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<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  <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>3</sub></font> be   <a href="graph_3.html#NM2">Vertex</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:79"/><i><font color="Green">E67</font></i>: 
 <a href="graph_3.html#K7" target="_self">Degree</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>and </b><br/><a NAME="E2:79"/><i><font color="Green">E68</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>2</sub></font> holds  <a href="graph_3.html#K7" target="_self">Degree</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> is <a href="abian.html#V1">even</a>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="graph_3.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font> <a href="graph_3.html#K13" target="_self">-CycleSet</a> <font color="Maroon">c<sub>3</sub></font>;<br/>

<a NAME="E3:79"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="graph_3.html#K13" target="_self">-CycleSet</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">c<sub>2</sub></font> <a href="graph_3.html#K9" target="_self">-CycleSet</a>  : (  <a href="relat_1.html#K2">rng</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font> &amp; not <font color="Olive">b<sub>1</sub></font> is <a href="xboole_0.html#V1">empty</a> &amp; ex <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="graph_2.html#R1">is_vertex_seq_of</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> ) ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="graph_3.html#E1:79"><i><font color="Green">E67</font></i></a>, <a class="txt" href="graph_3.html#E2:79"><i><font color="Green">E68</font></i></a>, <a class="ref" href="graph_3.html#D12" target="_self">Def12</a></i>;<br/>
<a NAME="E4:79"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="graph_3.html#K13" target="_self">-CycleSet</a> <font color="Maroon">c<sub>3</sub></font>
 
;<br/>
<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">c<sub>2</sub></font> <a href="graph_3.html#K9" target="_self">-CycleSet</a> <b> such that </b><br/><a NAME="E6:79"/><i><font color="Green">E70</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font> &amp; not <font color="Maroon">c<sub>5</sub></font> is <a href="xboole_0.html#V1">empty</a> &amp; ex <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="graph_2.html#R1">is_vertex_seq_of</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> ) )
 <b>by </b><i><a class="txt" href="graph_3.html#E3:79"><i><font color="Green">E69</font></i></a></i>;<br/>
<b>thus </b><a NAME="E7:79"/>
not <font color="Maroon">c<sub>4</sub></font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="txt" href="graph_3.html#E6:79"><i><font color="Green">E70</font></i></a></i>;<br/>

<b>thus </b><a NAME="E8:79"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="graph_3.html#E6:79"><i><font color="Green">E70</font></i></a></i>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="graph_1.html#NM4">Path</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="graph_3.html#D8" target="_self">Def8</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E11:79"/><i><font color="Green">E71</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="graph_2.html#R1">is_vertex_seq_of</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="graph_3.html#E6:79"><i><font color="Green">E70</font></i></a></i>;<br/>
<a NAME="E12:79"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>2</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>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="graph_3.html#E6:79"><i><font color="Green">E70</font></i></a>, <a class="txt" href="graph_3.html#E11:79"><i><font color="Green">E71</font></i></a>, <a class="ref" href="graph_2.html#T34">GRAPH_2:34</a></i>;<br/>
<a NAME="E13:79"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="graph_3.html#E11:79"><i><font color="Green">E71</font></i></a>, <a class="ref" href="graph_2.html#D7">GRAPH_2:def 7</a></i>;<br/>
<a NAME="E14:79"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E15:79"/><b>then </b>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>hence </b><a NAME="E16:79"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</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>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_3.html#E6:79"><i><font color="Green">E70</font></i></a>, <a class="txt" href="graph_3.html#E11:79"><i><font color="Green">E71</font></i></a>, <a class="txt" href="graph_3.html#E12:79"><i><font color="Green">E72</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


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