<?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>set </b><font color="Maroon">c<sub>2</sub></font> = the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<div><b>hereby </b>
<div class="add"><b>given </b><font color="Maroon">c<sub>3</sub></font> being  <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> such that </b><a NAME="E1:85_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="graph_3.html#V1" target="_self">Eulerian</a>
 ;<br/><b>consider </b><font color="Maroon">c<sub>4</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>1</sub></font><b> such that </b><br/><a NAME="E3:85_1"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="graph_2.html#R1">is_vertex_seq_of</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="msscyc_1.html#D2">MSSCYC_1:def 2</a></i>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="graph_3.html#NM2">Vertex</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><a NAME="E4:85_1"/><i><font color="Green">E75</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</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#E1:85_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="graph_3.html#D14" target="_self">Def14</a></i>;<br/><a NAME="E5:85_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="graph_3.html#E1:85_1"><i><font color="Green">E73</font></i></a>, <a class="txt" href="graph_3.html#E3:85_1"><i><font color="Green">E74</font></i></a>, <a class="ref" href="graph_3.html#T63" target="_self">Th63</a></i>;<br/><a NAME="E6:85_1"/><b>then </b>
 <a href="graph_3.html#K7" target="_self">Degree</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> is <a href="abian.html#V1">even</a>
 <b>by </b><i><a class="txt" href="graph_3.html#E3:85_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="graph_3.html#E64"><i><font color="Green">Lemma53</font></i></a></i>;<br/><b>hence </b><a NAME="E7:85_1"/>
 <a href="graph_1.html#K6">Degree</a> <font color="Maroon">c<sub>5</sub></font> is <a href="abian.html#V1">even</a>
 <b>by </b><i><a class="txt" href="graph_3.html#E4:85_1"><i><font color="Green">E75</font></i></a>, <a class="ref" href="graph_3.html#T29" target="_self">Th29</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:85"/><i><font color="Green">E73</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/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="85_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:85_2"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="msscyc_1.html#V2">empty</a> or  not <font color="Maroon">c<sub>1</sub></font> is <a href="msscyc_1.html#V2">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="85_2_1"><b>suppose </b></a><a NAME="E1:85_2_1"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="msscyc_1.html#V2">empty</a>
 ;<br/><div class="add"><a NAME="E2:85_2_1"/><b>then </b><i><font color="Green">E74</font></i>: 
the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">c<sub>1</sub></font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="ref" href="msscyc_1.html#D3">MSSCYC_1:def 3</a></i>;<br/><a NAME="E3:85_2_1"/>
 <a href="xboole_0.html#K1">{}</a>  is    <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> 
 <b>by </b><i><a class="ref" href="graph_3.html#T56" target="_self">Th56</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> =  <a href="xboole_0.html#K1">{}</a>  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/><b>take </b>
<font color="Maroon">c<sub>3</sub></font>
;<br/><b>thus </b><a NAME="E5:85_2_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</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#E2:85_2_1"><i><font color="Green">E74</font></i></a>, <a class="ref" href="finseq_1.html#T27">FINSEQ_1:27</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="graph_3.html#D14" target="_self">GRAPH_3:def 14</a><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="85_2_2"><b>suppose </b></a><a NAME="E1:85_2_2"/>
not <font color="Maroon">c<sub>1</sub></font> is <a href="msscyc_1.html#V2">empty</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>1</sub></font> as  <a href="graph_1.html#V6">connected</a> <a href="graph_1.html#V7">finite</a> non <a href="msscyc_1.html#V2">empty</a> <a href="graph_1.html#NM1">Graph</a> ;<br/><b>reconsider </b><font color="Maroon">c<sub>4</sub></font> =  <a href="subset_1.html#K10">choose</a> the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">c<sub>1</sub></font> as   <a href="graph_3.html#NM2">Vertex</a> of <font color="Maroon">c<sub>3</sub></font> ;<br/><a NAME="E4:85_2_2"/>
 <a href="subset_1.html#K10">choose</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="graph_3.html#K13" target="_self">-CycleSet</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <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> <a href="graph_3.html#K13" target="_self">-CycleSet</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> =  <a href="subset_1.html#K10">choose</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="graph_3.html#K13" target="_self">-CycleSet</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> as    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font> <a href="graph_3.html#K9" target="_self">-CycleSet</a>  ;<br/><div><i><font color="Green">E74</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="85_2_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="graph_3.html#NM2">Vertex</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><a NAME="E1:85_2_2_1"/>
 <a href="graph_1.html#K6">Degree</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="graph_3.html#K7" target="_self">Degree</a> <font color="Maroon">c<sub>6</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/><b>hence </b><a NAME="E2:85_2_2_1"/>
 <a href="graph_3.html#K7" target="_self">Degree</a> <font color="Maroon">c<sub>6</sub></font>,the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">c<sub>1</sub></font> is <a href="abian.html#V1">even</a>
 <b>by </b><i><a class="txt" href="graph_3.html#E2:85"><i><font color="Green">E73</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E7:85_2_2"/>
 <a href="graph_1.html#K6">Degree</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="graph_3.html#K7" target="_self">Degree</a> <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>
 <b>by </b><i><a class="ref" href="graph_3.html#T29" target="_self">Th29</a></i>;<br/><a NAME="E8:85_2_2"/><b>then </b><i><font color="Green">E75</font></i>: 
 <a href="graph_3.html#K7" target="_self">Degree</a> <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> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="ref" href="graph_3.html#T38" target="_self">Th38</a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1">Nat</a>,   <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font> <a href="graph_3.html#K9" target="_self">-CycleSet</a> ) <b>-&gt; </b>   <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font> <a href="graph_3.html#K9" target="_self">-CycleSet</a>  =  <a href="graph_3.html#K14" target="_self">ExtendCycle</a> <font color="Maroon">a<sub>2</sub></font>;<br/><b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,<font color="Maroon">c<sub>3</sub></font> <a href="graph_3.html#K9" target="_self">-CycleSet</a> <b> such that </b><br/><a NAME="E10:85_2_2"/><i><font color="Green">E76</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font>) ) )
 <b>from </b><i><a class="ref" href="recdef_1.html#S4">RECDEF_1:sch 4</a>();<br/></i><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_2_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:85_2_2_2"/><i><font color="Green">E77</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a><br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font> <a href="graph_3.html#K9" target="_self">-CycleSet</a>  holds <br/>(  not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> or  not  <a href="relat_1.html#K2">rng</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">c<sub>1</sub></font> )
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>7</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/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means ex <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font> <a href="graph_3.html#K9" target="_self">-CycleSet</a>  st <br/>( not <font color="Olive">b<sub>1</sub></font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&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="Olive">b<sub>1</sub></font></span>)</span> );<br/><a NAME="E3:85_2_2_2"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_2_2_2_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/>

<b>thus </b><a NAME="E1:85_2_2_2_1"/>
not <font color="Maroon">c<sub>5</sub></font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="txt" href="graph_3.html#E6:85_2_2"><i><font color="Green">E74</font></i></a>, <a class="txt" href="graph_3.html#E8:85_2_2"><i><font color="Green">E75</font></i></a>, <a class="ref" href="graph_3.html#T60" target="_self">Th60</a></i>;<br/>

<b>thus </b><a NAME="E2:85_2_2_2_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> 0
 <b>by </b><i><a class="txt" href="graph_3.html#E10:85_2_2"><i><font color="Green">E76</font></i></a></i>;<br/>

<b>thus </b><a NAME="E3:85_2_2_2_1"/>
0 <a href="xxreal_0.html#R1">&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>5</sub></font></span>)</span>
 ;<br/>


</div><b>end;</b></div><a NAME="E4:85_2_2_2"/><i><font color="Green">E79</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_2_2_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>given </b><font color="Maroon">c<sub>9</sub></font> being    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font> <a href="graph_3.html#K9" target="_self">-CycleSet</a> <b> such that </b><a NAME="E1:85_2_2_2_2"/><i><font color="Green">E80</font></i>: 
not <font color="Maroon">c<sub>9</sub></font> is <a href="xboole_0.html#V1">empty</a>
 <b>and </b><br/><a NAME="E2:85_2_2_2_2"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font>
 <b>and </b><br/><a NAME="E3:85_2_2_2_2"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&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>9</sub></font></span>)</span>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> =  <a href="graph_3.html#K14" target="_self">ExtendCycle</a> <font color="Maroon">c<sub>9</sub></font> as    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font> <a href="graph_3.html#K9" target="_self">-CycleSet</a>  ;<br/>
<b>take </b>
<font color="Maroon">c<sub>10</sub></font>
;<br/>

<a NAME="E5:85_2_2_2_2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="graph_3.html#E1:85_2_2_2"><i><font color="Green">E77</font></i></a>, <a class="txt" href="graph_3.html#E2:85_2_2_2_2"><i><font color="Green">E81</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:85_2_2_2_2"/>
not <font color="Maroon">c<sub>10</sub></font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="txt" href="graph_3.html#E2:85"><i><font color="Green">E73</font></i></a>, <a class="txt" href="graph_3.html#E1:85_2_2_2_2"><i><font color="Green">E80</font></i></a>, <a class="ref" href="graph_3.html#T62" target="_self">Th62</a></i>;<br/>

<b>thus </b><a NAME="E7:85_2_2_2_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="graph_3.html#E10:85_2_2"><i><font color="Green">E76</font></i></a>, <a class="txt" href="graph_3.html#E2:85_2_2_2_2"><i><font color="Green">E81</font></i></a></i>;<br/>

<a NAME="E8:85_2_2_2_2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="graph_3.html#E1:85_2_2_2"><i><font color="Green">E77</font></i></a>, <a class="txt" href="graph_3.html#E2:85_2_2_2_2"><i><font color="Green">E81</font></i></a></i>;<br/>
<a NAME="E9:85_2_2_2_2"/><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>9</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> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="graph_3.html#E2:85"><i><font color="Green">E73</font></i></a>, <a class="txt" href="graph_3.html#E1:85_2_2_2_2"><i><font color="Green">E80</font></i></a>, <a class="ref" href="graph_3.html#T62" target="_self">Th62</a></i>;<br/>
<a NAME="E10:85_2_2_2_2"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <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>10</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="graph_3.html#E3:85_2_2_2_2"><i><font color="Green">E82</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E11:85_2_2_2_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&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>10</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>


</div><b>end;</b></div><a NAME="E5:85_2_2_2"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="graph_3.html#E3:85_2_2_2"><i><font color="Green">E78</font></i></a>, <a class="txt" href="graph_3.html#E4:85_2_2_2"><i><font color="Green">E79</font></i></a>);<br/></i><b>then 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> <b> such that </b><br/><a NAME="E7:85_2_2_2"/><i><font color="Green">E80</font></i>: 
( not <font color="Maroon">c<sub>8</sub></font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&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/><a NAME="E8:85_2_2_2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/><a NAME="E9:85_2_2_2"/><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="xxreal_0.html#R1">&lt;=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="card_1.html#T80">CARD_1:80</a></i>;<br/><a NAME="E10:85_2_2_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 0
 <b>by </b><i><a class="txt" href="graph_3.html#E7:85_2_2_2"><i><font color="Green">E80</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E11:85_2_2_2"/>
contradiction
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/></div><b>end;</b></div><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="nat_1.html#NM1">Nat</a>, <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> <b> such that </b><br/><a NAME="E13:85_2_2"/><i><font color="Green">E77</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">c<sub>1</sub></font> )
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</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/><b>take </b>
<font color="Maroon">c<sub>9</sub></font>
;<br/><b>thus </b><a NAME="E15:85_2_2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</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#E13:85_2_2"><i><font color="Green">E77</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="graph_3.html#D14" target="_self">GRAPH_3:def 14</a><br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
