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

<b>let </b><font color="Maroon">G</font> be   <a href="graph_1.html#NM1">Graph</a>;<br/><b>let </b><font color="Maroon">sc</font> be  <a href="graph_2.html#V3" target="_self">simple</a>  <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">G</font>;<br/>



<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:77_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">sc</font> is   <a href="graph_1.html#NM4">Path</a> of <font color="Maroon">G</font>
 ;<br/><b>assume </b><a NAME="E2:77_1"/><i><font color="Green">E31</font></i>: 
not  <a href="finseq_1.html#K3">len</a> <font color="Maroon">sc</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><b>assume </b><a NAME="E3:77_1"/><i><font color="Green">E34</font></i>: 
not  <a href="finseq_1.html#K3">len</a> <font color="Maroon">sc</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><b>assume </b><a NAME="E4:77_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">sc</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">sc</font> <a href="funct_1.html#K1">.</a> 2
 ;<br/><a NAME="E5:77_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">sc</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 <b>by </b><i>, , <a class="ref" href="cqc_the1.html#T2">CQC_THE1:2</a></i>;<br/><a NAME="E6:77_1"/><b>then </b>
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">sc</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E7:77_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="graph_1.html#D13">GRAPH_1:def 13</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:77"/><i><font color="Green">E37</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">sc</font> <a href="hidden.html#R1">=</a> 0 or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">sc</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">sc</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">sc</font> <a href="funct_1.html#K1">.</a> 2 )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:77_2"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">sc</font> <a href="hidden.html#R1">=</a> 0 or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">sc</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">sc</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">sc</font> <a href="funct_1.html#K1">.</a> 2 )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_2_1"><b>suppose </b></a><a NAME="E1:77_2_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">sc</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:77_2_1"/><b>then </b>
for <font color="Olive">n</font>, <font color="Olive">m</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> &amp; <font color="Olive">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">m</font> &amp; <font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">sc</font> holds <br/><font color="Maroon">sc</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">sc</font> <a href="funct_1.html#K1">.</a> <font color="Olive">m</font>
 ;<br/><b>hence </b><a NAME="E3:77_2_1"/>
<font color="Maroon">sc</font> is   <a href="graph_1.html#NM4">Path</a> of <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="graph_1.html#D13">GRAPH_1:def 13</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_2_2"><b>suppose </b></a><a NAME="E1:77_2_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">sc</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:77_2_2"/><b>then </b>
for <font color="Olive">n</font>, <font color="Olive">m</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> &amp; <font color="Olive">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">m</font> &amp; <font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">sc</font> holds <br/><font color="Maroon">sc</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">sc</font> <a href="funct_1.html#K1">.</a> <font color="Olive">m</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E3:77_2_2"/>
<font color="Maroon">sc</font> is   <a href="graph_1.html#NM4">Path</a> of <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="graph_1.html#D13">GRAPH_1:def 13</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_2_3"><b>suppose </b></a><a NAME="E1:77_2_3"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">sc</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">sc</font> <a href="funct_1.html#K1">.</a> 2
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_2_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:77_2_3_1"/><i><font color="Green">E42</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E2:77_2_3_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font>
 <b>and </b><br/><a NAME="E3:77_2_3_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">sc</font>
 ;<br/><b>thus </b><a NAME="E4:77_2_3_1"/>
<font color="Maroon">sc</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">sc</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_2_3_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:77_2_3_1_1"/><i><font color="Green">E47</font></i>: 
not <font color="Maroon">sc</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">sc</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
 ;<br/>

<b>consider </b><font color="Maroon">vs</font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E3:77_2_3_1_1"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">vs</font> <a href="graph_2.html#R1" target="_self">is_vertex_seq_of</a> <font color="Maroon">sc</font> &amp; ( for <font color="Olive">n</font>, <font color="Olive">m</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> &amp; <font color="Olive">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">m</font> &amp; <font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> &amp; <font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> <font color="Olive">m</font> holds <br/>( <font color="Olive">n</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Olive">m</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> ) ) )
 <b>by </b><i><a class="txt" href="graph_2.html#E9:77_2_3_1_1"><i><font color="Green">E57</font></i></a></i>;<br/>
<a NAME="E4:77_2_3_1_1"/><i><font color="Green">E49</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">sc</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, <a class="ref" href="graph_2.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E5:77_2_3_1_1"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">sc</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> )
 
<b>by </b><i>, , , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E6:77_2_3_1_1"/><b>then </b><i><font color="Green">E52</font></i>: 
( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> )
 
<b>by </b><i>, , <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<a NAME="E7:77_2_3_1_1"/><i><font color="Green">E53</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> )
 
<b>by </b><i>, , , <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<b>set </b><font color="Maroon">vn</font> = <font color="Maroon">vs</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font>;<br/>
<b>set </b><font color="Maroon">vn1</font> = <font color="Maroon">vs</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/>
<b>set </b><font color="Maroon">vm</font> = <font color="Maroon">vs</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">m</font>;<br/>
<b>set </b><font color="Maroon">vm1</font> = <font color="Maroon">vs</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/>
<a NAME="E8:77_2_3_1_1"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">vs</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> &amp; <font color="Maroon">vs</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">vs</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> &amp; <font color="Maroon">vs</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 
<b>by </b><i>, , <a class="txt" href="graph_2.html#E4"><i><font color="Green">Lemma69</font></i></a>, , <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/>
<a NAME="E9:77_2_3_1_1"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">sc</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="graph_1.html#R2">joins</a> <font color="Maroon">vs</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font>,<font color="Maroon">vs</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">sc</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="graph_1.html#R2">joins</a> <font color="Maroon">vs</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">m</font>,<font color="Maroon">vs</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 
<b>by </b><i>, , , , <a class="ref" href="graph_2.html#T3" target="_self">Th3</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_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="E1:77_2_3_1_1_1"/>
( ( <font color="Maroon">vs</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">vs</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">m</font> &amp; <font color="Maroon">vs</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">vs</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) or ( <font color="Maroon">vs</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">vs</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">vs</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">vs</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">m</font> ) )
 </span><b>by </b><i>, , <a class="txt" href="graph_2.html#E1:77_2_3"><i><font color="Green">E38</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_2_3_1_1_1_1"><b>suppose </b></a><a NAME="E1:77_2_3_1_1_1_1"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">vs</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">vs</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">m</font> &amp; <font color="Maroon">vs</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">vs</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 ;<br/><div class="add"><a NAME="E2:77_2_3_1_1_1_1"/>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E3:77_2_3_1_1_1_1"/><b>then </b>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">m</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> &amp; <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> )
 <b>by </b><i>, , , <a class="txt" href="graph_2.html#E4"><i><font color="Green">Lemma69</font></i></a>, , , </i>;<br/><b>hence </b><a NAME="E4:77_2_3_1_1_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_2_3_1_1_1_2"><b>suppose </b></a><a NAME="E1:77_2_3_1_1_1_2"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">vs</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">vs</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">vs</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">vs</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">m</font> )
 ;<br/><div class="add"><b>thus </b><a NAME="E2:77_2_3_1_1_1_2"/>
contradiction
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_2_3_1_1_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:77_2_3_1_1_1_2_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E2:77_2_3_1_1_1_2_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_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="E1:77_2_3_1_1_1_2_1_1"/>
( <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> or <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> )
 </span><b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_2_3_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:77_2_3_1_1_1_2_1_1_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 ;<br/><div class="add"><a NAME="E2:77_2_3_1_1_1_2_1_1_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> )
 <b>by </b><i><a class="txt" href="graph_2.html#E1:77_2_3_1_1_1_2_1_1_1"><i><font color="Green">E62</font></i></a>, , , , , <a class="ref" href="graph_2.html#T4" target="_self">Th4</a></i>;<br/><b>hence </b><a NAME="E3:77_2_3_1_1_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="graph_2.html#E2:77_2_3_1_1_1_2_1"><i><font color="Green">E61</font></i></a>, , <a class="ref" href="graph_2.html#T5" target="_self">Th5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_2_3_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:77_2_3_1_1_1_2_1_1_2"/>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font>
 ;<br/><div class="add"><a NAME="E2:77_2_3_1_1_1_2_1_1_2"/><b>then </b>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> &amp; <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">m</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> )
 <b>by </b><i>, , <a class="txt" href="graph_2.html#E4"><i><font color="Green">Lemma69</font></i></a>, , , , <a class="ref" href="graph_2.html#T4" target="_self">Th4</a></i>;<br/><b>hence </b><a NAME="E3:77_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="E3:77_2_3"/>
<font color="Maroon">sc</font> is   <a href="graph_1.html#NM4">Path</a> of <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="graph_1.html#D13">GRAPH_1:def 13</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
