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

<b>let </b><font color="Maroon" title="c1">G</font> be  <a href="graph_1.html#V7" title="GRAPH_1:attr.7">finite</a> <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Graph</a>;<br/><b>let </b><font color="Maroon" title="c2">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/><b>let </b><font color="Maroon" title="c3">vs</font> be    <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>;<br/><b>let </b><font color="Maroon" title="c4">c</font> be   <a href="graph_1.html#NM4" title="GRAPH_1:NM.4">Path</a> of <font color="Maroon" title="c1">G</font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:68"/><i><font color="Green" title="E56">A1</font></i>: 
not <font color="Maroon" title="c4">c</font> is <a href="msscyc_1.html#V1" title="MSSCYC_1:attr.1">cyclic</a>
 <b>and </b><br/><a NAME="E2:68"/><i><font color="Green" title="E57">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="c4">c</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/>
<a NAME="E3:68"/>
 <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="c4">c</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#E2:68"><i><font color="Green" title="E57">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="E4:68"/><b>then </b><i><font color="Green" title="E58">A3</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="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="E5:68"/><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="c6">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="c7">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/>
<a NAME="E7:68"/><i><font color="Green" title="E59">A4</font></i>: 
<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#E1:68"><i><font color="Green" title="E56">A1</font></i></a>, <a class="txt" href="graph_3.html#E2:68"><i><font color="Green" title="E57">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>set </b><font color="Maroon" title="c8">G'</font> =  <a href="graph_3.html#K8" title="GRAPH_3:func.8">AddNewEdge</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c7">v2</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">v'</font> = <font color="Maroon" title="c2">v</font>, <font color="Maroon" title="c10">v1'</font> = <font color="Maroon" title="c6">v1</font>, <font color="Maroon" title="c11">v2'</font> = <font color="Maroon" title="c7">v2</font> as   <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="c6">v1</font>,<font color="Maroon" title="c7">v2</font></span>)</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>reconsider </b><font color="Maroon" title="c12">c'</font> = <font color="Maroon" title="c4">c</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="c6">v1</font>,<font color="Maroon" title="c7">v2</font> <b>by </b><i><a class="ref" href="graph_3.html#T42" target="_self" title="GRAPH_3:th.42">Th42</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">vs'</font> = <font color="Maroon" title="c3">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="c6">v1</font>,<font color="Maroon" title="c7">v2</font></span>)</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="c14">e</font> = the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c1">G</font>;<br/>
<a NAME="E11:68"/><i><font color="Green" title="E60">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="c6">v1</font>,<font color="Maroon" title="c7">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/>
<a NAME="E12:68"/><i><font color="Green" title="E61">A6</font></i>: 
not 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 <font color="Maroon" title="c1">G</font>
 
;<br/>
<a NAME="E13:68"/><i><font color="Green" title="E62">A7</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> <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="E14:68"/><b>then </b>
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="c6">v1</font>,<font color="Maroon" title="c7">v2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="graph_3.html#E11:68"><i><font color="Green" title="E60">A5</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 reconsider </b><font color="Maroon" title="c15">ep</font> = <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> 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="c6">v1</font>,<font color="Maroon" title="c7">v2</font> <b>by </b><i><a class="ref" href="graph_3.html#T7" target="_self" title="GRAPH_3:th.7">Th7</a></i>;<br/>
<b>set </b><font color="Maroon" title="c16">v21'</font> = <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c11">v2'</font>,<font color="Maroon" title="c10">v1'</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>;<br/>
<b>set </b><font color="Maroon" title="c17">vs'e</font> = <font color="Maroon" title="c13">vs'</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c11">v2'</font>,<font color="Maroon" title="c10">v1'</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>;<br/>
<a NAME="E16:68"/><i><font color="Green" title="E63">A8</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">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="E17:68"/><b>then </b><i><font color="Green" title="E64">A9</font></i>: 
not 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="c4">c</font>
 
<b>by </b><i><a class="txt" href="graph_3.html#E12:68"><i><font color="Green" title="E61">A6</font></i></a></i>;<br/>
<a NAME="E18:68"/><i><font color="Green" title="E65">A10</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">ep</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>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T56" title="FINSEQ_1:th.56">FINSEQ_1:56</a></i>;<br/>
<a NAME="E19:68"/><i><font color="Green" title="E66">A11</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">c'</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">ep</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:68_1"/>
not  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">c'</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">ep</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c18">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:68_1"/><i><font color="Green" title="E67">A12</font></i>: 
( <font color="Maroon" title="c18">x</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">c'</font> &amp; <font color="Maroon" title="c18">x</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="c15">ep</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<b>thus </b><a NAME="E4:68_1"/>
contradiction
 <b>by </b><i><a class="txt" href="graph_3.html#E17:68"><i><font color="Green" title="E64">A9</font></i></a>, <a class="txt" href="graph_3.html#E18:68"><i><font color="Green" title="E65">A10</font></i></a>, <a class="txt" href="graph_3.html#E3:68_1"><i><font color="Green" title="E67">A12</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E20:68"/><i><font color="Green" title="E67">A13</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">c'</font>
 
<b>by </b><i><a class="txt" href="graph_3.html#E2:68"><i><font color="Green" title="E57">A2</font></i></a>, <a class="ref" href="graph_3.html#T41" target="_self" title="GRAPH_3:th.41">Th41</a></i>;<br/>
<a NAME="E21:68"/><i><font color="Green" title="E68">A14</font></i>: 
<font color="Maroon" title="c13">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="c13">vs'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c11">v2'</font>,<font color="Maroon" title="c10">v1'</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>;<br/>
<a NAME="E22:68"/>
( 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="c6">v1</font>,<font color="Maroon" title="c7">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="c10">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="c6">v1</font>,<font color="Maroon" title="c7">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="c11">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="E23:68"/><b>then </b><i><font color="Green" title="E69">A15</font></i>: 
<span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c11">v2'</font>,<font color="Maroon" title="c10">v1'</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Maroon" title="c15">ep</font>
 
<b>by </b><i><a class="ref" href="graph_3.html#T16" target="_self" title="GRAPH_3:th.16">Th16</a></i>;<br/>
<a NAME="E24:68"/><i><font color="Green" title="E70">A16</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">vs'</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <span class="p2"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c11">v2'</font>,<font color="Maroon" title="c10">v1'</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span></span>)</span> <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">vs</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#E4:68"><i><font color="Green" title="E58">A3</font></i></a>, <a class="ref" href="graph_2.html#T14" title="GRAPH_2:th.14">GRAPH_2:14</a></i>;<br/>
<a NAME="E25:68"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c11">v2'</font>,<font color="Maroon" title="c10">v1'</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 
<b>by </b><i><a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>;<br/>
<a NAME="E26:68"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c13">vs'</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <span class="p2"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c11">v2'</font>,<font color="Maroon" title="c10">v1'</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span></span>)</span> <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> <span class="p2">(<span class="default"><font color="Maroon" title="c13">vs'</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <span class="p3"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c11">v2'</font>,<font color="Maroon" title="c10">v1'</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c11">v2'</font>,<font color="Maroon" title="c10">v1'</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2
 
<b>by </b><i><a class="ref" href="graph_2.html#T16" title="GRAPH_2:th.16">GRAPH_2:16</a></i>;<br/>
<a NAME="E27:68"/><b>then </b><i><font color="Green" title="E71">A17</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">vs'</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <span class="p2"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c11">v2'</font>,<font color="Maroon" title="c10">v1'</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">vs'</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <span class="p2"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c11">v2'</font>,<font color="Maroon" title="c10">v1'</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span></span>)</span> <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> <span class="p2">(<span class="default"><font color="Maroon" title="c13">vs'</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <span class="p3"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c11">v2'</font>,<font color="Maroon" title="c10">v1'</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="graph_3.html#E24:68"><i><font color="Green" title="E70">A16</font></i></a>, <a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c18">c'e</font> = <font color="Maroon" title="c12">c'</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c15">ep</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="c6">v1</font>,<font color="Maroon" title="c7">v2</font> <b>by </b><i><a class="txt" href="graph_3.html#E19:68"><i><font color="Green" title="E66">A11</font></i></a>, <a class="txt" href="graph_3.html#E20:68"><i><font color="Green" title="E67">A13</font></i></a>, <a class="txt" href="graph_3.html#E21:68"><i><font color="Green" title="E68">A14</font></i></a>, <a class="txt" href="graph_3.html#E23:68"><i><font color="Green" title="E69">A15</font></i></a>, <a class="ref" href="graph_3.html#T9" target="_self" title="GRAPH_3:th.9">Th9</a></i>;<br/>
<a NAME="E29:68"/>
<font color="Maroon" title="c13">vs'</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c11">v2'</font>,<font color="Maroon" title="c10">v1'</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Maroon" title="c18">c'e</font>
 
<b>by </b><i><a class="txt" href="graph_3.html#E20:68"><i><font color="Green" title="E67">A13</font></i></a>, <a class="txt" href="graph_3.html#E21:68"><i><font color="Green" title="E68">A14</font></i></a>, <a class="txt" href="graph_3.html#E23:68"><i><font color="Green" title="E69">A15</font></i></a>, <a class="ref" href="graph_2.html#T47" title="GRAPH_2:th.47">GRAPH_2:47</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c19">c'e</font> = <font color="Maroon" title="c18">c'e</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  <a href="graph_3.html#K8" title="GRAPH_3:func.8">AddNewEdge</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c7">v2</font> <b>by </b><i><a class="txt" href="graph_3.html#E27:68"><i><font color="Green" title="E71">A17</font></i></a>, <a class="ref" href="msscyc_1.html#D2" title="MSSCYC_1:def.2">MSSCYC_1:def 2</a></i>;<br/>
<a NAME="E31:68"/><i><font color="Green" title="E72">A18</font></i>: 
 <a href="graph_3.html#K7" title="GRAPH_3:func.7">Degree</a> <font color="Maroon" title="c9">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="c19">c'e</font></span>)</span> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a>
 
<b>by </b><i><a class="ref" href="graph_3.html#T54" target="_self" title="GRAPH_3:th.54">Th54</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c20">dv'</font> =  <a href="graph_3.html#K7" title="GRAPH_3:func.7">Degree</a> <font color="Maroon" title="c9">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="c19">c'e</font></span>)</span> as  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  <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>by </b><i><a class="ref" href="graph_3.html#T54" target="_self" title="GRAPH_3:th.54">Th54</a></i>;<br/>
<a NAME="E33:68"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">ep</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c1">G</font>
 
<b>by </b><i><a class="txt" href="graph_3.html#E12:68"><i><font color="Green" title="E61">A6</font></i></a>, <a class="txt" href="graph_3.html#E18:68"><i><font color="Green" title="E65">A10</font></i></a>, <a class="ref" href="zfmisc_1.html#T56" title="ZFMISC_1:th.56">ZFMISC_1:56</a></i>;<br/>
<a NAME="E34:68"/><b>then </b><i><font color="Green" title="E73">A19</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">ep</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<i><font color="Green" title="E74">A20</font></i>: <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c19">c'e</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> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">c</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">ep</font></span>)</span></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="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">c</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></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
<b>by </b><i><a class="txt" href="graph_3.html#E34:68"><i><font color="Green" title="E73">A19</font></i></a>, <a class="ref" href="xboole_1.html#T23" title="XBOOLE_1:th.23">XBOOLE_1:23</a></i>
<br/>.= 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">c</font>
<b>by </b><i><a class="txt" href="graph_3.html#E16:68"><i><font color="Green" title="E63">A8</font></i></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>
;<br/>
<a NAME="E36:68"/><b>then </b><i><font color="Green" title="E75">A21</font></i>: 
 <a href="graph_3.html#K7" title="GRAPH_3:func.7">Degree</a> <font color="Maroon" title="c2">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="c19">c'e</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="c2">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="c4">c</font></span>)</span>
 
<b>by </b><i><a class="ref" href="graph_3.html#T36" target="_self" title="GRAPH_3:th.36">Th36</a></i>;<br/>
<a NAME="E37:68"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c19">c'e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">c</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">ep</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></i>;<br/>
<a NAME="E38:68"/><b>then </b><i><font color="Green" title="E76">A22</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>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c19">c'e</font>
 
<b>by </b><i><a class="txt" href="graph_3.html#E13:68"><i><font color="Green" title="E62">A7</font></i></a>, <a class="txt" href="graph_3.html#E18:68"><i><font color="Green" title="E65">A10</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:68_3"/>
( ( <font color="Maroon" title="c2">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="c2">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">v2</font> ) or <font color="Maroon" title="c2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v1</font> or <font color="Maroon" title="c2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_3_1"><b>suppose </b></a><a NAME="E1:68_3_1"/>
( <font color="Maroon" title="c2">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="c2">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">v2</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:68_3_1"/>
(  <a href="graph_3.html#K7" title="GRAPH_3:func.7">Degree</a> <font color="Maroon" title="c2">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="c4">c</font></span>)</span> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> iff ( <font color="Maroon" title="c2">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 &amp; <font color="Maroon" title="c2">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <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> ) )
 <b>by </b><i><a class="txt" href="graph_3.html#E31:68"><i><font color="Green" title="E72">A18</font></i></a>, <a class="txt" href="graph_3.html#E36:68"><i><font color="Green" title="E75">A21</font></i></a>, <a class="ref" href="graph_3.html#T53" target="_self" title="GRAPH_3:th.53">Th53</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_3_2"><b>suppose </b></a><a NAME="E1:68_3_2"/><i><font color="Green" title="E77">A23</font></i>: 
( <font color="Maroon" title="c2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v1</font> or <font color="Maroon" title="c2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v2</font> )
 ;<br/><div class="add"><a NAME="E2:68_3_2"/><b>then </b>
 <a href="graph_3.html#K7" title="GRAPH_3:func.7">Degree</a> <font color="Maroon" title="c9">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="c19">c'e</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="graph_3.html#K7" title="GRAPH_3:func.7">Degree</a> <font color="Maroon" title="c2">v</font>,<span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c19">c'e</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#E7:68"><i><font color="Green" title="E59">A4</font></i></a>, <a class="txt" href="graph_3.html#E38:68"><i><font color="Green" title="E76">A22</font></i></a>, <a class="ref" href="graph_3.html#T52" target="_self" title="GRAPH_3:th.52">Th52</a></i>;<br/><a NAME="E3:68_3_2"/><b>then </b>
 <a href="graph_3.html#K7" title="GRAPH_3:func.7">Degree</a> <font color="Maroon" title="c2">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="c19">c'e</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">dv'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 ;<br/><b>hence </b><a NAME="E4:68_3_2"/>
(  <a href="graph_3.html#K7" title="GRAPH_3:func.7">Degree</a> <font color="Maroon" title="c2">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="c4">c</font></span>)</span> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> iff ( <font color="Maroon" title="c2">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 &amp; <font color="Maroon" title="c2">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <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> ) )
 <b>by </b><i><a class="txt" href="graph_3.html#E35:68"><i><font color="Green" title="E74">A20</font></i></a>, <a class="txt" href="graph_3.html#E1:68_3_2"><i><font color="Green" title="E77">A23</font></i></a>, <a class="ref" href="graph_3.html#T36" target="_self" title="GRAPH_3:th.36">Th36</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
