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<div class="add">

<b>let </b><font color="Maroon" title="c1">G</font> be   <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Graph</a>;<br/><b>let </b><font color="Maroon" title="c2">sc</font> be  <a href="graph_2.html#V3" title="GRAPH_2:attr.3">simple</a>  <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c1">G</font>;<br/>



<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:76_1"/><i><font color="Green" title="E60">A1</font></i>: 
<font color="Maroon" title="c2">sc</font> is   <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><a NAME="E2:76_1"/><i><font color="Green" title="E61">A2</font></i>: 
not  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">sc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><b>assume </b><a NAME="E3:76_1"/><i><font color="Green" title="E62">A3</font></i>: 
not  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">sc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><b>assume </b><a NAME="E4:76_1"/><i><font color="Green" title="E63">A4</font></i>: 
<font color="Maroon" title="c2">sc</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">sc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2
 ;<br/><a NAME="E5:76_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">sc</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <b>by </b><i><a class="txt" href="graph_2.html#E2:76_1"><i><font color="Green" title="E61">A2</font></i></a>, <a class="txt" href="graph_2.html#E3:76_1"><i><font color="Green" title="E62">A3</font></i></a>, <a class="ref" href="nat_1.html#T26" title="NAT_1:th.26">NAT_1:26</a></i>;<br/><a NAME="E6:76_1"/><b>then </b>
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="c2">sc</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>hence </b><a NAME="E7:76_1"/>
contradiction
 <b>by </b><i><a class="txt" href="graph_2.html#E1:76_1"><i><font color="Green" title="E60">A1</font></i></a>, <a class="txt" href="graph_2.html#E4:76_1"><i><font color="Green" title="E63">A4</font></i></a>, <a class="ref" href="graph_1.html#D13" title="GRAPH_1:def.13">GRAPH_1:def 13</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:76"/><i><font color="Green" title="E60">A5</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">sc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">sc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c2">sc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">sc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:76_2"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">sc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">sc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c2">sc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">sc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 )
 </span><b>by </b><i><a class="txt" href="graph_2.html#E2:76"><i><font color="Green" title="E60">A5</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_2_1"><b>suppose </b></a><a NAME="E1:76_2_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">sc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:76_2_1"/><b>then </b>
for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">m</font> &amp; <font color="Olive" title="b2">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="c2">sc</font> holds <br/><font color="Maroon" title="c2">sc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">sc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">m</font>
 ;<br/><b>hence </b><a NAME="E3:76_2_1"/>
<font color="Maroon" title="c2">sc</font> is   <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_1.html#D13" title="GRAPH_1:def.13">GRAPH_1:def 13</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_2_2"><b>suppose </b></a><a NAME="E1:76_2_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">sc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:76_2_2"/><b>then </b>
for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">m</font> &amp; <font color="Olive" title="b2">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="c2">sc</font> holds <br/><font color="Maroon" title="c2">sc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">sc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">m</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E3:76_2_2"/>
<font color="Maroon" title="c2">sc</font> is   <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_1.html#D13" title="GRAPH_1:def.13">GRAPH_1:def 13</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_2_3"><b>suppose </b></a><a NAME="E1:76_2_3"/><i><font color="Green" title="E61">A6</font></i>: 
<font color="Maroon" title="c2">sc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">sc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_2_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">n</font>, <font color="Maroon" title="c4">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:76_2_3_1"/><i><font color="Green" title="E62">A7</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font>
 <b>and </b><br/><a NAME="E2:76_2_3_1"/><i><font color="Green" title="E63">A8</font></i>: 
<font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">m</font>
 <b>and </b><br/><a NAME="E3:76_2_3_1"/><i><font color="Green" title="E64">A9</font></i>: 
<font color="Maroon" title="c4">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="c2">sc</font>
 ;<br/><b>thus </b><a NAME="E4:76_2_3_1"/>
<font color="Maroon" title="c2">sc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">sc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">m</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_2_3_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:76_2_3_1_1"/><i><font color="Green" title="E65">A10</font></i>: 
not <font color="Maroon" title="c2">sc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">sc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">m</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c5">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="E3:76_2_3_1_1"/><i><font color="Green" title="E66">A11</font></i>: 
( <font color="Maroon" title="c5">vs</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Maroon" title="c2">sc</font> &amp; ( for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">m</font> &amp; <font color="Olive" title="b2">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="c5">vs</font> &amp; <font color="Maroon" title="c5">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">m</font> holds <br/>( <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Olive" title="b2">m</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="c5">vs</font> ) ) )
 <b>by </b><i><a class="ref" href="graph_2.html#D10" target="_self" title="GRAPH_2:def.10">Def10</a></i>;<br/>
<a NAME="E4:76_2_3_1_1"/><i><font color="Green" title="E67">A12</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">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">sc</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_2.html#E3:76_2_3_1_1"><i><font color="Green" title="E66">A11</font></i></a>, <a class="ref" href="graph_2.html#D7" target="_self" title="GRAPH_2:def.7">Def7</a></i>;<br/>
<a NAME="E5:76_2_3_1_1"/><i><font color="Green" title="E68">A13</font></i>: 
( <font color="Maroon" title="c3">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="c2">sc</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">m</font> )
 
<b>by </b><i><a class="txt" href="graph_2.html#E1:76_2_3_1"><i><font color="Green" title="E62">A7</font></i></a>, <a class="txt" href="graph_2.html#E2:76_2_3_1"><i><font color="Green" title="E63">A8</font></i></a>, <a class="txt" href="graph_2.html#E3:76_2_3_1"><i><font color="Green" title="E64">A9</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E6:76_2_3_1_1"/><b>then </b><i><font color="Green" title="E69">A14</font></i>: 
( <font color="Maroon" title="c3">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="c5">vs</font> &amp; <font color="Maroon" title="c4">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="c5">vs</font> )
 
<b>by </b><i><a class="txt" href="graph_2.html#E3:76_2_3_1"><i><font color="Green" title="E64">A9</font></i></a>, <a class="txt" href="graph_2.html#E4:76_2_3_1_1"><i><font color="Green" title="E67">A12</font></i></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/>
<a NAME="E7:76_2_3_1_1"/><i><font color="Green" title="E70">A15</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c3">n</font> <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="c5">vs</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c4">m</font> <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="c5">vs</font> )
 
<b>by </b><i><a class="txt" href="graph_2.html#E3:76_2_3_1"><i><font color="Green" title="E64">A9</font></i></a>, <a class="txt" href="graph_2.html#E4:76_2_3_1_1"><i><font color="Green" title="E67">A12</font></i></a>, <a class="txt" href="graph_2.html#E5:76_2_3_1_1"><i><font color="Green" title="E68">A13</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/>
<b>set </b><font color="Maroon" title="c6">vn</font> = <font color="Maroon" title="c5">vs</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c3">n</font>;<br/>
<b>set </b><font color="Maroon" title="c7">vn1</font> = <font color="Maroon" title="c5">vs</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c8">vm</font> = <font color="Maroon" title="c5">vs</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c4">m</font>;<br/>
<b>set </b><font color="Maroon" title="c9">vm1</font> = <font color="Maroon" title="c5">vs</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>;<br/>
<a NAME="E8:76_2_3_1_1"/><i><font color="Green" title="E71">A16</font></i>: 
( <font color="Maroon" title="c5">vs</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font> &amp; <font color="Maroon" title="c5">vs</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <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="c5">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c5">vs</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">m</font> &amp; <font color="Maroon" title="c5">vs</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m</font> <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="c5">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 
<b>by </b><i><a class="txt" href="graph_2.html#E1:76_2_3_1"><i><font color="Green" title="E62">A7</font></i></a>, <a class="txt" href="graph_2.html#E5:76_2_3_1_1"><i><font color="Green" title="E68">A13</font></i></a>, <a class="txt" href="graph_2.html#E6:76_2_3_1_1"><i><font color="Green" title="E69">A14</font></i></a>, <a class="txt" href="graph_2.html#E7:76_2_3_1_1"><i><font color="Green" title="E70">A15</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/>
<a NAME="E9:76_2_3_1_1"/><i><font color="Green" title="E72">A17</font></i>: 
( <font color="Maroon" title="c2">sc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font> <a href="graph_1.html#R2" title="GRAPH_1:pred.2">joins</a> <font color="Maroon" title="c5">vs</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c5">vs</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c2">sc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">m</font> <a href="graph_1.html#R2" title="GRAPH_1:pred.2">joins</a> <font color="Maroon" title="c5">vs</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c4">m</font>,<font color="Maroon" title="c5">vs</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 
<b>by </b><i><a class="txt" href="graph_2.html#E1:76_2_3_1"><i><font color="Green" title="E62">A7</font></i></a>, <a class="txt" href="graph_2.html#E3:76_2_3_1"><i><font color="Green" title="E64">A9</font></i></a>, <a class="txt" href="graph_2.html#E3:76_2_3_1_1"><i><font color="Green" title="E66">A11</font></i></a>, <a class="txt" href="graph_2.html#E5:76_2_3_1_1"><i><font color="Green" title="E68">A13</font></i></a>, <a class="ref" href="graph_2.html#D7" target="_self" title="GRAPH_2:def.7">Def7</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_2_3_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:76_2_3_1_1_1"/>
( ( <font color="Maroon" title="c5">vs</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">vs</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c4">m</font> &amp; <font color="Maroon" title="c5">vs</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <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="c5">vs</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> ) or ( <font color="Maroon" title="c5">vs</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">vs</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c5">vs</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <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="c5">vs</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c4">m</font> ) )
 </span><b>by </b><i><a class="txt" href="graph_2.html#E1:76_2_3_1_1"><i><font color="Green" title="E65">A10</font></i></a>, <a class="txt" href="graph_2.html#E9:76_2_3_1_1"><i><font color="Green" title="E72">A17</font></i></a>, <a class="ref" href="graph_2.html#T32" target="_self" title="GRAPH_2:th.32">Th32</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_2_3_1_1_1_1"><b>suppose </b></a><a NAME="E1:76_2_3_1_1_1_1"/><i><font color="Green" title="E73">A18</font></i>: 
( <font color="Maroon" title="c5">vs</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">vs</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c4">m</font> &amp; <font color="Maroon" title="c5">vs</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <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="c5">vs</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 ;<br/><div class="add"><a NAME="E2:76_2_3_1_1_1_1"/>
<font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="graph_2.html#E2:76_2_3_1"><i><font color="Green" title="E63">A8</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E3:76_2_3_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c4">m</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="c5">vs</font> &amp; <font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c4">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <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="c5">vs</font> )
 <b>by </b><i><a class="txt" href="graph_2.html#E1:76_2_3_1"><i><font color="Green" title="E62">A7</font></i></a>, <a class="txt" href="graph_2.html#E2:76_2_3_1"><i><font color="Green" title="E63">A8</font></i></a>, <a class="txt" href="graph_2.html#E3:76_2_3_1_1"><i><font color="Green" title="E66">A11</font></i></a>, <a class="txt" href="graph_2.html#E6:76_2_3_1_1"><i><font color="Green" title="E69">A14</font></i></a>, <a class="txt" href="graph_2.html#E7:76_2_3_1_1"><i><font color="Green" title="E70">A15</font></i></a>, <a class="txt" href="graph_2.html#E8:76_2_3_1_1"><i><font color="Green" title="E71">A16</font></i></a>, <a class="txt" href="graph_2.html#E1:76_2_3_1_1_1_1"><i><font color="Green" title="E73">A18</font></i></a></i>;<br/><b>hence </b><a NAME="E4:76_2_3_1_1_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_2_3_1_1_1_2"><b>suppose </b></a><a NAME="E1:76_2_3_1_1_1_2"/><i><font color="Green" title="E73">A19</font></i>: 
( <font color="Maroon" title="c5">vs</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">vs</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c5">vs</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <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="c5">vs</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c4">m</font> )
 ;<br/><div class="add"><b>thus </b><a NAME="E2:76_2_3_1_1_1_2"/>
contradiction
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_2_3_1_1_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:76_2_3_1_1_1_2_1"/><i><font color="Green" title="E74">A20</font></i>: 
<font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="graph_2.html#E2:76_2_3_1"><i><font color="Green" title="E63">A8</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/>
<a NAME="E2:76_2_3_1_1_1_2_1"/><i><font color="Green" title="E75">A21</font></i>: 
<font color="Maroon" title="c3">n</font> <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="c4">m</font>
 
<b>by </b><i><a class="txt" href="graph_2.html#E2:76_2_3_1"><i><font color="Green" title="E63">A8</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_2_3_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:76_2_3_1_1_1_2_1_1"/>
( <font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">m</font> or <font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">m</font> )
 </span><b>by </b><i><a class="txt" href="graph_2.html#E2:76_2_3_1_1_1_2_1"><i><font color="Green" title="E75">A21</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_2_3_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:76_2_3_1_1_1_2_1_1_1"/><i><font color="Green" title="E76">A22</font></i>: 
<font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">m</font>
 ;<br/><div class="add"><a NAME="E2:76_2_3_1_1_1_2_1_1_1"/>
( <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c4">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <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="c5">vs</font> )
 <b>by </b><i><a class="txt" href="graph_2.html#E1:76_2_3_1"><i><font color="Green" title="E62">A7</font></i></a>, <a class="txt" href="graph_2.html#E3:76_2_3_1_1"><i><font color="Green" title="E66">A11</font></i></a>, <a class="txt" href="graph_2.html#E7:76_2_3_1_1"><i><font color="Green" title="E70">A15</font></i></a>, <a class="txt" href="graph_2.html#E8:76_2_3_1_1"><i><font color="Green" title="E71">A16</font></i></a>, <a class="txt" href="graph_2.html#E1:76_2_3_1_1_1_2"><i><font color="Green" title="E73">A19</font></i></a>, <a class="txt" href="graph_2.html#E1:76_2_3_1_1_1_2_1"><i><font color="Green" title="E74">A20</font></i></a></i>;<br/><b>hence </b><a NAME="E3:76_2_3_1_1_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="graph_2.html#E1:76_2_3"><i><font color="Green" title="E61">A6</font></i></a>, <a class="txt" href="graph_2.html#E1:76_2_3_1_1"><i><font color="Green" title="E65">A10</font></i></a>, <a class="txt" href="graph_2.html#E1:76_2_3_1_1_1_2_1_1_1"><i><font color="Green" title="E76">A22</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_2_3_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:76_2_3_1_1_1_2_1_1_2"/>
<font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">m</font>
 ;<br/><div class="add"><a NAME="E2:76_2_3_1_1_1_2_1_1_2"/><b>then </b>
( <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c4">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <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="c5">vs</font> &amp; <font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c4">m</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="c5">vs</font> )
 <b>by </b><i><a class="txt" href="graph_2.html#E1:76_2_3_1"><i><font color="Green" title="E62">A7</font></i></a>, <a class="txt" href="graph_2.html#E3:76_2_3_1_1"><i><font color="Green" title="E66">A11</font></i></a>, <a class="txt" href="graph_2.html#E6:76_2_3_1_1"><i><font color="Green" title="E69">A14</font></i></a>, <a class="txt" href="graph_2.html#E7:76_2_3_1_1"><i><font color="Green" title="E70">A15</font></i></a>, <a class="txt" href="graph_2.html#E8:76_2_3_1_1"><i><font color="Green" title="E71">A16</font></i></a>, <a class="txt" href="graph_2.html#E1:76_2_3_1_1_1_2"><i><font color="Green" title="E73">A19</font></i></a>, <a class="txt" href="graph_2.html#E1:76_2_3_1_1_1_2_1"><i><font color="Green" title="E74">A20</font></i></a></i>;<br/><b>hence </b><a NAME="E3:76_2_3_1_1_1_2_1_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:76_2_3"/>
<font color="Maroon" title="c2">sc</font> is   <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_1.html#D13" title="GRAPH_1:def.13">GRAPH_1:def 13</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
