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<b>let </b><font color="Maroon">G</font> be   <a href="graph_1.html#NM1">Graph</a>;<br/><b>let </b><font color="Maroon">vs1</font> be    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">vs2</font> be    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G</font>;<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/>







<b>assume </b><b>that </b><br/><a NAME="E1:69"/><i><font color="Green">E30</font></i>: 
2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">sc</font>
 <b>and </b><br/><a NAME="E2:69"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">vs1</font> <a href="graph_2.html#R1" target="_self">is_vertex_seq_of</a> <font color="Maroon">sc</font>
 <b>and </b><br/><a NAME="E3:69"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">vs2</font> <a href="graph_2.html#R1" target="_self">is_vertex_seq_of</a> <font color="Maroon">sc</font>
 ;<br/>

<b>assume </b><a NAME="E4:69"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">vs1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">vs2</font>
 ;<br/>

<b>set </b><font color="Maroon">SG</font> = the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font>;<br/>
<b>set </b><font color="Maroon">TG</font> = the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font>;<br/>
<a NAME="E5:69"/><i><font color="Green">E37</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs1</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 &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs2</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/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">vs1</font> &amp; <font color="Maroon">vs1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">vs2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> );<br/>
<a NAME="E6:69"/><i><font color="Green">E38</font></i>: 
(  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">vs1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">vs1</font> &amp;  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">vs2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">vs2</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E7:69"/><b>then </b><i><font color="Green">E42</font></i>: 
 ex <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">j</font>]
 
<b>by </b><i>, , <a class="ref" href="finseq_2.html#T10">FINSEQ_2:10</a></i>;<br/>
<b>consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E9:69"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 <b>and </b><br/><a NAME="E10:69"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font>
 <b>from </b><i><a class="ref" href="nat_1.html#S5">NAT_1:sch 5</a>();<br/></i>
<a NAME="E11:69"/><i><font color="Green">E47</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</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="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:69_1"/>
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1 or 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</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="69_1_1"><b>suppose </b></a><a NAME="E1:69_1_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:69_1_1"/><i><font color="Green">E49</font></i>: 
<span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <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">vs1</font>
 <b>by </b><i><a class="txt" href="graph_2.html#E13:69_1_1"><i><font color="Green">E59</font></i></a>, , <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E3:69_1_1"/><b>then </b><i><font color="Green">E50</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs1</font> &amp; 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">vs1</font> )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>set </b><font color="Maroon">v11</font> = <font color="Maroon">vs1</font> <a href="finseq_4.html#K4">/.</a> 1;<br/><b>set </b><font color="Maroon">v12</font> = <font color="Maroon">vs1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><b>set </b><font color="Maroon">v13</font> = <font color="Maroon">vs1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><b>set </b><font color="Maroon">v21</font> = <font color="Maroon">vs2</font> <a href="finseq_4.html#K4">/.</a> 1;<br/><b>set </b><font color="Maroon">v22</font> = <font color="Maroon">vs2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><b>set </b><font color="Maroon">v23</font> = <font color="Maroon">vs2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><a NAME="E4:69_1_1"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">vs1</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">vs1</font> <a href="funct_1.html#K1">.</a> 1 &amp; <font color="Maroon">vs1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">vs1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">vs1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">vs1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">vs2</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">vs2</font> <a href="funct_1.html#K1">.</a> 1 &amp; <font color="Maroon">vs2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">vs2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">vs2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">vs2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i>, , , <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/><a NAME="E5:69_1_1"/>
( 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> &amp; 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="txt" href="graph_2.html#E13:69_1_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E6:69_1_1"/><b>then </b><i><font color="Green">E53</font></i>: 
( <font color="Maroon">sc</font> <a href="funct_1.html#K1">.</a> 1 <a href="graph_1.html#R2">joins</a> <font color="Maroon">vs1</font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">vs1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">sc</font> <a href="funct_1.html#K1">.</a> 1 <a href="graph_1.html#R2">joins</a> <font color="Maroon">vs2</font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">vs2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">sc</font> <a href="funct_1.html#K1">.</a> 2 <a href="graph_1.html#R2">joins</a> <font color="Maroon">vs1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">vs1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">sc</font> <a href="funct_1.html#K1">.</a> 2 <a href="graph_1.html#R2">joins</a> <font color="Maroon">vs2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">vs2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <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/><a NAME="E7:69_1_1"/><b>then </b>
( ( <font color="Maroon">vs1</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">vs2</font> <a href="finseq_4.html#K4">/.</a> 1 &amp; <font color="Maroon">vs1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">vs2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> ) or ( <font color="Maroon">vs1</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">vs2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">vs1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">vs2</font> <a href="finseq_4.html#K4">/.</a> 1 ) )
 <b>by </b><i><a class="txt" href="graph_2.html#E6:69"><i><font color="Green">E38</font></i></a></i>;<br/><a NAME="E8:69_1_1"/><b>then </b><i><font color="Green">E54</font></i>: 
( <font color="Maroon">vs1</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">vs1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">vs2</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">vs2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <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="txt" href="graph_2.html#E6:69"><i><font color="Green">E38</font></i></a></i>;<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="E10:69_1_1"/><i><font color="Green">E57</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#E10:69_1_1"><i><font color="Green">E57</font></i></a></i>;<br/><a NAME="E11:69_1_1"/>
<font color="Maroon">sc</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="graph_2.html#E13:69_1_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/><a NAME="E12:69_1_1"/><b>then </b><i><font color="Green">E58</font></i>: 
( <font color="Maroon">vs</font> <a href="hidden.html#R1">=</a> <font color="Maroon">vs1</font> or <font color="Maroon">vs</font> <a href="hidden.html#R1">=</a> <font color="Maroon">vs2</font> )
 <b>by </b><i>, , , , </i>;<br/><a NAME="E13:69_1_1"/><b>then </b><i><font color="Green">E59</font></i>: 
<span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <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="ref" href="graph_2.html#T4" target="_self">Th4</a>, </i>;<br/><a NAME="E14:69_1_1"/>
<span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <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="txt" href="graph_2.html#E13:69_1_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E15:69_1_1"/>
contradiction
 <b>by </b><i>, <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="69_1_2"><b>suppose </b></a><a NAME="E1:69_1_2"/>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><a NAME="E2:69_1_2"/><b>then </b>
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>then consider </b><font color="Maroon">k1</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:69_1_2"/><i><font color="Green">E60</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k1</font> &amp; <font color="Maroon">k1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i/>;<br/><a NAME="E5:69_1_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs1</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E6:69_1_2"/><b>then </b><i><font color="Green">E62</font></i>: 
<font color="Maroon">k1</font> <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="txt" href="graph_2.html#E7:69_1_2"><i><font color="Green">E63</font></i></a>, <a class="txt" href="graph_2.html#E4"><i><font color="Green">Lemma69</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E7:69_1_2"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">k1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs1</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="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E8:69_1_2"/><b>then </b><i><font color="Green">E64</font></i>: 
<font color="Maroon">k1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">vs1</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="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E9:69_1_2"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">vs1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">vs1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k1</font> &amp; <font color="Maroon">vs1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">vs1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> &amp; <font color="Maroon">vs2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">vs2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k1</font> &amp; <font color="Maroon">vs2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">vs2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> )
 <b>by </b><i><a class="txt" href="graph_2.html#E7:69_1_2"><i><font color="Green">E63</font></i></a>, <a class="txt" href="graph_2.html#E8:69_1_2"><i><font color="Green">E64</font></i></a>, , <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>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E10:69_1_2"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">sc</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k1</font> <a href="graph_1.html#R2">joins</a> <font color="Maroon">vs1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k1</font>,<font color="Maroon">vs1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="graph_2.html#E4:69_1_2"><i><font color="Green">E60</font></i></a>, <a class="txt" href="graph_2.html#E4"><i><font color="Green">Lemma69</font></i></a>, , <a class="ref" href="graph_2.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E11:69_1_2"/>
<font color="Maroon">sc</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k1</font> <a href="graph_1.html#R2">joins</a> <font color="Maroon">vs2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k1</font>,<font color="Maroon">vs2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="graph_2.html#E5:69_1_2"><i><font color="Green">E61</font></i></a>, <a class="txt" href="graph_2.html#E4"><i><font color="Green">Lemma69</font></i></a>, , <a class="ref" href="graph_2.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E12:69_1_2"/><b>then </b>
( ( ( the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">sc</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">vs1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k1</font> &amp; the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">sc</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">vs1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> ) or ( the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">sc</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">vs1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> &amp; the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">sc</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">vs1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k1</font> ) ) &amp; ( ( the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">sc</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">vs2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k1</font> &amp; the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">sc</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">vs2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> ) or ( the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">sc</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">vs2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> &amp; the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">sc</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">vs2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k1</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_1.html#D9">GRAPH_1:def 9</a></i>;<br/><b>hence </b><a NAME="E13:69_1_2"/>
contradiction
 <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/></div><b>end;</b></div></div><b>end;</b></div>

</div>
