color="Maroon" title="c13">v'</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="c14">e</font> ) )
 <b>by </b><i><a class="txt" href="graph_3.html#E6:80"><i><font color="Green" title="E71">A4</font></i></a>, <a class="txt" href="graph_3.html#E1:80_1_1_2"><i><font color="Green" title="E74">A8</font></i></a>, <a class="ref" href="graph_3.html#T24" target="_self" title="GRAPH_3:th.24">Th24</a></i>;<br/><a NAME="E8:80_1_1_2"/>
 <a href="graph_1.html#K6" title="GRAPH_1:func.6">Degree</a> <font color="Maroon" title="c13">v'</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="c13">v'</font>,<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">c</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_3.html#E7:80"><i><font color="Green" title="E72">A5</font></i></a>, <a class="txt" href="graph_3.html#E4:80_1_1_2"><i><font color="Green" title="E75">A9</font></i></a>, <a class="txt" href="graph_3.html#E7:80_1_1_2"><i><font color="Green" title="E76">A10</font></i></a>, <a class="ref" href="graph_3.html#T31" target="_self" title="GRAPH_3:th.31">Th31</a></i>;<br/><b>hence </b><a NAME="E9:80_1_1_2"/>
 ex <font color="Olive" title="b1">v</font> being  <a href="graph_3.html#NM2" title="GRAPH_3:NM.2">Vertex</a> of <font color="Maroon" title="c1">G</font> st <br/>( <font color="Olive" title="b1">v</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="c5">vs</font> &amp;  <a href="graph_1.html#K6" title="GRAPH_1:func.6">Degree</a> <font color="Olive" title="b1">v</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="Olive" title="b1">v</font>,<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">c</font></span>)</span> )
 <b>by </b><i><a class="txt" href="graph_3.html#E7:80_1_1_2"><i><font color="Green" title="E76">A10</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c9">v</font> being   <a href="graph_3.html#NM2" title="GRAPH_3:NM.2">Vertex</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E10:80"/><i><font color="Green" title="E73">A11</font></i>: 
( <font color="Maroon" title="c9">v</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="c5">vs</font> &amp;  <a href="graph_1.html#K6" title="GRAPH_1:func.6">Degree</a> <font color="Maroon" title="c9">v</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="c9">v</font>,<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">c</font></span>)</span> )
 ;<br/>
<a NAME="E11:80"/><i><font color="Green" title="E74">A12</font></i>: 
<font color="Maroon" title="c9">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">v'</font> where <font color="Olive" title="b1">v'</font> is   <a href="graph_3.html#NM2" title="GRAPH_3:NM.2">Vertex</a> of <font color="Maroon" title="c1">G</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">v'</font>] </span>}</span> 
 
<b>by </b><i><a class="txt" href="graph_3.html#E6:80"><i><font color="Green" title="E71">A4</font></i></a>, <a class="txt" href="graph_3.html#E10:80"><i><font color="Green" title="E73">A11</font></i></a></i>;<br/>
<a NAME="E12:80"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">v'</font> where <font color="Olive" title="b1">v'</font> is   <a href="graph_3.html#NM2" title="GRAPH_3:NM.2">Vertex</a> of <font color="Maroon" title="c1">G</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">v'</font>] </span>}</span>  is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</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>from </b><i><a class="ref" href="domain_1.html#S7" title="DOMAIN_1:sch.7">DOMAIN_1:sch 7</a>();<br/></i>
<b>hence </b><a NAME="E13:80"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">v'</font> where <font color="Olive" title="b1">v'</font> is   <a href="graph_3.html#NM2" title="GRAPH_3:NM.2">Vertex</a> of <font color="Maroon" title="c1">G</font> : ( <font color="Olive" title="b1">v'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="graph_3.html#K1" title="GRAPH_3:func.1">-VSet</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">c</font></span>)</span> &amp;  <a href="graph_1.html#K6" title="GRAPH_1:func.6">Degree</a> <font color="Olive" title="b1">v'</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="Olive" title="b1">v'</font>,<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">c</font></span>)</span> ) </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 the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="graph_3.html#E11:80"><i><font color="Green" title="E74">A12</font></i></a></i>;<br/>


</div>
