<?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>set </b><font color="Maroon" title="c2">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/>
<div><b>hereby </b>
<div class="add"><b>given </b><font color="Maroon" title="c3">c</font> being  <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> such that </b><a NAME="E2:85_1"/><i><font color="Green" title="E73">A1</font></i>: 
<font color="Maroon" title="c3">c</font> is <a href="graph_3.html#V1" title="GRAPH_3:attr.1">Eulerian</a>
 ;<br/><b>consider </b><font color="Maroon" title="c4">vs</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E4:85_1"/><i><font color="Green" title="E74">A2</font></i>: 
( <font color="Maroon" title="c4">vs</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Maroon" title="c3">c</font> &amp; <font color="Maroon" title="c4">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">vs</font></span>)</span> )
 <b>by </b><i><a class="ref" href="msscyc_1.html#D2" title="MSSCYC_1:def.2">MSSCYC_1:def 2</a></i>;<br/><b>let </b><font color="Maroon" title="c5">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="E5:85_1"/><i><font color="Green" title="E75">A3</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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#E2:85_1"><i><font color="Green" title="E73">A1</font></i></a>, <a class="ref" href="graph_3.html#D14" target="_self" title="GRAPH_3:def.14">Def14</a></i>;<br/><a NAME="E6:85_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">vs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="graph_3.html#E2:85_1"><i><font color="Green" title="E73">A1</font></i></a>, <a class="txt" href="graph_3.html#E4:85_1"><i><font color="Green" title="E74">A2</font></i></a>, <a class="ref" href="graph_3.html#T63" target="_self" title="GRAPH_3:th.63">Th63</a></i>;<br/><a NAME="E7:85_1"/><b>then </b>
 <a href="graph_3.html#K7" title="GRAPH_3:func.7">Degree</a> <font color="Maroon" title="c5">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="c3">c</font></span>)</span> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a>
 <b>by </b><i><a class="txt" href="graph_3.html#E4:85_1"><i><font color="Green" title="E74">A2</font></i></a>, <a class="txt" href="graph_3.html#E64"><i><font color="Green" title="E53">Lm4</font></i></a></i>;<br/><b>hence </b><a NAME="E8:85_1"/>
 <a href="graph_1.html#K6" title="GRAPH_1:func.6">Degree</a> <font color="Maroon" title="c5">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#E5:85_1"><i><font color="Green" title="E75">A3</font></i></a>, <a class="ref" href="graph_3.html#T29" target="_self" title="GRAPH_3:th.29">Th29</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:85"/><i><font color="Green" title="E73">A4</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/>

<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" title="c1">G</font> is <a href="msscyc_1.html#V2" title="MSSCYC_1:attr.2">empty</a> or  not <font color="Maroon" title="c1">G</font> is <a href="msscyc_1.html#V2" title="MSSCYC_1:attr.2">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" title="c1">G</font> is <a href="msscyc_1.html#V2" title="MSSCYC_1:attr.2">empty</a>
 ;<br/><div class="add"><a NAME="E2:85_2_1"/><b>then </b><i><font color="Green" title="E74">A5</font></i>: 
the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c1">G</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>by </b><i><a class="ref" href="msscyc_1.html#D3" title="MSSCYC_1:def.3">MSSCYC_1:def 3</a></i>;<br/><a NAME="E3:85_2_1"/>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  is    <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> 
 <b>by </b><i><a class="ref" href="graph_3.html#T56" target="_self" title="GRAPH_3:th.56">Th56</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c3">ec</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  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/><b>take </b>
<font color="Maroon" title="c3">ec</font>
;<br/><b>thus </b><a NAME="E5:85_2_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">ec</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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#E2:85_2_1"><i><font color="Green" title="E74">A5</font></i></a>, <a class="ref" href="finseq_1.html#T27" title="FINSEQ_1:th.27">FINSEQ_1:27</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="graph_3.html#D14" target="_self" title="GRAPH_3:def.14">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" title="c1">G</font> is <a href="msscyc_1.html#V2" title="MSSCYC_1:attr.2">empty</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c3">G'</font> = <font color="Maroon" title="c1">G</font> as  <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> non <a href="msscyc_1.html#V2" title="MSSCYC_1:attr.2">empty</a> <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Graph</a> ;<br/><b>reconsider </b><font color="Maroon" title="c4">V</font> =  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c1">G</font> as   <a href="graph_3.html#NM2" title="GRAPH_3:NM.2">Vertex</a> of <font color="Maroon" title="c3">G'</font> ;<br/><a NAME="E4:85_2_2"/>
 <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</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="graph_3.html#K13" title="GRAPH_3:func.13">-CycleSet</a> <font color="Maroon" title="c4">V</font></span>)</span> <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> <a href="graph_3.html#K13" title="GRAPH_3:func.13">-CycleSet</a> <font color="Maroon" title="c4">V</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c5">ec</font> =  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</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="graph_3.html#K13" title="GRAPH_3:func.13">-CycleSet</a> <font color="Maroon" title="c4">V</font></span>)</span> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c3">G'</font> <a href="graph_3.html#K9" title="GRAPH_3:func.9">-CycleSet</a>  ;<br/><div><i><font color="Green" title="E74">A6</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" title="c6">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="E1:85_2_2_1"/>
 <a href="graph_1.html#K6" title="GRAPH_1:func.6">Degree</a> <font color="Maroon" title="c6">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="c6">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/><b>hence </b><a NAME="E2:85_2_2_1"/>
 <a href="graph_3.html#K7" title="GRAPH_3:func.7">Degree</a> <font color="Maroon" title="c6">v</font>,the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c1">G</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a>
 <b>by </b><i><a class="txt" href="graph_3.html#E2:85"><i><font color="Green" title="E73">A4</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E7:85_2_2"/>
 <a href="graph_1.html#K6" title="GRAPH_1:func.6">Degree</a> <font color="Maroon" title="c4">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="c4">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="E8:85_2_2"/><b>then </b><i><font color="Green" title="E75">A7</font></i>: 
 <a href="graph_3.html#K7" title="GRAPH_3:func.7">Degree</a> <font color="Maroon" title="c4">V</font>,the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c1">G</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="graph_3.html#T38" target="_self" title="GRAPH_3:th.38">Th38</a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c3">G'</font> <a href="graph_3.html#K9" title="GRAPH_3:func.9">-CycleSet</a> ) <b>-&gt; </b>   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c3">G'</font> <a href="graph_3.html#K9" title="GRAPH_3:func.9">-CycleSet</a>  =  <a href="graph_3.html#K14" title="GRAPH_3:func.14">ExtendCycle</a> $2;<br/><b>consider </b><font color="Maroon" title="c6">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<font color="Maroon" title="c3">G'</font> <a href="graph_3.html#K9" title="GRAPH_3:func.9">-CycleSet</a> <b> such that </b><br/><a NAME="E10:85_2_2"/><i><font color="Green" title="E76">A8</font></i>: 
( <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">ec</font> &amp; ( for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">n</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font>) ) )
 <b>from </b><i><a class="ref" href="recdef_1.html#S4" title="RECDEF_1:sch.4">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" title="E77">A9</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b2">c</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c3">G'</font> <a href="graph_3.html#K9" title="GRAPH_3:func.9">-CycleSet</a>   holds <br/>(  not <font color="Olive" title="b2">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font> or  not  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b2">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c1">G</font> )
 ;<br/><b>reconsider </b><font color="Maroon" title="c7">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/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b> ex <font color="Olive" title="b1">c</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c3">G'</font> <a href="graph_3.html#K9" title="GRAPH_3:func.9">-CycleSet</a>  st <br/>( not <font color="Olive" title="b1">c</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1 &amp; $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">c</font></span>)</span> );<br/><a NAME="E3:85_2_2_2"/><i><font color="Green" title="E78">A10</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 <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" title="c5">ec</font>
;<br/>

<b>thus </b><a NAME="E1:85_2_2_2_1"/>
not <font color="Maroon" title="c5">ec</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#E6:85_2_2"><i><font color="Green" title="E74">A6</font></i></a>, <a class="txt" href="graph_3.html#E8:85_2_2"><i><font color="Green" title="E75">A7</font></i></a>, <a class="ref" href="graph_3.html#T60" target="_self" title="GRAPH_3:th.60">Th60</a></i>;<br/>

<b>thus </b><a NAME="E2:85_2_2_2_1"/>
<font color="Maroon" title="c5">ec</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="graph_3.html#E10:85_2_2"><i><font color="Green" title="E76">A8</font></i></a></i>;<br/>

<b>thus </b><a NAME="E3:85_2_2_2_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">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="c5">ec</font></span>)</span>
 ;<br/>


</div><b>end;</b></div><a NAME="E4:85_2_2_2"/><i><font color="Green" title="E79">A11</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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" title="c8">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>given </b><font color="Maroon" title="c9">c</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c3">G'</font> <a href="graph_3.html#K9" title="GRAPH_3:func.9">-CycleSet</a> <b> such that </b><a NAME="E2:85_2_2_2_2"/><i><font color="Green" title="E80">A12</font></i>: 
not <font color="Maroon" title="c9">c</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>and </b><br/><a NAME="E3:85_2_2_2_2"/><i><font color="Green" title="E81">A13</font></i>: 
<font color="Maroon" title="c9">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font>
 <b>and </b><br/><a NAME="E4:85_2_2_2_2"/><i><font color="Green" title="E82">A14</font></i>: 
<font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">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="c9">c</font></span>)</span>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c10">r</font> =  <a href="graph_3.html#K14" title="GRAPH_3:func.14">ExtendCycle</a> <font color="Maroon" title="c9">c</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c3">G'</font> <a href="graph_3.html#K9" title="GRAPH_3:func.9">-CycleSet</a>  ;<br/>
<b>take </b>
<font color="Maroon" title="c10">r</font>
;<br/>

<a NAME="E6:85_2_2_2_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">E</font>
 
<b>by </b><i><a class="txt" href="graph_3.html#E1:85_2_2_2"><i><font color="Green" title="E77">A9</font></i></a>, <a class="txt" href="graph_3.html#E3:85_2_2_2_2"><i><font color="Green" title="E81">A13</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:85_2_2_2_2"/>
not <font color="Maroon" title="c10">r</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#E2:85"><i><font color="Green" title="E73">A4</font></i></a>, <a class="txt" href="graph_3.html#E2:85_2_2_2_2"><i><font color="Green" title="E80">A12</font></i></a>, <a class="ref" href="graph_3.html#T62" target="_self" title="GRAPH_3:th.62">Th62</a></i>;<br/>

<b>thus </b><a NAME="E8:85_2_2_2_2"/>
<font color="Maroon" title="c10">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="graph_3.html#E10:85_2_2"><i><font color="Green" title="E76">A8</font></i></a>, <a class="txt" href="graph_3.html#E3:85_2_2_2_2"><i><font color="Green" title="E81">A13</font></i></a></i>;<br/>

<a NAME="E9:85_2_2_2_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">E</font>
 
<b>by </b><i><a class="txt" href="graph_3.html#E1:85_2_2_2"><i><font color="Green" title="E77">A9</font></i></a>, <a class="txt" href="graph_3.html#E3:85_2_2_2_2"><i><font color="Green" title="E81">A13</font></i></a></i>;<br/>
<a NAME="E10:85_2_2_2_2"/><b>then </b>
 <a href="card_1.html#K5" title="CARD_1:func.5">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="c9">c</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="card_1.html#K5" title="CARD_1:func.5">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="c10">r</font></span>)</span>
 
<b>by </b><i><a class="txt" href="graph_3.html#E2:85"><i><font color="Green" title="E73">A4</font></i></a>, <a class="txt" href="graph_3.html#E2:85_2_2_2_2"><i><font color="Green" title="E80">A12</font></i></a>, <a class="ref" href="graph_3.html#T62" target="_self" title="GRAPH_3:th.62">Th62</a></i>;<br/>
<a NAME="E11:85_2_2_2_2"/><b>then </b>
<font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="card_1.html#K5" title="CARD_1:func.5">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="c10">r</font></span>)</span>
 
<b>by </b><i><a class="txt" href="graph_3.html#E4:85_2_2_2_2"><i><font color="Green" title="E82">A14</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E12:85_2_2_2_2"/>
<font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">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="c10">r</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>


</div><b>end;</b></div><a NAME="E5:85_2_2_2"/>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="graph_3.html#E3:85_2_2_2"><i><font color="Green" title="E78">A10</font></i></a>, <a class="txt" href="graph_3.html#E4:85_2_2_2"><i><font color="Green" title="E79">A11</font></i></a>);<br/></i><b>then 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> <b> such that </b><br/><a NAME="E7:85_2_2_2"/><i><font color="Green" title="E80">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; <font color="Maroon" title="c8">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c7">E</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c7">E</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">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/><a NAME="E8:85_2_2_2"/>
 <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> <font color="Maroon" title="c7">E</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/><a NAME="E9:85_2_2_2"/><b>then </b>
 <a href="card_1.html#K5" title="CARD_1:func.5">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="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c7">E</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T44" title="NAT_1:th.44">NAT_1:44</a></i>;<br/><a NAME="E10:85_2_2_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c7">E</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c7">E</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="graph_3.html#E7:85_2_2_2"><i><font color="Green" title="E80">A15</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">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" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/></div><b>end;</b></div><b>then consider </b><font color="Maroon" title="c7">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <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> <b> such that </b><br/><a NAME="E13:85_2_2"/><i><font color="Green" title="E77">A16</font></i>: 
( <font color="Maroon" title="c8">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">n</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c1">G</font> )
 ;<br/><b>reconsider </b><font color="Maroon" title="c9">c</font> = <font color="Maroon" title="c8">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/><b>take </b>
<font color="Maroon" title="c9">c</font>
;<br/><b>thus </b><a NAME="E15:85_2_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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#E13:85_2_2"><i><font color="Green" title="E77">A16</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="graph_3.html#D14" target="_self" title="GRAPH_3:def.14">GRAPH_3:def 14</a><br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
