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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="graph_1.html#V6">connected</a> <a href="graph_1.html#V7">finite</a> <a href="graph_1.html#NM1">Graph</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <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>assume </b><b>that </b><br/><a NAME="E1:82"/><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="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E2:82"/><i><font color="Green">E71</font></i>: 
not <font color="Maroon">c<sub>2</sub></font> is <a href="xboole_0.html#V1">empty</a>
 <b>and </b><br/><a NAME="E3:82"/><i><font color="Green">E72</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="graph_3.html#NM2">Vertex</a> of <font color="Maroon">c<sub>1</sub></font> holds  <a href="graph_1.html#K6">Degree</a> <font color="Olive">b<sub>1</sub></font> is <a href="abian.html#V1">even</a>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>3</sub></font> = the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E4:82"/><i><font color="Green">E73</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>reconsider </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> 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="E6:82"/>
 <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#E1:82"><i><font color="Green">E70</font></i></a>, <a class="txt" href="graph_3.html#E4:82"><i><font color="Green">E73</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>5</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:82"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">c<sub>5</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>5</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>6</sub></font> = <font color="Maroon">c<sub>4</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#E8:82"><i><font color="Green">E74</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</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>  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#E1:82"><i><font color="Green">E70</font></i></a>, <a class="txt" href="graph_3.html#E2:82"><i><font color="Green">E71</font></i></a>, <a class="ref" href="graph_3.html#T61" target="_self">Th61</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></font> being    <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> , <font color="Maroon">c<sub>9</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="E12:82"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K10">choose</a> <font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E13:82"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="graph_3.html#K13" target="_self">-CycleSet</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E14:82"/><i><font color="Green">E77</font></i>: 
 <a href="graph_3.html#K14" target="_self">ExtendCycle</a> <font color="Maroon">c<sub>2</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>8</sub></font>
 <b>by </b><i><a class="txt" href="graph_3.html#E1:82"><i><font color="Green">E70</font></i></a>, <a class="txt" href="graph_3.html#E2:82"><i><font color="Green">E71</font></i></a>, <a class="ref" href="graph_3.html#D13" target="_self">Def13</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>2</sub></font> as  <a href="msscyc_1.html#V1">cyclic</a> <a href="graph_1.html#NM4">Path</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="graph_3.html#D8" target="_self">Def8</a></i>;<br/>
<div><i><font color="Green">E78</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>11</sub></font> be   <a href="graph_3.html#NM2">Vertex</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><a NAME="E1:82_1"/><i><font color="Green">E79</font></i>: 
 <a href="graph_1.html#K6">Degree</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="graph_3.html#K7" target="_self">Degree</a> <font color="Maroon">c<sub>11</sub></font>,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="graph_3.html#T29" target="_self">Th29</a></i>;<br/><a NAME="E2:82_1"/><i><font color="Green">E80</font></i>: 
 <a href="graph_1.html#K6">Degree</a> <font color="Maroon">c<sub>11</sub></font> is <a href="abian.html#V1">even</a>
 <b>by </b><i><a class="txt" href="graph_3.html#E3:82"><i><font color="Green">E72</font></i></a></i>;<br/><a NAME="E3:82_1"/><i><font color="Green">E81</font></i>: 
 <a href="graph_3.html#K7" target="_self">Degree</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>6</sub></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">c<sub>11</sub></font>,<font color="Maroon">c<sub>4</sub></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">c<sub>11</sub></font>,<span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="graph_3.html#E4:82"><i><font color="Green">E73</font></i></a>, <a class="ref" href="graph_3.html#T34" target="_self">Th34</a></i>;<br/><a NAME="E4:82_1"/>
 <a href="graph_3.html#K7" target="_self">Degree</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> is <a href="abian.html#V1">even</a>
 <b>by </b><i><a class="ref" href="graph_3.html#T54" target="_self">Th54</a></i>;<br/><b>hence </b><a NAME="E5:82_1"/>
 <a href="graph_3.html#K7" target="_self">Degree</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>6</sub></font> is <a href="abian.html#V1">even</a>
 <b>by </b><i><a class="txt" href="graph_3.html#E1:82_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="graph_3.html#E2:82_1"><i><font color="Green">E80</font></i></a>, <a class="txt" href="graph_3.html#E3:82_1"><i><font color="Green">E81</font></i></a>, <a class="ref" href="graph_3.html#T1" target="_self">Th1</a></i>;<br/></div><b>end;</b></div>
<a NAME="E17:82"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="graph_3.html#E12:82"><i><font color="Green">E75</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>11</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="E19:82"/><i><font color="Green">E79</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</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>11</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>11</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/>
<a NAME="E20:82"/><i><font color="Green">E80</font></i>: 
 <a href="graph_3.html#K7" target="_self">Degree</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>6</sub></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">c<sub>9</sub></font>,<font color="Maroon">c<sub>4</sub></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">c<sub>9</sub></font>,<span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="graph_3.html#E4:82"><i><font color="Green">E73</font></i></a>, <a class="ref" href="graph_3.html#T34" target="_self">Th34</a></i>;<br/>
<a NAME="E21:82"/>
 <a href="graph_1.html#K6">Degree</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="graph_3.html#K7" target="_self">Degree</a> <font color="Maroon">c<sub>9</sub></font>,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#E19:82"><i><font color="Green">E79</font></i></a>, <a class="ref" href="graph_3.html#T29" target="_self">Th29</a></i>;<br/>
<a NAME="E22:82"/><b>then </b>
 <a href="graph_3.html#K7" target="_self">Degree</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="graph_3.html#E19:82"><i><font color="Green">E79</font></i></a>, <a class="txt" href="graph_3.html#E20:82"><i><font color="Green">E80</font></i></a></i>;<br/>
<a NAME="E23:82"/><b>then </b><i><font color="Green">E81</font></i>: 
( not <font color="Maroon">c<sub>8</sub></font> is <a href="xboole_0.html#V1">empty</a> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>8</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> <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> &amp; <font color="Maroon">c<sub>9</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>8</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="graph_3.html#E13:82"><i><font color="Green">E76</font></i></a>, <a class="txt" href="graph_3.html#E16:82"><i><font color="Green">E78</font></i></a>, <a class="ref" href="graph_3.html#T60" target="_self">Th60</a></i>;<br/>
<a NAME="E24:82"/><b>then </b><i><font color="Green">E82</font></i>: 
<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> <a href="xboole_0.html#NR5" target="_self">meets</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>8</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="graph_3.html#E19:82"><i><font color="Green">E79</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E25:82"/><b>then </b><i><font color="Green">E83</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="graph_3.html#K1" target="_self">-VSet</a> <span class="p2">(<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="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="graph_3.html#K1" target="_self">-VSet</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E26:82"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#R1">misses</a> 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="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/>
<a NAME="E27:82"/><b>then </b><i><font color="Green">E84</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<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="p2">(<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>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K9">/\</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="subset_1.html#K8">/\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="graph_3.html#E23:82"><i><font color="Green">E81</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
<br/>.= 
<a href="xboole_0.html#K1">{}</a>  <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="graph_3.html#E27:82"><i><font color="Green">E84</font></i></a>, <a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>
<br/>.= 
 <a href="xboole_0.html#K1">{}</a> 

;<br/>
<a NAME="E29:82"/><b>then </b><i><font color="Green">E85</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>12</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="E31:82"/><i><font color="Green">E86</font></i>: 
<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"><font color="Maroon">c<sub>1</sub></font> <a href="graph_3.html#K1" target="_self">-VSet</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="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="graph_3.html#K1" target="_self">-VSet</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E32:82"/><i><font color="Green">E87</font></i>: 
 <a href="graph_3.html#K11" target="_self">CatCycles</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="graph_3.html#K10" target="_self">Rotate</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><a href="graph_3.html#K10" target="_self">Rotate</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_3.html#E24:82"><i><font color="Green">E82</font></i></a>, <a class="ref" href="graph_3.html#D10" target="_self">Def10</a></i>;<br/>
<a NAME="E33:82"/>
<font color="Maroon">c<sub>12</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>
 
<b>by </b><i><a class="txt" href="graph_3.html#E25:82"><i><font color="Green">E83</font></i></a>, <a class="txt" href="graph_3.html#E31:82"><i><font color="Green">E86</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E34:82"/><b>then </b><i><font color="Green">E88</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="graph_3.html#K10" target="_self">Rotate</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="graph_3.html#E71"><i><font color="Green">Lemma60</font></i></a></i>;<br/>
<a NAME="E35:82"/>
<font color="Maroon">c<sub>12</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>8</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="graph_3.html#E25:82"><i><font color="Green">E83</font></i></a>, <a class="txt" href="graph_3.html#E31:82"><i><font color="Green">E86</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E36:82"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="graph_3.html#K10" target="_self">Rotate</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="graph_3.html#E71"><i><font color="Green">Lemma60</font></i></a></i>;<br/>
<a NAME="E37:82"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="graph_3.html#K14" target="_self">ExtendCycle</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</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> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="graph_3.html#E14:82"><i><font color="Green">E77</font></i></a>, <a class="txt" href="graph_3.html#E32:82"><i><font color="Green">E87</font></i></a>, <a class="txt" href="graph_3.html#E34:82"><i><font color="Green">E88</font></i></a>, <a class="ref" href="finseq_1.html#T44">FINSEQ_1:44</a></i>;<br/>
<a NAME="E38:82"/><b>then </b><i><font color="Green">E89</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="graph_3.html#K14" target="_self">ExtendCycle</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<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="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="graph_3.html#E29:82"><i><font color="Green">E85</font></i></a>, <a class="ref" href="card_2.html#T53">CARD_2:53</a></i>;<br/>
<a NAME="E39:82"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="graph_3.html#E23:82"><i><font color="Green">E81</font></i></a>, <a class="ref" href="relat_1.html#T64">RELAT_1:64</a></i>;<br/>
<a NAME="E40:82"/><b>then </b>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="card_2.html#T59">CARD_2:59</a></i>;<br/>
<a NAME="E41:82"/><b>then </b><i><font color="Green">E90</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 
;<br/>
<b>thus </b><a NAME="E42:82"/>
not  <a href="graph_3.html#K14" target="_self">ExtendCycle</a> <font color="Maroon">c<sub>2</sub></font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="txt" href="graph_3.html#E2:82"><i><font color="Green">E71</font></i></a>, <a class="txt" href="graph_3.html#E14:82"><i><font color="Green">E77</font></i></a>, <a class="txt" href="graph_3.html#E24:82"><i><font color="Green">E82</font></i></a>, <a class="txt" href="graph_3.html#E29:82"><i><font color="Green">E85</font></i></a>, <a class="ref" href="graph_3.html#T58" target="_self">Th58</a></i>;<br/>

<b>thus </b><a NAME="E43:82"/>
 <a href="card_1.html#K4">card</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> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="graph_3.html#K14" target="_self">ExtendCycle</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="graph_3.html#E38:82"><i><font color="Green">E89</font></i></a>, <a class="txt" href="graph_3.html#E41:82"><i><font color="Green">E90</font></i></a>, <a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/>


</div>
