<?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/>

<div><b>hereby </b>
<div class="add"><b>given </b><font color="Maroon" title="c2">p</font> being   <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="E1:86_1"/><i><font color="Green" title="E73">A1</font></i>: 
( not <font color="Maroon" title="c2">p</font> is <a href="msscyc_1.html#V1" title="MSSCYC_1:attr.1">cyclic</a> &amp; <font color="Maroon" title="c2">p</font> is <a href="graph_3.html#V1" title="GRAPH_3:attr.1">Eulerian</a> )
 ;<br/><b>consider </b><font color="Maroon" title="c3">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="E3:86_1"/><i><font color="Green" title="E74">A2</font></i>: 
<font color="Maroon" title="c3">vs</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Maroon" title="c2">p</font>
 <b>by </b><i><a class="ref" href="graph_2.html#T36" title="GRAPH_2:th.36">GRAPH_2:36</a></i>;<br/><a NAME="E4:86_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">vs</font> <a href="hidden.html#R1" title="HIDDEN:pred.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="c2">p</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="graph_3.html#E3:86_1"><i><font color="Green" title="E74">A2</font></i></a>, <a class="ref" href="graph_2.html#D7" title="GRAPH_2:def.7">GRAPH_2:def 7</a></i>;<br/><a NAME="E5:86_1"/><b>then </b>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">vs</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E6:86_1"/><b>then </b>
( 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">vs</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">vs</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">vs</font> )
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c4">v1</font> = <font color="Maroon" title="c3">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1, <font color="Maroon" title="c5">v2</font> = <font color="Maroon" title="c3">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="c3">vs</font></span>)</span> as   <a href="graph_3.html#NM2" title="GRAPH_3:NM.2">Vertex</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="ref" href="finseq_2.html#T13" title="FINSEQ_2:th.13">FINSEQ_2:13</a></i>;<br/><b>take </b><font color="Maroon" title="c6">v1</font> = <font color="Maroon" title="c4">v1</font>;<br/><b>take </b><font color="Maroon" title="c7">v2</font> = <font color="Maroon" title="c5">v2</font>;<br/><b>thus </b><a NAME="E8:86_1"/>
<font color="Maroon" title="c6">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">v2</font>
 <b>by </b><i><a class="txt" href="graph_3.html#E1:86_1"><i><font color="Green" title="E73">A1</font></i></a>, <a class="txt" href="graph_3.html#E3:86_1"><i><font color="Green" title="E74">A2</font></i></a>, <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="c8">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 href="graph_1.html#K6" title="GRAPH_1:func.6">Degree</a> <font color="Maroon" title="c8">v</font> = 
 <a href="graph_3.html#K7" title="GRAPH_3:func.7">Degree</a> <font color="Maroon" title="c8">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 href="graph_3.html#K7" title="GRAPH_3:func.7">Degree</a> <font color="Maroon" title="c8">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">p</font></span>)</span>
<b>by </b><i><a class="txt" href="graph_3.html#E1:86_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/><b>hence </b><a NAME="E10:86_1"/>
(  <a href="graph_1.html#K6" title="GRAPH_1:func.6">Degree</a> <font color="Maroon" title="c8">v</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> iff ( <font color="Maroon" title="c8">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">v1</font> &amp; <font color="Maroon" title="c8">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">v2</font> ) )
 <b>by </b><i><a class="txt" href="graph_3.html#E1:86_1"><i><font color="Green" title="E73">A1</font></i></a>, <a class="txt" href="graph_3.html#E3:86_1"><i><font color="Green" title="E74">A2</font></i></a>, <a class="ref" href="graph_3.html#T55" target="_self" title="GRAPH_3:th.55">Th55</a></i>;<br/></div>
<b>end;</b></div>

<b>given </b><font color="Maroon" title="c2">v1</font>, <font color="Maroon" title="c3">v2</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><a NAME="E2:86"/><i><font color="Green" title="E73">A3</font></i>: 
<font color="Maroon" title="c2">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">v2</font>
 <b>and </b><br/><a NAME="E3:86"/><i><font color="Green" title="E74">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 <br/> (  <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> iff ( <font color="Olive" title="b1">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">v1</font> &amp; <font color="Olive" title="b1">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">v2</font> ) )
 ;<br/>

<b>set </b><font color="Maroon" title="c4">V</font> = the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c1">G</font>;<br/>
<b>set </b><font color="Maroon" title="c5">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/>
<b>set </b><font color="Maroon" title="c6">G'</font> =  <a href="graph_3.html#K8" title="GRAPH_3:func.8">AddNewEdge</a> <font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font>;<br/>
<a NAME="E4:86"/><i><font color="Green" title="E75">A5</font></i>: 
the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <span class="p1">(<span class="default"><a href="graph_3.html#K8" title="GRAPH_3:func.8">AddNewEdge</a> <font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font></span>)</span> <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> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><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></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="graph_3.html#D7" target="_self" title="GRAPH_3:def.7">Def7</a></i>;<br/>
<b>set </b><font color="Maroon" title="c7">E'</font> = the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <span class="p1">(<span class="default"><a href="graph_3.html#K8" title="GRAPH_3:func.8">AddNewEdge</a> <font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font></span>)</span>;<br/>
<a NAME="E5:86"/>
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#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><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></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E6:86"/><b>then </b><i><font color="Green" title="E76">A6</font></i>: 
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#R2" title="HIDDEN:pred.2">in</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <span class="p1">(<span class="default"><a href="graph_3.html#K8" title="GRAPH_3:func.8">AddNewEdge</a> <font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="graph_3.html#E4:86"><i><font color="Green" title="E75">A5</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E7:86"/><i><font color="Green" title="E77">A7</font></i>: 
( the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <span class="p1">(<span class="default"><a href="graph_3.html#K8" title="GRAPH_3:func.8">AddNewEdge</a> <font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">v1</font> &amp; the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <span class="p1">(<span class="default"><a href="graph_3.html#K8" title="GRAPH_3:func.8">AddNewEdge</a> <font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v2</font> )
 
<b>by </b><i><a class="ref" href="graph_3.html#T39" target="_self" title="GRAPH_3:th.39">Th39</a></i>;<br/>
<a NAME="E8:86"/><i><font color="Green" title="E78">A8</font></i>: 
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#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <span class="p1">(<span class="default"><a href="graph_3.html#K8" title="GRAPH_3:func.8">AddNewEdge</a> <font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</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#E4:86"><i><font color="Green" title="E75">A5</font></i></a>, <a class="ref" href="xboole_1.html#T21" title="XBOOLE_1:th.21">XBOOLE_1:21</a></i>;<br/>
<a NAME="E9:86"/>
for <font color="Olive" title="b1">v</font> being  <a href="graph_3.html#NM2" title="GRAPH_3:NM.2">Vertex</a> of <span class="p1">(<span class="default"><a href="graph_3.html#K8" title="GRAPH_3:func.8">AddNewEdge</a> <font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font></span>)</span> 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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="86_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">v'</font> be   <a href="graph_3.html#NM2" title="GRAPH_3:NM.2">Vertex</a> of <span class="p1">(<span class="default"><a href="graph_3.html#K8" title="GRAPH_3:func.8">AddNewEdge</a> <font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon" title="c9">v</font> = <font color="Maroon" title="c8">v'</font> as   <a href="graph_3.html#NM2" title="GRAPH_3:NM.2">Vertex</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="ref" href="graph_3.html#D7" target="_self" title="GRAPH_3:def.7">Def7</a></i>;<br/>
<a NAME="E2:86_2"/><i><font color="Green" title="E79">A9</font></i>: 
 <a href="graph_1.html#K6" title="GRAPH_1:func.6">Degree</a> <font color="Maroon" title="c8">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="c8">v'</font>,the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <span class="p1">(<span class="default"><a href="graph_3.html#K8" title="GRAPH_3:func.8">AddNewEdge</a> <font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font></span>)</span>
 
<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="E3:86_2"/><i><font color="Green" title="E80">A10</font></i>: 
 <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 <span class="p1">(<span class="default"><a href="graph_3.html#K8" title="GRAPH_3:func.8">AddNewEdge</a> <font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font></span>)</span> <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#E8:86"><i><font color="Green" title="E78">A8</font></i></a>, <a class="ref" href="graph_3.html#T36" target="_self" title="GRAPH_3:th.36">Th36</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="86_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:86_2_1"/>
( ( <font color="Maroon" title="c8">v'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">v1</font> &amp; <font color="Maroon" title="c8">v'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">v2</font> ) or <font color="Maroon" title="c9">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">v1</font> or <font color="Maroon" title="c9">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="86_2_1_1"><b>suppose </b></a><a NAME="E1:86_2_1_1"/><i><font color="Green" title="E81">A11</font></i>: 
( <font color="Maroon" title="c8">v'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">v1</font> &amp; <font color="Maroon" title="c8">v'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">v2</font> )
 ;<br/><div class="add"><b>then </b> <a href="graph_1.html#K6" title="GRAPH_1:func.6">Degree</a> <font color="Maroon" title="c8">v'</font> = 
 <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 <span class="p1">(<span class="default"><a href="graph_3.html#K8" title="GRAPH_3:func.8">AddNewEdge</a> <font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font></span>)</span>
<b>by </b><i><a class="txt" href="graph_3.html#E2:86_2"><i><font color="Green" title="E79">A9</font></i></a>, <a class="ref" href="graph_3.html#T53" target="_self" title="GRAPH_3:th.53">Th53</a></i>
<br/>.= 
 <a href="graph_1.html#K6" title="GRAPH_1:func.6">Degree</a> <font color="Maroon" title="c9">v</font>
<b>by </b><i><a class="txt" href="graph_3.html#E3:86_2"><i><font color="Green" title="E80">A10</font></i></a>, <a class="ref" href="graph_3.html#T29" target="_self" title="GRAPH_3:th.29">Th29</a></i>
;<br/><b>hence </b><a NAME="E3:86_2_1_1"/>
 <a href="graph_1.html#K6" title="GRAPH_1:func.6">Degree</a> <font color="Maroon" title="c8">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#E3:86"><i><font color="Green" title="E74">A4</font></i></a>, <a class="txt" href="graph_3.html#E1:86_2_1_1"><i><font color="Green" title="E81">A11</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="86_2_1_2"><b>suppose </b></a><a NAME="E1:86_2_1_2"/><i><font color="Green" title="E81">A12</font></i>: 
( <font color="Maroon" title="c9">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">v1</font> or <font color="Maroon" title="c9">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v2</font> )
 ;<br/><div class="add"><b>then </b><i><font color="Green" title="E82">A13</font></i>:  <a href="graph_1.html#K6" title="GRAPH_1:func.6">Degree</a> <font color="Maroon" title="c8">v'</font> = 
<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>,the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <span class="p2">(<span class="default"><a href="graph_3.html#K8" title="GRAPH_3:func.8">AddNewEdge</a> <font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
<b>by </b><i><a class="txt" href="graph_3.html#E2:86"><i><font color="Green" title="E73">A3</font></i></a>, <a class="txt" href="graph_3.html#E6:86"><i><font color="Green" title="E76">A6</font></i></a>, <a class="txt" href="graph_3.html#E2:86_2"><i><font color="Green" title="E79">A9</font></i></a>, <a class="ref" href="graph_3.html#T52" target="_self" title="GRAPH_3:th.52">Th52</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="graph_1.html#K6" title="GRAPH_1:func.6">Degree</a> <font color="Maroon" title="c9">v</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
<b>by </b><i><a class="txt" href="graph_3.html#E3:86_2"><i><font color="Green" title="E80">A10</font></i></a>, <a class="ref" href="graph_3.html#T29" target="_self" title="GRAPH_3:th.29">Th29</a></i>
;<br/><b>reconsider </b><font color="Maroon" title="c10">dv</font> =  <a href="graph_1.html#K6" title="GRAPH_1:func.6">Degree</a> <font color="Maroon" title="c9">v</font> as  <a href="abian.html#NV3" title="ABIAN:NV.3">odd</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="graph_3.html#E3:86"><i><font color="Green" title="E74">A4</font></i></a>, <a class="txt" href="graph_3.html#E1:86_2_1_2"><i><font color="Green" title="E81">A12</font></i></a></i>;<br/><a NAME="E4:86_2_1_2"/>
<font color="Maroon" title="c10">dv</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 is <a href="abian.html#V1" title="ABIAN:attr.1">even</a>
 ;<br/><b>hence </b><a NAME="E5:86_2_1_2"/>
 <a href="graph_1.html#K6" title="GRAPH_1:func.6">Degree</a> <font color="Maroon" title="c8">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#E2:86_2_1_2"><i><font color="Green" title="E82">A13</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c8">P'</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  <a href="graph_3.html#K8" title="GRAPH_3:func.8">AddNewEdge</a> <font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font><b> such that </b><br/><a NAME="E11:86"/><i><font color="Green" title="E79">A14</font></i>: 
<font color="Maroon" title="c8">P'</font> is <a href="graph_3.html#V1" title="GRAPH_3:attr.1">Eulerian</a>
 <b>by </b><i><a class="ref" href="graph_3.html#T64" target="_self" title="GRAPH_3:th.64">Th64</a></i>;<br/>
<a NAME="E12:86"/><i><font color="Green" title="E80">A15</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">P'</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 <span class="p1">(<span class="default"><a href="graph_3.html#K8" title="GRAPH_3:func.8">AddNewEdge</a> <font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="graph_3.html#E11:86"><i><font color="Green" title="E79">A14</font></i></a>, <a class="ref" href="graph_3.html#D14" target="_self" title="GRAPH_3:def.14">Def14</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E14:86"/><i><font color="Green" title="E81">A16</font></i>: 
( <font color="Maroon" title="c9">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">P'</font> &amp; <font color="Maroon" title="c8">P'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</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#E6:86"><i><font color="Green" title="E76">A6</font></i></a>, <a class="ref" href="finseq_2.html#T11" title="FINSEQ_2:th.11">FINSEQ_2:11</a></i>;<br/>
<a NAME="E15:86"/><i><font color="Green" title="E82">A17</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n</font> &amp; <font color="Maroon" title="c9">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">P'</font> )
 
<b>by </b><i><a class="txt" href="graph_3.html#E14:86"><i><font color="Green" title="E81">A16</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E16:86"/><b>then </b><i><font color="Green" title="E83">A18</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">P'</font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="86_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:86_3"/><i><font color="Green" title="E84">A19</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">P'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><b>consider </b><font color="Maroon" title="c10">c</font> being    <a href="graph_1.html#M1" title="GRAPH_1:mode.1">Chain</a> of <font color="Maroon" title="c1">G</font>, <font color="Maroon" title="c11">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="E3:86_3"/><i><font color="Green" title="E85">A20</font></i>: 
not <font color="Maroon" title="c10">c</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>and </b><br/><a NAME="E4:86_3"/>
( <font color="Maroon" title="c11">vs</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Maroon" title="c10">c</font> &amp; <font color="Maroon" title="c11">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="c2">v1</font> &amp; <font color="Maroon" title="c11">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="c11">vs</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v2</font> )
 <b>by </b><i><a class="txt" href="graph_3.html#E2:86"><i><font color="Green" title="E73">A3</font></i></a>, <a class="ref" href="graph_3.html#T23" target="_self" title="GRAPH_3:th.23">Th23</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c12">c</font> = <font color="Maroon" title="c10">c</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of 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="msscyc_1.html#D1" title="MSSCYC_1:def.1">MSSCYC_1:def 1</a></i>;<br/><a NAME="E6:86_3"/><i><font color="Green" title="E86">A21</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">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/><a NAME="E7:86_3"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">c</font>
 <b>by </b><i><a class="txt" href="graph_3.html#E3:86_3"><i><font color="Green" title="E85">A20</font></i></a>, <a class="ref" href="finseq_5.html#T6" title="FINSEQ_5:th.6">FINSEQ_5:6</a></i>;<br/><a NAME="E8:86_3"/><b>then </b><i><font color="Green" title="E87">A22</font></i>: 
<font color="Maroon" title="c12">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <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="c12">c</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E9:86_3"/><b>then </b><i><font color="Green" title="E88">A23</font></i>: 
<font color="Maroon" title="c12">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <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>
 <b>by </b><i><a class="txt" href="graph_3.html#E6:86_3"><i><font color="Green" title="E86">A21</font></i></a></i>;<br/><a NAME="E10:86_3"/>
<font color="Maroon" title="c12">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <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 <span class="p1">(<span class="default"><a href="graph_3.html#K8" title="GRAPH_3:func.8">AddNewEdge</a> <font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_3.html#E4:86"><i><font color="Green" title="E75">A5</font></i></a>, <a class="txt" href="graph_3.html#E6:86_3"><i><font color="Green" title="E86">A21</font></i></a>, <a class="txt" href="graph_3.html#E8:86_3"><i><font color="Green" title="E87">A22</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><b>then consider </b><font color="Maroon" title="c13">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E12:86_3"/><i><font color="Green" title="E89">A24</font></i>: 
( <font color="Maroon" title="c13">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">P'</font> &amp; <font color="Maroon" title="c8">P'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 )
 <b>by </b><i><a class="txt" href="graph_3.html#E12:86"><i><font color="Green" title="E80">A15</font></i></a>, <a class="ref" href="finseq_2.html#T11" title="FINSEQ_2:th.11">FINSEQ_2:11</a></i>;<br/><a NAME="E13:86_3"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">m</font> &amp; <font color="Maroon" title="c13">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">P'</font> )
 <b>by </b><i><a class="txt" href="graph_3.html#E12:86_3"><i><font color="Green" title="E89">A24</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E14:86_3"/><b>then </b>
( <font color="Maroon" title="c9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c13">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="txt" href="graph_3.html#E15:86"><i><font color="Green" title="E82">A17</font></i></a>, <a class="txt" href="graph_3.html#E1:86_3"><i><font color="Green" title="E84">A19</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><b>hence </b><a NAME="E15:86_3"/>
contradiction
 <b>by </b><i><a class="txt" href="graph_3.html#E14:86"><i><font color="Green" title="E81">A16</font></i></a>, <a class="txt" href="graph_3.html#E9:86_3"><i><font color="Green" title="E88">A23</font></i></a>, <a class="txt" href="graph_3.html#E12:86_3"><i><font color="Green" title="E89">A24</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E18:86"/><b>then </b>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">P'</font>
 
<b>by </b><i><a class="txt" href="graph_3.html#E16:86"><i><font color="Green" title="E83">A18</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/>
<a NAME="E19:86"/><b>then </b><i><font color="Green" title="E84">A25</font></i>: 
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">P'</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">p'</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  <a href="graph_3.html#K8" title="GRAPH_3:func.8">AddNewEdge</a> <font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font><b> such that </b><br/><a NAME="E21:86"/><i><font color="Green" title="E85">A26</font></i>: 
( <font color="Maroon" title="c10">p'</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="c8">P'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">P'</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">p'</font> <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">P'</font> )
 <b>by </b><i><a class="txt" href="graph_3.html#E14:86"><i><font color="Green" title="E81">A16</font></i></a>, <a class="ref" href="graph_3.html#T15" target="_self" title="GRAPH_3:th.15">Th15</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">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 <span class="p1">(<span class="default"><a href="graph_3.html#K8" title="GRAPH_3:func.8">AddNewEdge</a> <font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font></span>)</span><b> such that </b><br/><a NAME="E23:86"/><i><font color="Green" title="E86">A27</font></i>: 
<font color="Maroon" title="c11">vs'</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Maroon" title="c10">p'</font>
 <b>by </b><i><a class="ref" href="graph_2.html#T36" title="GRAPH_2:th.36">GRAPH_2:36</a></i>;<br/>
<a NAME="E24:86"/><i><font color="Green" title="E87">A28</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">vs'</font> <a href="hidden.html#R1" title="HIDDEN:pred.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="c10">p'</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="graph_3.html#E23:86"><i><font color="Green" title="E86">A27</font></i></a>, <a class="ref" href="graph_2.html#D7" title="GRAPH_2:def.7">GRAPH_2:def 7</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">p</font> = <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">p'</font></span>)</span> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c10">p'</font> as   <a href="graph_1.html#NM4" title="GRAPH_1:NM.4">Path</a> of  <a href="graph_3.html#K8" title="GRAPH_3:func.8">AddNewEdge</a> <font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font> <b>by </b><i><a class="ref" href="graph_3.html#T8" target="_self" title="GRAPH_3:th.8">Th8</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">vs</font> = <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">vs'</font></span>)</span> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c11">vs'</font> as    <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 <span class="p1">(<span class="default"><a href="graph_3.html#K8" title="GRAPH_3:func.8">AddNewEdge</a> <font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font></span>)</span> ;<br/>
<a NAME="E27:86"/><i><font color="Green" title="E88">A29</font></i>: 
<font color="Maroon" title="c13">vs</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Maroon" title="c12">p</font>
 
<b>by </b><i><a class="txt" href="graph_3.html#E19:86"><i><font color="Green" title="E84">A25</font></i></a>, <a class="txt" href="graph_3.html#E21:86"><i><font color="Green" title="E85">A26</font></i></a>, <a class="txt" href="graph_3.html#E23:86"><i><font color="Green" title="E86">A27</font></i></a>, <a class="txt" href="graph_3.html#E24:86"><i><font color="Green" title="E87">A28</font></i></a>, <a class="ref" href="graph_2.html#T45" title="GRAPH_2:th.45">GRAPH_2:45</a></i>;<br/>
<div><i><font color="Green" title="E89">A30</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="86_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:86_4"/>
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#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">p</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c14">a</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:86_4"/><i><font color="Green" title="E90">A31</font></i>: 
( <font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">p</font> &amp; <font color="Maroon" title="c12">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">a</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="ref" href="finseq_2.html#T11" title="FINSEQ_2:th.11">FINSEQ_2:11</a></i>;<br/><b>consider </b><font color="Maroon" title="c15">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E5:86_4"/><i><font color="Green" title="E91">A32</font></i>: 
( <font color="Maroon" title="c15">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c10">p'</font> &amp; <font color="Maroon" title="c10">p'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">a</font> &amp; <font color="Maroon" title="c15">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c14">a</font> &amp; 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">k</font> &amp; <font color="Maroon" title="c15">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">p'</font> )
 <b>by </b><i><a class="txt" href="graph_3.html#E3:86_4"><i><font color="Green" title="E90">A31</font></i></a>, <a class="ref" href="graph_3.html#T2" target="_self" title="GRAPH_3:th.2">Th2</a></i>;<br/><a NAME="E6:86_4"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c10">p'</font>
 <b>by </b><i><a class="txt" href="graph_3.html#E5:86_4"><i><font color="Green" title="E91">A32</font></i></a>, <a class="ref" href="finseq_5.html#T6" title="FINSEQ_5:th.6">FINSEQ_5:6</a>, <a class="ref" href="relat_1.html#T60" title="RELAT_1:th.60">RELAT_1:60</a></i>;<br/><a NAME="E7:86_4"/><b>then </b>
<font color="Maroon" title="c15">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="graph_3.html#E14:86"><i><font color="Green" title="E81">A16</font></i></a>, <a class="txt" href="graph_3.html#E21:86"><i><font color="Green" title="E85">A26</font></i></a>, <a class="txt" href="graph_3.html#E3:86_4"><i><font color="Green" title="E90">A31</font></i></a>, <a class="txt" href="graph_3.html#E5:86_4"><i><font color="Green" title="E91">A32</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/><b>hence </b><a NAME="E8:86_4"/>
contradiction
 <b>by </b><i><a class="txt" href="graph_3.html#E5:86_4"><i><font color="Green" title="E91">A32</font></i></a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c14">p</font> = <font color="Maroon" title="c12">p</font> as   <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#T50" target="_self" title="GRAPH_3:th.50">Th50</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">vs</font> = <font color="Maroon" title="c13">vs</font> as    <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>by </b><i><a class="ref" href="graph_3.html#D7" target="_self" title="GRAPH_3:def.7">Def7</a></i>;<br/>
<a NAME="E31:86"/><i><font color="Green" title="E90">A33</font></i>: 
<font color="Maroon" title="c15">vs</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Maroon" title="c14">p</font>
 
<b>by </b><i><a class="txt" href="graph_3.html#E27:86"><i><font color="Green" title="E88">A29</font></i></a>, <a class="txt" href="graph_3.html#E28:86"><i><font color="Green" title="E89">A30</font></i></a>, <a class="ref" href="graph_3.html#T51" target="_self" title="GRAPH_3:th.51">Th51</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c14">p</font>
;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="86_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:86_5"/><i><font color="Green" title="E91">A34</font></i>: 
<font color="Maroon" title="c15">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="c15">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="c15">vs</font></span>)</span>
 ;<br/><a NAME="E2:86_5"/>
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">p'</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="graph_3.html#E19:86"><i><font color="Green" title="E84">A25</font></i></a>, <a class="txt" href="graph_3.html#E21:86"><i><font color="Green" title="E85">A26</font></i></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/><a NAME="E3:86_5"/><b>then </b><i><font color="Green" title="E92">A35</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">vs'</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="txt" href="graph_3.html#E24:86"><i><font color="Green" title="E87">A28</font></i></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/><a NAME="E4:86_5"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">vs</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.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="c10">p'</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="graph_3.html#E24:86"><i><font color="Green" title="E87">A28</font></i></a>, <a class="txt" href="graph_3.html#E2"><i><font color="Green" title="E2">Lm1</font></i></a></i>;<br/><a NAME="E5:86_5"/><b>then </b><i><font color="Green" title="E93">A36</font></i>: 
( 0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">vs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">p'</font> )
 ;<br/><a NAME="E6:86_5"/><b>then </b>
0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">vs</font>
 <b>by </b><i><a class="txt" href="graph_3.html#E15:86"><i><font color="Green" title="E82">A17</font></i></a>, <a class="txt" href="graph_3.html#E21:86"><i><font color="Green" title="E85">A26</font></i></a></i>;<br/><a NAME="E7:86_5"/><b>then </b><i><font color="Green" title="E94">A37</font></i>: 
<font color="Maroon" title="c15">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">vs'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 0</span>)</span>
 <b>by </b><i><a class="txt" href="graph_3.html#E3:86_5"><i><font color="Green" title="E92">A35</font></i></a>, <a class="txt" href="graph_3.html#E2"><i><font color="Green" title="E2">Lm1</font></i></a></i>;<br/><a NAME="E8:86_5"/>
0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">vs</font>
 <b>by </b><i><a class="txt" href="graph_3.html#E15:86"><i><font color="Green" title="E82">A17</font></i></a>, <a class="txt" href="graph_3.html#E21:86"><i><font color="Green" title="E85">A26</font></i></a>, <a class="txt" href="graph_3.html#E5:86_5"><i><font color="Green" title="E93">A36</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>then consider </b><font color="Maroon" title="c16">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E10:86_5"/><i><font color="Green" title="E95">A38</font></i>: 
( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">i</font> &amp; <font color="Maroon" title="c16">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">vs</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">vs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="ref" href="graph_2.html#T1" title="GRAPH_2:th.1">GRAPH_2:1</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">vs</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">vs</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">vs'</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
<b>by </b><i><a class="txt" href="graph_3.html#E3:86_5"><i><font color="Green" title="E92">A35</font></i></a>, <a class="txt" href="graph_3.html#E2"><i><font color="Green" title="E2">Lm1</font></i></a></i>
;<br/><a NAME="E12:86_5"/><b>then </b><i><font color="Green" title="E96">A39</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">vs'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="graph_3.html#E10:86_5"><i><font color="Green" title="E95">A38</font></i></a></i>;<br/><a NAME="E13:86_5"/><i><font color="Green" title="E97">A40</font></i>: 
<font color="Maroon" title="c11">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="c11">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="c11">vs'</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_3.html#E23:86"><i><font color="Green" title="E86">A27</font></i></a>, <a class="ref" href="msscyc_1.html#T6" title="MSSCYC_1:th.6">MSSCYC_1:6</a></i>;<br/><a NAME="E14:86_5"/>
( ( <font color="Maroon" title="c11">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="c2">v1</font> &amp; <font color="Maroon" title="c11">vs'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v2</font> ) or ( <font color="Maroon" title="c11">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="c3">v2</font> &amp; <font color="Maroon" title="c11">vs'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">v1</font> ) )
 <b>by </b><i><a class="txt" href="graph_3.html#E7:86"><i><font color="Green" title="E77">A7</font></i></a>, <a class="txt" href="graph_3.html#E14:86"><i><font color="Green" title="E81">A16</font></i></a>, <a class="txt" href="graph_3.html#E16:86"><i><font color="Green" title="E83">A18</font></i></a>, <a class="txt" href="graph_3.html#E21:86"><i><font color="Green" title="E85">A26</font></i></a>, <a class="txt" href="graph_3.html#E23:86"><i><font color="Green" title="E86">A27</font></i></a>, <a class="txt" href="graph_3.html#E21"><i><font color="Green" title="E16">Lm3</font></i></a></i>;<br/><b>hence </b><a NAME="E15:86_5"/>
contradiction
 <b>by </b><i><a class="txt" href="graph_3.html#E2:86"><i><font color="Green" title="E73">A3</font></i></a>, <a class="txt" href="graph_3.html#E1:86_5"><i><font color="Green" title="E91">A34</font></i></a>, <a class="txt" href="graph_3.html#E3:86_5"><i><font color="Green" title="E92">A35</font></i></a>, <a class="txt" href="graph_3.html#E7:86_5"><i><font color="Green" title="E94">A37</font></i></a>, <a class="txt" href="graph_3.html#E10:86_5"><i><font color="Green" title="E95">A38</font></i></a>, <a class="txt" href="graph_3.html#E12:86_5"><i><font color="Green" title="E96">A39</font></i></a>, <a class="txt" href="graph_3.html#E13:86_5"><i><font color="Green" title="E97">A40</font></i></a>, <a class="txt" href="graph_3.html#E2"><i><font color="Green" title="E2">Lm1</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E33:86"/>
not <font color="Maroon" title="c14">p</font> is <a href="msscyc_1.html#V1" title="MSSCYC_1:attr.1">cyclic</a>
 <b>by </b><i><a class="txt" href="graph_3.html#E31:86"><i><font color="Green" title="E90">A33</font></i></a>, <a class="ref" href="msscyc_1.html#T6" title="MSSCYC_1:th.6">MSSCYC_1:6</a></i>;<br/>

<a NAME="E34:86"/><i><font color="Green" title="E91">A41</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><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></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><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></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T56" title="FINSEQ_1:th.56">FINSEQ_1:56</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c16">p1</font> = 1,1 <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c10">p'</font> as    <a href="graph_1.html#M1" title="GRAPH_1:mode.1">Chain</a> of  <a href="graph_3.html#K8" title="GRAPH_3:func.8">AddNewEdge</a> <font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font> <b>by </b><i><a class="txt" href="graph_3.html#E16:86"><i><font color="Green" title="E83">A18</font></i></a>, <a class="txt" href="graph_3.html#E21:86"><i><font color="Green" title="E85">A26</font></i></a>, <a class="ref" href="graph_2.html#T44" title="GRAPH_2:th.44">GRAPH_2:44</a></i>;<br/>
<a NAME="E36:86"/><i><font color="Green" title="E92">A42</font></i>: 
<font color="Maroon" title="c16">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><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></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="graph_3.html#E14:86"><i><font color="Green" title="E81">A16</font></i></a>, <a class="txt" href="graph_3.html#E16:86"><i><font color="Green" title="E83">A18</font></i></a>, <a class="txt" href="graph_3.html#E21:86"><i><font color="Green" title="E85">A26</font></i></a>, <a class="ref" href="graph_2.html#T6" title="GRAPH_2:th.6">GRAPH_2:6</a></i>;<br/>
<a NAME="E37:86"/><i><font color="Green" title="E93">A43</font></i>: 
<font color="Maroon" title="c10">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">p1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">p'</font></span>)</span> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c10">p'</font></span>)</span>
 
<b>by </b><i><a class="txt" href="graph_3.html#E16:86"><i><font color="Green" title="E83">A18</font></i></a>, <a class="txt" href="graph_3.html#E21:86"><i><font color="Green" title="E85">A26</font></i></a>, <a class="ref" href="graph_2.html#T9" title="GRAPH_2:th.9">GRAPH_2:9</a></i>;<br/>
<a NAME="E38:86"/><b>then </b><i><font color="Green" title="E94">A44</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><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></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></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="c14">p</font></span>)</span>
 
<b>by </b><i><a class="txt" href="graph_3.html#E34:86"><i><font color="Green" title="E91">A41</font></i></a>, <a class="txt" href="graph_3.html#E36:86"><i><font color="Green" title="E92">A42</font></i></a>, <a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></i>;<br/>
<a NAME="E39:86"/><i><font color="Green" title="E95">A45</font></i>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><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></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <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="c14">p</font>
 
<b>by </b><i><a class="txt" href="graph_3.html#E34:86"><i><font color="Green" title="E91">A41</font></i></a>, <a class="txt" href="graph_3.html#E36:86"><i><font color="Green" title="E92">A42</font></i></a>, <a class="txt" href="graph_3.html#E37:86"><i><font color="Green" title="E93">A43</font></i></a>, <a class="ref" href="finseq_3.html#T98" title="FINSEQ_3:th.98">FINSEQ_3:98</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="86_6"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:86_6"/>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><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></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</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>then consider </b><font color="Maroon" title="c17">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:86_6"/><i><font color="Green" title="E96">A46</font></i>: 
( <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><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></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c17">x</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> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/><a NAME="E4:86_6"/>
<font color="Maroon" title="c17">x</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#E3:86_6"><i><font color="Green" title="E96">A46</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E5:86_6"/>
contradiction
 <b>by </b><i><a class="txt" href="graph_3.html#E3:86_6"><i><font color="Green" title="E96">A46</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E41:86"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c14">p</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#E4:86"><i><font color="Green" title="E75">A5</font></i></a>, <a class="txt" href="graph_3.html#E12:86"><i><font color="Green" title="E80">A15</font></i></a>, <a class="txt" href="graph_3.html#E21:86"><i><font color="Green" title="E85">A26</font></i></a>, <a class="txt" href="graph_3.html#E38:86"><i><font color="Green" title="E94">A44</font></i></a>, <a class="txt" href="graph_3.html#E39:86"><i><font color="Green" title="E95">A45</font></i></a>, <a class="ref" href="xboole_1.html#T72" title="XBOOLE_1:th.72">XBOOLE_1:72</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>
