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

<b>let </b><font color="Maroon" title="c1">G</font> be  <a href="graph_1.html#V6" title="GRAPH_1:attr.6">connected</a> <a href="graph_1.html#V7" title="GRAPH_1:attr.7">finite</a> <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Graph</a>;<br/><b>let </b><font color="Maroon" title="c2">c</font> be    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">G</font> <a href="graph_3.html#K9" title="GRAPH_3:func.9">-CycleSet</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:82"/><i><font color="Green" title="E70">A1</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c1">G</font>
 <b>and </b><br/><a NAME="E2:82"/><i><font color="Green" title="E71">A2</font></i>: 
not <font color="Maroon" title="c2">c</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>and </b><br/><a NAME="E3:82"/><i><font color="Green" title="E72">A3</font></i>: 
for <font color="Olive" title="b1">v</font> being  <a href="graph_3.html#NM2" title="GRAPH_3:NM.2">Vertex</a> of <font color="Maroon" title="c1">G</font> holds   <a href="graph_1.html#K6" title="GRAPH_1:func.6">Degree</a> <font color="Olive" title="b1">v</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a>
 ;<br/>

<b>set </b><font color="Maroon" title="c3">E</font> = the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c1">G</font>;<br/>
<a NAME="E4:82"/><i><font color="Green" title="E73">A4</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">c</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c1">G</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">E'</font> = the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c1">G</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="ref" href="graph_1.html#D8" title="GRAPH_1:def.8">GRAPH_1:def 8</a></i>;<br/>
<a NAME="E6:82"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">c</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c1">G</font>
 
<b>by </b><i><a class="txt" href="graph_3.html#E1:82"><i><font color="Green" title="E70">A1</font></i></a>, <a class="txt" href="graph_3.html#E4:82"><i><font color="Green" title="E73">A4</font></i></a>, <a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c5">xx</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E8:82"/><i><font color="Green" title="E74">A5</font></i>: 
( <font color="Maroon" title="c5">xx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c1">G</font> &amp; not <font color="Maroon" title="c5">xx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">c</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T6" title="XBOOLE_0:th.6">XBOOLE_0:6</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">Erc</font> = <font color="Maroon" title="c4">E'</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">c</font></span>)</span> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="txt" href="graph_3.html#E8:82"><i><font color="Green" title="E74">A5</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">X</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">v'</font> where <font color="Olive" title="b1">v'</font> is   <a href="graph_3.html#NM2" title="GRAPH_3:NM.2">Vertex</a> of <font color="Maroon" title="c1">G</font> : ( <font color="Olive" title="b1">v'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="graph_3.html#K1" title="GRAPH_3:func.1">-VSet</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">c</font></span>)</span> &amp;  <a href="graph_1.html#K6" title="GRAPH_1:func.6">Degree</a> <font color="Olive" title="b1">v'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="graph_3.html#K7" title="GRAPH_3:func.7">Degree</a> <font color="Olive" title="b1">v'</font>,<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">c</font></span>)</span> ) </span>}</span>  as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="txt" href="graph_3.html#E1:82"><i><font color="Green" title="E70">A1</font></i></a>, <a class="txt" href="graph_3.html#E2:82"><i><font color="Green" title="E71">A2</font></i></a>, <a class="ref" href="graph_3.html#T61" target="_self" title="GRAPH_3:th.61">Th61</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">c'</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">G</font> <a href="graph_3.html#K9" title="GRAPH_3:func.9">-CycleSet</a> , <font color="Maroon" title="c9">v</font> being   <a href="graph_3.html#NM2" title="GRAPH_3:NM.2">Vertex</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E12:82"/><i><font color="Green" title="E75">A6</font></i>: 
<font color="Maroon" title="c9">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <font color="Maroon" title="c7">X</font>
 <b>and </b><br/><a NAME="E13:82"/><i><font color="Green" title="E76">A7</font></i>: 
<font color="Maroon" title="c8">c'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Erc</font> <a href="graph_3.html#K13" title="GRAPH_3:func.13">-CycleSet</a> <font color="Maroon" title="c9">v</font></span>)</span>
 <b>and </b><br/><a NAME="E14:82"/><i><font color="Green" title="E77">A8</font></i>: 
 <a href="graph_3.html#K14" title="GRAPH_3:func.14">ExtendCycle</a> <font color="Maroon" title="c2">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_3.html#K11" title="GRAPH_3:func.11">CatCycles</a> <font color="Maroon" title="c2">c</font>,<font color="Maroon" title="c8">c'</font>
 <b>by </b><i><a class="txt" href="graph_3.html#E1:82"><i><font color="Green" title="E70">A1</font></i></a>, <a class="txt" href="graph_3.html#E2:82"><i><font color="Green" title="E71">A2</font></i></a>, <a class="ref" href="graph_3.html#D13" target="_self" title="GRAPH_3:def.13">Def13</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">ccp</font> = <font color="Maroon" title="c2">c</font> as  <a href="msscyc_1.html#V1" title="MSSCYC_1:attr.1">cyclic</a> <a href="graph_1.html#NM4" title="GRAPH_1:NM.4">Path</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="ref" href="graph_3.html#D8" target="_self" title="GRAPH_3:def.8">Def8</a></i>;<br/>
<div><i><font color="Green" title="E78">A9</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" title="c11">v</font> be   <a href="graph_3.html#NM2" title="GRAPH_3:NM.2">Vertex</a> of <font color="Maroon" title="c1">G</font>;<br/><a NAME="E1:82_1"/><i><font color="Green" title="E79">A10</font></i>: 
 <a href="graph_1.html#K6" title="GRAPH_1:func.6">Degree</a> <font color="Maroon" title="c11">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_3.html#K7" title="GRAPH_3:func.7">Degree</a> <font color="Maroon" title="c11">v</font>,the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="ref" href="graph_3.html#T29" target="_self" title="GRAPH_3:th.29">Th29</a></i>;<br/><a NAME="E2:82_1"/><i><font color="Green" title="E80">A11</font></i>: 
 <a href="graph_1.html#K6" title="GRAPH_1:func.6">Degree</a> <font color="Maroon" title="c11">v</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a>
 <b>by </b><i><a class="txt" href="graph_3.html#E3:82"><i><font color="Green" title="E72">A3</font></i></a></i>;<br/><a NAME="E3:82_1"/><i><font color="Green" title="E81">A12</font></i>: 
 <a href="graph_3.html#K7" title="GRAPH_3:func.7">Degree</a> <font color="Maroon" title="c11">v</font>,<font color="Maroon" title="c6">Erc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="graph_3.html#K7" title="GRAPH_3:func.7">Degree</a> <font color="Maroon" title="c11">v</font>,<font color="Maroon" title="c4">E'</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="graph_3.html#K7" title="GRAPH_3:func.7">Degree</a> <font color="Maroon" title="c11">v</font>,<span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">c</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="graph_3.html#E4:82"><i><font color="Green" title="E73">A4</font></i></a>, <a class="ref" href="graph_3.html#T34" target="_self" title="GRAPH_3:th.34">Th34</a></i>;<br/><a NAME="E4:82_1"/>
 <a href="graph_3.html#K7" title="GRAPH_3:func.7">Degree</a> <font color="Maroon" title="c11">v</font>,<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">ccp</font></span>)</span> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a>
 <b>by </b><i><a class="ref" href="graph_3.html#T54" target="_self" title="GRAPH_3:th.54">Th54</a></i>;<br/><b>hence </b><a NAME="E5:82_1"/>
 <a href="graph_3.html#K7" title="GRAPH_3:func.7">Degree</a> <font color="Maroon" title="c11">v</font>,<font color="Maroon" title="c6">Erc</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a>
 <b>by </b><i><a class="txt" href="graph_3.html#E1:82_1"><i><font color="Green" title="E79">A10</font></i></a>, <a class="txt" href="graph_3.html#E2:82_1"><i><font color="Green" title="E80">A11</font></i></a>, <a class="txt" href="graph_3.html#E3:82_1"><i><font color="Green" title="E81">A12</font></i></a>, <a class="ref" href="graph_3.html#T1" target="_self" title="GRAPH_3:th.1">Th1</a></i>;<br/></div><b>end;</b></div>
<a NAME="E17:82"/>
<font color="Maroon" title="c9">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font>
 
<b>by </b><i><a class="txt" href="graph_3.html#E12:82"><i><font color="Green" title="E75">A6</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c11">v'</font> being   <a href="graph_3.html#NM2" title="GRAPH_3:NM.2">Vertex</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E19:82"/><i><font color="Green" title="E79">A13</font></i>: 
( <font color="Maroon" title="c9">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">v'</font> &amp; <font color="Maroon" title="c11">v'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="graph_3.html#K1" title="GRAPH_3:func.1">-VSet</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">c</font></span>)</span> &amp;  <a href="graph_1.html#K6" title="GRAPH_1:func.6">Degree</a> <font color="Maroon" title="c11">v'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="graph_3.html#K7" title="GRAPH_3:func.7">Degree</a> <font color="Maroon" title="c11">v'</font>,<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">c</font></span>)</span> )
 ;<br/>
<a NAME="E20:82"/><i><font color="Green" title="E80">A14</font></i>: 
 <a href="graph_3.html#K7" title="GRAPH_3:func.7">Degree</a> <font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c6">Erc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="graph_3.html#K7" title="GRAPH_3:func.7">Degree</a> <font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c4">E'</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="graph_3.html#K7" title="GRAPH_3:func.7">Degree</a> <font color="Maroon" title="c9">v</font>,<span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">c</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="graph_3.html#E4:82"><i><font color="Green" title="E73">A4</font></i></a>, <a class="ref" href="graph_3.html#T34" target="_self" title="GRAPH_3:th.34">Th34</a></i>;<br/>
<a NAME="E21:82"/>
 <a href="graph_1.html#K6" title="GRAPH_1:func.6">Degree</a> <font color="Maroon" title="c11">v'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_3.html#K7" title="GRAPH_3:func.7">Degree</a> <font color="Maroon" title="c9">v</font>,the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c1">G</font>
 
<b>by </b><i><a class="txt" href="graph_3.html#E19:82"><i><font color="Green" title="E79">A13</font></i></a>, <a class="ref" href="graph_3.html#T29" target="_self" title="GRAPH_3:th.29">Th29</a></i>;<br/>
<a NAME="E22:82"/><b>then </b>
 <a href="graph_3.html#K7" title="GRAPH_3:func.7">Degree</a> <font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c6">Erc</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="graph_3.html#E19:82"><i><font color="Green" title="E79">A13</font></i></a>, <a class="txt" href="graph_3.html#E20:82"><i><font color="Green" title="E80">A14</font></i></a></i>;<br/>
<a NAME="E23:82"/><b>then </b><i><font color="Green" title="E81">A15</font></i>: 
( not <font color="Maroon" title="c8">c'</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">c'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c1">G</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">c</font></span>)</span> &amp; <font color="Maroon" title="c9">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="graph_3.html#K1" title="GRAPH_3:func.1">-VSet</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">c'</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="graph_3.html#E13:82"><i><font color="Green" title="E76">A7</font></i></a>, <a class="txt" href="graph_3.html#E16:82"><i><font color="Green" title="E78">A9</font></i></a>, <a class="ref" href="graph_3.html#T60" target="_self" title="GRAPH_3:th.60">Th60</a></i>;<br/>
<a NAME="E24:82"/><b>then </b><i><font color="Green" title="E82">A16</font></i>: 
<font color="Maroon" title="c1">G</font> <a href="graph_3.html#K1" title="GRAPH_3:func.1">-VSet</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">c</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">G</font> <a href="graph_3.html#K1" title="GRAPH_3:func.1">-VSet</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">c'</font></span>)</span>
 
<b>by </b><i><a class="txt" href="graph_3.html#E19:82"><i><font color="Green" title="E79">A13</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E25:82"/><b>then </b><i><font color="Green" title="E83">A17</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="graph_3.html#K1" title="GRAPH_3:func.1">-VSet</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">c</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="graph_3.html#K1" title="GRAPH_3:func.1">-VSet</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">c'</font></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E26:82"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">c</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c1">G</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">c</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T79" title="XBOOLE_1:th.79">XBOOLE_1:79</a></i>;<br/>
<a NAME="E27:82"/><b>then </b><i><font color="Green" title="E84">A18</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">c</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default">the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c1">G</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">c</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">c</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">c'</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">c</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c1">G</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">c</font></span>)</span></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">c'</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="graph_3.html#E23:82"><i><font color="Green" title="E81">A15</font></i></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>
<br/>.= 
<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">c'</font></span>)</span>
<b>by </b><i><a class="txt" href="graph_3.html#E27:82"><i><font color="Green" title="E84">A18</font></i></a>, <a class="ref" href="xboole_1.html#T16" title="XBOOLE_1:th.16">XBOOLE_1:16</a></i>
<br/>.= 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 

;<br/>
<a NAME="E29:82"/><b>then </b><i><font color="Green" title="E85">A19</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">c</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">c'</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c12">vr</font> being   <a href="graph_3.html#NM2" title="GRAPH_3:NM.2">Vertex</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E31:82"/><i><font color="Green" title="E86">A20</font></i>: 
<font color="Maroon" title="c12">vr</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="graph_3.html#K1" title="GRAPH_3:func.1">-VSet</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">c</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="graph_3.html#K1" title="GRAPH_3:func.1">-VSet</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">c'</font></span>)</span></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E32:82"/><i><font color="Green" title="E87">A21</font></i>: 
 <a href="graph_3.html#K11" title="GRAPH_3:func.11">CatCycles</a> <font color="Maroon" title="c2">c</font>,<font color="Maroon" title="c8">c'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="graph_3.html#K10" title="GRAPH_3:func.10">Rotate</a> <font color="Maroon" title="c2">c</font>,<font color="Maroon" title="c12">vr</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><a href="graph_3.html#K10" title="GRAPH_3:func.10">Rotate</a> <font color="Maroon" title="c8">c'</font>,<font color="Maroon" title="c12">vr</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_3.html#E24:82"><i><font color="Green" title="E82">A16</font></i></a>, <a class="ref" href="graph_3.html#D10" target="_self" title="GRAPH_3:def.10">Def10</a></i>;<br/>
<a NAME="E33:82"/>
<font color="Maroon" title="c12">vr</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="graph_3.html#K1" title="GRAPH_3:func.1">-VSet</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">c</font></span>)</span>
 
<b>by </b><i><a class="txt" href="graph_3.html#E25:82"><i><font color="Green" title="E83">A17</font></i></a>, <a class="txt" href="graph_3.html#E31:82"><i><font color="Green" title="E86">A20</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E34:82"/><b>then </b><i><font color="Green" title="E88">A22</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="graph_3.html#K10" title="GRAPH_3:func.10">Rotate</a> <font color="Maroon" title="c2">c</font>,<font color="Maroon" title="c12">vr</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">c</font>
 
<b>by </b><i><a class="txt" href="graph_3.html#E71"><i><font color="Green" title="E60">Lm5</font></i></a></i>;<br/>
<a NAME="E35:82"/>
<font color="Maroon" title="c12">vr</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="graph_3.html#K1" title="GRAPH_3:func.1">-VSet</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">c'</font></span>)</span>
 
<b>by </b><i><a class="txt" href="graph_3.html#E25:82"><i><font color="Green" title="E83">A17</font></i></a>, <a class="txt" href="graph_3.html#E31:82"><i><font color="Green" title="E86">A20</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E36:82"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="graph_3.html#K10" title="GRAPH_3:func.10">Rotate</a> <font color="Maroon" title="c8">c'</font>,<font color="Maroon" title="c12">vr</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">c'</font>
 
<b>by </b><i><a class="txt" href="graph_3.html#E71"><i><font color="Green" title="E60">Lm5</font></i></a></i>;<br/>
<a NAME="E37:82"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="graph_3.html#K14" title="GRAPH_3:func.14">ExtendCycle</a> <font color="Maroon" title="c2">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">c</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">c'</font></span>)</span>
 
<b>by </b><i><a class="txt" href="graph_3.html#E14:82"><i><font color="Green" title="E77">A8</font></i></a>, <a class="txt" href="graph_3.html#E32:82"><i><font color="Green" title="E87">A21</font></i></a>, <a class="txt" href="graph_3.html#E34:82"><i><font color="Green" title="E88">A22</font></i></a>, <a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></i>;<br/>
<a NAME="E38:82"/><b>then </b><i><font color="Green" title="E89">A23</font></i>: 
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="graph_3.html#K14" title="GRAPH_3:func.14">ExtendCycle</a> <font color="Maroon" title="c2">c</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">c</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">c'</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="graph_3.html#E29:82"><i><font color="Green" title="E85">A19</font></i></a>, <a class="ref" href="card_2.html#T53" title="CARD_2:th.53">CARD_2:53</a></i>;<br/>
<a NAME="E39:82"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">c'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="graph_3.html#E23:82"><i><font color="Green" title="E81">A15</font></i></a>, <a class="ref" href="relat_1.html#T64" title="RELAT_1:th.64">RELAT_1:64</a></i>;<br/>
<a NAME="E40:82"/><b>then </b>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">c'</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="card_2.html#T59" title="CARD_2:th.59">CARD_2:59</a></i>;<br/>
<a NAME="E41:82"/><b>then </b><i><font color="Green" title="E90">A24</font></i>: 
0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">c'</font></span>)</span>
 
;<br/>
<b>thus </b><a NAME="E42:82"/>
not  <a href="graph_3.html#K14" title="GRAPH_3:func.14">ExtendCycle</a> <font color="Maroon" title="c2">c</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>by </b><i><a class="txt" href="graph_3.html#E2:82"><i><font color="Green" title="E71">A2</font></i></a>, <a class="txt" href="graph_3.html#E14:82"><i><font color="Green" title="E77">A8</font></i></a>, <a class="txt" href="graph_3.html#E24:82"><i><font color="Green" title="E82">A16</font></i></a>, <a class="txt" href="graph_3.html#E29:82"><i><font color="Green" title="E85">A19</font></i></a>, <a class="ref" href="graph_3.html#T58" target="_self" title="GRAPH_3:th.58">Th58</a></i>;<br/>

<b>thus </b><a NAME="E43:82"/>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">c</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="graph_3.html#K14" title="GRAPH_3:func.14">ExtendCycle</a> <font color="Maroon" title="c2">c</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="graph_3.html#E38:82"><i><font color="Green" title="E89">A23</font></i></a>, <a class="txt" href="graph_3.html#E41:82"><i><font color="Green" title="E90">A24</font></i></a>, <a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/>


</div>
