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

<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="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> ;<br/>
<b>set </b><font color="Maroon" title="c4">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="E1:81_1_1"/><i><font color="Green" title="E70">A3</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/>
<b>set </b><font color="Maroon" title="c5">v</font> =  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</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="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> ;<br/>
<a NAME="E2:81_1_1"/>
not <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 <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
<b>by </b><i><a class="txt" href="graph_3.html#E1:81"><i><font color="Green" title="E68">A1</font></i></a>, <a class="txt" href="graph_3.html#E2:81"><i><font color="Green" title="E69">A2</font></i></a>, <a class="ref" href="graph_3.html#T61" target="_self" title="GRAPH_3:th.61">Th61</a></i>;<br/>
<a NAME="E3:81_1_1"/><b>then </b>
 <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</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="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>  <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="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> 
 
;<br/>
<b>then consider </b><font color="Maroon" title="c6">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="E5:81_1_1"/><i><font color="Green" title="E71">A4</font></i>: 
(  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</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="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>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v'</font> &amp; <font color="Maroon" title="c6">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="Maroon" title="c6">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="c6">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/>
<b>reconsider </b><font color="Maroon" title="c7">v</font> =  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</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="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>  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#E5:81_1_1"><i><font color="Green" title="E71">A4</font></i></a></i>;<br/>
<b>reconsider </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> 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="E8:81_1_1"/>
 <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#R2" title="XBOOLE_0:pred.2">c&lt;</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#E1:81"><i><font color="Green" title="E68">A1</font></i></a>, <a class="txt" href="graph_3.html#E1:81_1_1"><i><font color="Green" title="E70">A3</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="c9">xx</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E10:81_1_1"/><i><font color="Green" title="E72">A5</font></i>: 
( <font color="Maroon" title="c9">xx</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">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="c2">c</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="c10">Erc</font> = <font color="Maroon" title="c8">E'</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="c2">c</font></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="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="txt" href="graph_3.html#E10:81_1_1"><i><font color="Green" title="E72">A5</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="c11">c'</font> =  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">Erc</font> <a href="graph_3.html#K13" title="GRAPH_3:func.13">-CycleSet</a> <font color="Maroon" title="c7">v</font></span>)</span>;<br/>
<a NAME="E12:81_1_1"/>
 <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">Erc</font> <a href="graph_3.html#K13" title="GRAPH_3:func.13">-CycleSet</a> <font color="Maroon" title="c7">v</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Erc</font> <a href="graph_3.html#K13" title="GRAPH_3:func.13">-CycleSet</a> <font color="Maroon" title="c7">v</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c12">c'</font> =  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">Erc</font> <a href="graph_3.html#K13" title="GRAPH_3:func.13">-CycleSet</a> <font color="Maroon" title="c7">v</font></span>)</span> 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>  ;<br/>
<b>reconsider </b><font color="Maroon" title="c13">IT</font> =  <a href="graph_3.html#K11" title="GRAPH_3:func.11">CatCycles</a> <font color="Maroon" title="c2">c</font>,<font color="Maroon" title="c12">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>  ;<br/>
<b>take </b>
<font color="Maroon" title="c13">IT</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c12">c'</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c7">v</font>
;<br/>

<b>thus </b><a NAME="E15:81_1_1"/>
( <font color="Maroon" title="c7">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</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="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>  &amp; <font color="Maroon" title="c12">c'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c1">G</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3">(<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> <a href="graph_3.html#K13" title="GRAPH_3:func.13">-CycleSet</a> <font color="Maroon" title="c7">v</font></span>)</span> &amp; <font color="Maroon" title="c13">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_3.html#K11" title="GRAPH_3:func.11">CatCycles</a> <font color="Maroon" title="c2">c</font>,<font color="Maroon" title="c12">c'</font> )
 ;<br/>


</div>
