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

<b>set </b><font color="Maroon">c<sub>3</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="graph_3.html#NM2">Vertex</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="graph_3.html#K1" target="_self">-VSet</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp;  <a href="graph_1.html#K6">Degree</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="graph_3.html#K7" target="_self">Degree</a> <font color="Olive">b<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> = the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E1:81_1_1"/><i><font color="Green">E70</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="subset_1.html#K10">choose</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="graph_3.html#NM2">Vertex</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="graph_3.html#K1" target="_self">-VSet</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp;  <a href="graph_1.html#K6">Degree</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="graph_3.html#K7" target="_self">Degree</a> <font color="Olive">b<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> ) </span>}</span> ;<br/>
<a NAME="E2:81_1_1"/>
not <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="graph_3.html#NM2">Vertex</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="graph_3.html#K1" target="_self">-VSet</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp;  <a href="graph_1.html#K6">Degree</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="graph_3.html#K7" target="_self">Degree</a> <font color="Olive">b<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> ) </span>}</span>  is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="txt" href="graph_3.html#E1:81"><i><font color="Green">E68</font></i></a>, <a class="txt" href="graph_3.html#E2:81"><i><font color="Green">E69</font></i></a>, <a class="ref" href="graph_3.html#T61" target="_self">Th61</a></i>;<br/>
<a NAME="E3:81_1_1"/><b>then </b>
 <a href="subset_1.html#K10">choose</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="graph_3.html#NM2">Vertex</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="graph_3.html#K1" target="_self">-VSet</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp;  <a href="graph_1.html#K6">Degree</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="graph_3.html#K7" target="_self">Degree</a> <font color="Olive">b<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> ) </span>}</span>  <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="graph_3.html#NM2">Vertex</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="graph_3.html#K1" target="_self">-VSet</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp;  <a href="graph_1.html#K6">Degree</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="graph_3.html#K7" target="_self">Degree</a> <font color="Olive">b<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> ) </span>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="graph_3.html#NM2">Vertex</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E5:81_1_1"/><i><font color="Green">E71</font></i>: 
(  <a href="subset_1.html#K10">choose</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="graph_3.html#NM2">Vertex</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="graph_3.html#K1" target="_self">-VSet</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp;  <a href="graph_1.html#K6">Degree</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="graph_3.html#K7" target="_self">Degree</a> <font color="Olive">b<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> ) </span>}</span>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="graph_3.html#K1" target="_self">-VSet</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp;  <a href="graph_1.html#K6">Degree</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="graph_3.html#K7" target="_self">Degree</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> )
 ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> =  <a href="subset_1.html#K10">choose</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="graph_3.html#NM2">Vertex</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="graph_3.html#K1" target="_self">-VSet</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp;  <a href="graph_1.html#K6">Degree</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="graph_3.html#K7" target="_self">Degree</a> <font color="Olive">b<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> ) </span>}</span>  as   <a href="graph_3.html#NM2">Vertex</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="graph_3.html#E5:81_1_1"><i><font color="Green">E71</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">c<sub>1</sub></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="E8:81_1_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#R2">c&lt;</a> the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="graph_3.html#E1:81"><i><font color="Green">E68</font></i></a>, <a class="txt" href="graph_3.html#E1:81_1_1"><i><font color="Green">E70</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">c<sub>9</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E10:81_1_1"/><i><font color="Green">E72</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">c<sub>1</sub></font> &amp; not <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></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">c<sub>10</sub></font> = <font color="Maroon">c<sub>8</sub></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">c<sub>2</sub></font></span>)</span> as  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="txt" href="graph_3.html#E10:81_1_1"><i><font color="Green">E72</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>11</sub></font> =  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="graph_3.html#K13" target="_self">-CycleSet</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>;<br/>
<a NAME="E12:81_1_1"/>
 <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="graph_3.html#K13" target="_self">-CycleSet</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> <a href="graph_3.html#K13" target="_self">-CycleSet</a> <font color="Maroon">c<sub>7</sub></font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> =  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="graph_3.html#K13" target="_self">-CycleSet</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> as    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">c<sub>1</sub></font> <a href="graph_3.html#K9" target="_self">-CycleSet</a>  ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>13</sub></font> =  <a href="graph_3.html#K11" target="_self">CatCycles</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>12</sub></font> as    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">c<sub>1</sub></font> <a href="graph_3.html#K9" target="_self">-CycleSet</a>  ;<br/>
<b>take </b>
<font color="Maroon">c<sub>13</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>12</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>7</sub></font>
;<br/>

<b>thus </b><a NAME="E15:81_1_1"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K10">choose</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="graph_3.html#NM2">Vertex</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="graph_3.html#K1" target="_self">-VSet</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp;  <a href="graph_1.html#K6">Degree</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="graph_3.html#K7" target="_self">Degree</a> <font color="Olive">b<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> ) </span>}</span>  &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="graph_3.html#K13" target="_self">-CycleSet</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="graph_3.html#K11" target="_self">CatCycles</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>12</sub></font> )
 ;<br/>


</div>
