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

<b>let </b><font color="Maroon">c<sub>1</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>2</sub></font> be   <a href="graph_3.html#NM2">Vertex</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <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>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="graph_1.html#NM4">Path</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:68"/><i><font color="Green">E56</font></i>: 
not <font color="Maroon">c<sub>4</sub></font> is <a href="msscyc_1.html#V1">cyclic</a>
 <b>and </b><br/><a NAME="E2:68"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="graph_2.html#R1">is_vertex_seq_of</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>5</sub></font> = the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E3:68"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</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>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="graph_3.html#E2:68"><i><font color="Green">E57</font></i></a>, <a class="ref" href="graph_2.html#D7">GRAPH_2:def 7</a></i>;<br/>
<a NAME="E4:68"/><b>then </b><i><font color="Green">E58</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E5:68"/><b>then </b>
( 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> 1, <font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>3</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>3</sub></font></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="ref" href="finseq_2.html#T13">FINSEQ_2:13</a></i>;<br/>
<a NAME="E7:68"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>6</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#E1:68"><i><font color="Green">E56</font></i></a>, <a class="txt" href="graph_3.html#E2:68"><i><font color="Green">E57</font></i></a>, <a class="ref" href="msscyc_1.html#D2">MSSCYC_1:def 2</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> =  <a href="graph_3.html#K8" target="_self">AddNewEdge</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="graph_3.html#NM2">Vertex</a> of <span class="p1">(<span class="default"><a href="graph_3.html#K8" target="_self">AddNewEdge</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <b>by </b><i><a class="ref" href="graph_3.html#D7" target="_self">Def7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="graph_1.html#NM4">Path</a> of  <a href="graph_3.html#K8" target="_self">AddNewEdge</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> <b>by </b><i><a class="ref" href="graph_3.html#T42" target="_self">Th42</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>3</sub></font> as    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U1">Vertices</a> of <span class="p1">(<span class="default"><a href="graph_3.html#K8" target="_self">AddNewEdge</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <b>by </b><i><a class="ref" href="graph_3.html#D7" target="_self">Def7</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>14</sub></font> = the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E11:68"/><i><font color="Green">E60</font></i>: 
the <a href="graph_1.html#U2">Edges</a> of <span class="p1">(<span class="default"><a href="graph_3.html#K8" target="_self">AddNewEdge</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="graph_3.html#D7" target="_self">Def7</a></i>;<br/>
<a NAME="E12:68"/><i><font color="Green">E61</font></i>: 
not the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">c<sub>1</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>
 
;<br/>
<a NAME="E13:68"/><i><font color="Green">E62</font></i>: 
the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E14:68"/><b>then </b>
the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2">Edges</a> of <span class="p1">(<span class="default"><a href="graph_3.html#K8" target="_self">AddNewEdge</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="graph_3.html#E11:68"><i><font color="Green">E60</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>15</sub></font> = <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default">the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="finseq_1.html#K9">*&gt;</a></span> as   <a href="graph_1.html#NM4">Path</a> of  <a href="graph_3.html#K8" target="_self">AddNewEdge</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> <b>by </b><i><a class="ref" href="graph_3.html#T7" target="_self">Th7</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>16</sub></font> = <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="finseq_4.html#K2">*&gt;</a></span>;<br/>
<b>set </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>13</sub></font> <a href="graph_2.html#K4">^'</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="finseq_4.html#K2">*&gt;</a></span>;<br/>
<a NAME="E16:68"/><i><font color="Green">E63</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</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/>
<a NAME="E17:68"/><b>then </b><i><font color="Green">E64</font></i>: 
not the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="graph_3.html#E12:68"><i><font color="Green">E61</font></i></a></i>;<br/>
<a NAME="E18:68"/><i><font color="Green">E65</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/>
<a NAME="E19:68"/><i><font color="Green">E66</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>12</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:68_1"/>
not  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>12</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>18</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:68_1"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>thus </b><a NAME="E4:68_1"/>
contradiction
 <b>by </b><i><a class="txt" href="graph_3.html#E17:68"><i><font color="Green">E64</font></i></a>, <a class="txt" href="graph_3.html#E18:68"><i><font color="Green">E65</font></i></a>, <a class="txt" href="graph_3.html#E3:68_1"><i><font color="Green">E67</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E20:68"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="graph_2.html#R1">is_vertex_seq_of</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="graph_3.html#E2:68"><i><font color="Green">E57</font></i></a>, <a class="ref" href="graph_3.html#T41" target="_self">Th41</a></i>;<br/>
<a NAME="E21:68"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">c<sub>13</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>13</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/>
<a NAME="E22:68"/>
( the <a href="graph_1.html#U3">Source</a> of <span class="p1">(<span class="default"><a href="graph_3.html#K8" target="_self">AddNewEdge</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; the <a href="graph_1.html#U4">Target</a> of <span class="p1">(<span class="default"><a href="graph_3.html#K8" target="_self">AddNewEdge</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> )
 
<b>by </b><i><a class="ref" href="graph_3.html#T39" target="_self">Th39</a></i>;<br/>
<a NAME="E23:68"/><b>then </b><i><font color="Green">E69</font></i>: 
<span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="graph_2.html#R1">is_vertex_seq_of</a> <font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="ref" href="graph_3.html#T16" target="_self">Th16</a></i>;<br/>
<a NAME="E24:68"/><i><font color="Green">E70</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="graph_2.html#K4">^'</a> <span class="p2"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="finseq_4.html#K2">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> 1
 
<b>by </b><i><a class="txt" href="graph_3.html#E4:68"><i><font color="Green">E58</font></i></a>, <a class="ref" href="graph_2.html#T14">GRAPH_2:14</a></i>;<br/>
<a NAME="E25:68"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="hidden.html#R1">=</a> 2
 
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/>
<a NAME="E26:68"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="graph_2.html#K4">^'</a> <span class="p2"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="finseq_4.html#K2">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="graph_2.html#K4">^'</a> <span class="p3"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="finseq_4.html#K2">*&gt;</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 2
 
<b>by </b><i><a class="ref" href="graph_2.html#T16">GRAPH_2:16</a></i>;<br/>
<a NAME="E27:68"/><b>then </b><i><font color="Green">E71</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="graph_2.html#K4">^'</a> <span class="p2"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="finseq_4.html#K2">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="graph_2.html#K4">^'</a> <span class="p2"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="finseq_4.html#K2">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="graph_2.html#K4">^'</a> <span class="p3"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="finseq_4.html#K2">*&gt;</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="graph_3.html#E24:68"><i><font color="Green">E70</font></i></a>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>12</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>15</sub></font> as   <a href="graph_1.html#NM4">Path</a> of  <a href="graph_3.html#K8" target="_self">AddNewEdge</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> <b>by </b><i><a class="txt" href="graph_3.html#E19:68"><i><font color="Green">E66</font></i></a>, <a class="txt" href="graph_3.html#E20:68"><i><font color="Green">E67</font></i></a>, <a class="txt" href="graph_3.html#E21:68"><i><font color="Green">E68</font></i></a>, <a class="txt" href="graph_3.html#E23:68"><i><font color="Green">E69</font></i></a>, <a class="ref" href="graph_3.html#T9" target="_self">Th9</a></i>;<br/>
<a NAME="E29:68"/>
<font color="Maroon">c<sub>13</sub></font> <a href="graph_2.html#K4">^'</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="graph_2.html#R1">is_vertex_seq_of</a> <font color="Maroon">c<sub>18</sub></font>
 
<b>by </b><i><a class="txt" href="graph_3.html#E20:68"><i><font color="Green">E67</font></i></a>, <a class="txt" href="graph_3.html#E21:68"><i><font color="Green">E68</font></i></a>, <a class="txt" href="graph_3.html#E23:68"><i><font color="Green">E69</font></i></a>, <a class="ref" href="graph_2.html#T47">GRAPH_2:47</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>18</sub></font> as  <a href="msscyc_1.html#V1">cyclic</a> <a href="graph_1.html#NM4">Path</a> of  <a href="graph_3.html#K8" target="_self">AddNewEdge</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> <b>by </b><i><a class="txt" href="graph_3.html#E27:68"><i><font color="Green">E71</font></i></a>, <a class="ref" href="msscyc_1.html#D2">MSSCYC_1:def 2</a></i>;<br/>
<a NAME="E31:68"/><i><font color="Green">E72</font></i>: 
 <a href="graph_3.html#K7" target="_self">Degree</a> <font color="Maroon">c<sub>9</sub></font>,<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>19</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>reconsider </b><font color="Maroon">c<sub>20</sub></font> =  <a href="graph_3.html#K7" target="_self">Degree</a> <font color="Maroon">c<sub>9</sub></font>,<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> as  <a href="abian.html#V1">even</a> <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="graph_3.html#T54" target="_self">Th54</a></i>;<br/>
<a NAME="E33:68"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</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>
 
<b>by </b><i><a class="txt" href="graph_3.html#E12:68"><i><font color="Green">E61</font></i></a>, <a class="txt" href="graph_3.html#E18:68"><i><font color="Green">E65</font></i></a>, <a class="ref" href="zfmisc_1.html#T56">ZFMISC_1:56</a></i>;<br/>
<a NAME="E34:68"/><b>then </b><i><font color="Green">E73</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">c<sub>1</sub></font> <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/>
<i><font color="Green">E74</font></i>: <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">c<sub>1</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</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#T44">FINSEQ_1:44</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="txt" href="graph_3.html#E34:68"><i><font color="Green">E73</font></i></a>, <a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
<br/>.= 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="graph_3.html#E16:68"><i><font color="Green">E63</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/>
<a NAME="E36:68"/><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>2</sub></font>,<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="graph_3.html#K7" target="_self">Degree</a> <font color="Maroon">c<sub>2</sub></font>,<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="ref" href="graph_3.html#T36" target="_self">Th36</a></i>;<br/>
<a NAME="E37:68"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>19</sub></font> <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>4</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>15</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T44">FINSEQ_1:44</a></i>;<br/>
<a NAME="E38:68"/><b>then </b><i><font color="Green">E76</font></i>: 
the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>19</sub></font>
 
<b>by </b><i><a class="txt" href="graph_3.html#E13:68"><i><font color="Green">E62</font></i></a>, <a class="txt" href="graph_3.html#E18:68"><i><font color="Green">E65</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:68_3"/>
( ( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> ) or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_3_1"><b>suppose </b></a><a NAME="E1:68_3_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:68_3_1"/>
(  <a href="graph_3.html#K7" target="_self">Degree</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> is <a href="abian.html#V1">even</a> iff ( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> 1 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</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>3</sub></font></span>)</span> ) )
 <b>by </b><i><a class="txt" href="graph_3.html#E31:68"><i><font color="Green">E72</font></i></a>, <a class="txt" href="graph_3.html#E36:68"><i><font color="Green">E75</font></i></a>, <a class="ref" href="graph_3.html#T53" target="_self">Th53</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_3_2"><b>suppose </b></a><a NAME="E1:68_3_2"/><i><font color="Green">E77</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> )
 ;<br/><div class="add"><a NAME="E2:68_3_2"/><b>then </b>
 <a href="graph_3.html#K7" target="_self">Degree</a> <font color="Maroon">c<sub>9</sub></font>,<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <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>2</sub></font>,<span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="graph_3.html#E7:68"><i><font color="Green">E59</font></i></a>, <a class="txt" href="graph_3.html#E38:68"><i><font color="Green">E76</font></i></a>, <a class="ref" href="graph_3.html#T52" target="_self">Th52</a></i>;<br/><a NAME="E3:68_3_2"/><b>then </b>
 <a href="graph_3.html#K7" target="_self">Degree</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> <a href="xcmplx_0.html#K6">-</a> 1
 ;<br/><b>hence </b><a NAME="E4:68_3_2"/>
(  <a href="graph_3.html#K7" target="_self">Degree</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> is <a href="abian.html#V1">even</a> iff ( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> 1 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</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>3</sub></font></span>)</span> ) )
 <b>by </b><i><a class="txt" href="graph_3.html#E35:68"><i><font color="Green">E74</font></i></a>, <a class="txt" href="graph_3.html#E1:68_3_2"><i><font color="Green">E77</font></i></a>, <a class="ref" href="graph_3.html#T36" target="_self">Th36</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
