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<b>let </b><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c2">G</font> be   <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Graph</a>;<br/><b>let </b><font color="Maroon" title="c3">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="c2">G</font>;<br/>





<b>consider </b><font color="Maroon" title="c4">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="c2">G</font><b> such that </b><br/><a NAME="E2:66"/><i><font color="Green" title="E55">A1</font></i>: 
( <font color="Maroon" title="c4">vs</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Maroon" title="c3">sc</font> &amp; ( for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   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="c4">vs</font> &amp; <font color="Maroon" title="c4">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="c4">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="c4">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/>
<b>reconsider </b><font color="Maroon" title="c5">q'</font> = <font color="Maroon" title="c3">sc</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span> as    <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c2">G</font> <b>by </b><i><a class="ref" href="graph_1.html#T4" title="GRAPH_1:th.4">GRAPH_1:4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">p'</font> = <font color="Maroon" title="c4">vs</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c2">G</font> <b>by </b><i><a class="ref" href="finseq_1.html#T23" title="FINSEQ_1:th.23">FINSEQ_1:23</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon" title="c7">p'</font> = <font color="Maroon" title="c6">p'</font>;<br/><b>thus </b><a NAME="E1:66_1"/>
<font color="Maroon" title="c7">p'</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Maroon" title="c5">q'</font>
 <b>by </b><i><a class="txt" href="graph_2.html#E2:66"><i><font color="Green" title="E55">A1</font></i></a>, <a class="ref" href="graph_2.html#T43" target="_self" title="GRAPH_2:th.43">Th43</a></i>;<br/><b>let </b><font color="Maroon" title="c8">k</font>, <font color="Maroon" title="c9">m</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E2:66_1"/><i><font color="Green" title="E56">A2</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font>
 <b>and </b><br/><a NAME="E3:66_1"/><i><font color="Green" title="E57">A3</font></i>: 
<font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">m</font>
 <b>and </b><br/><a NAME="E4:66_1"/><i><font color="Green" title="E58">A4</font></i>: 
<font color="Maroon" title="c9">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="c7">p'</font>
 <b>and </b><br/><a NAME="E5:66_1"/><i><font color="Green" title="E59">A5</font></i>: 
<font color="Maroon" title="c7">p'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">m</font>
 ;<br/><a NAME="E6:66_1"/>
( <font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p'</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">m</font> )
 <b>by </b><i><a class="txt" href="graph_2.html#E2:66_1"><i><font color="Green" title="E56">A2</font></i></a>, <a class="txt" href="graph_2.html#E3:66_1"><i><font color="Green" title="E57">A3</font></i></a>, <a class="txt" href="graph_2.html#E4:66_1"><i><font color="Green" title="E58">A4</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E7:66_1"/><b>then </b><i><font color="Green" title="E60">A6</font></i>: 
( <font color="Maroon" title="c7">p'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k</font> &amp; <font color="Maroon" title="c7">p'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">m</font> )
 <b>by </b><i><a class="txt" href="graph_2.html#E2:66_1"><i><font color="Green" title="E56">A2</font></i></a>, <a class="txt" href="graph_2.html#E4:66_1"><i><font color="Green" title="E58">A4</font></i></a>, <a class="ref" href="graph_2.html#T2" target="_self" title="GRAPH_2:th.2">Th2</a></i>;<br/><a NAME="E8:66_1"/><i><font color="Green" title="E61">A7</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p'</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="c4">vs</font>
 <b>by </b><i><a class="ref" href="graph_2.html#T2" target="_self" title="GRAPH_2:th.2">Th2</a></i>;<br/><a NAME="E9:66_1"/><b>then </b><i><font color="Green" title="E62">A8</font></i>: 
<font color="Maroon" title="c9">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="c4">vs</font>
 <b>by </b><i><a class="txt" href="graph_2.html#E4:66_1"><i><font color="Green" title="E58">A4</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E10:66_1"/>
<font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="graph_2.html#E2:66"><i><font color="Green" title="E55">A1</font></i></a>, <a class="txt" href="graph_2.html#E2:66_1"><i><font color="Green" title="E56">A2</font></i></a>, <a class="txt" href="graph_2.html#E3:66_1"><i><font color="Green" title="E57">A3</font></i></a>, <a class="txt" href="graph_2.html#E5:66_1"><i><font color="Green" title="E59">A5</font></i></a>, <a class="txt" href="graph_2.html#E7:66_1"><i><font color="Green" title="E60">A6</font></i></a></i>;<br/><a NAME="E11:66_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">vs</font> or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p'</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">vs</font> )
 <b>by </b><i><a class="txt" href="graph_2.html#E8:66_1"><i><font color="Green" title="E61">A7</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><b>hence </b><a NAME="E12:66_1"/>
<font color="Maroon" title="c9">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="c7">p'</font>
 <b>by </b><i><a class="txt" href="graph_2.html#E2:66"><i><font color="Green" title="E55">A1</font></i></a>, <a class="txt" href="graph_2.html#E2:66_1"><i><font color="Green" title="E56">A2</font></i></a>, <a class="txt" href="graph_2.html#E3:66_1"><i><font color="Green" title="E57">A3</font></i></a>, <a class="txt" href="graph_2.html#E4:66_1"><i><font color="Green" title="E58">A4</font></i></a>, <a class="txt" href="graph_2.html#E5:66_1"><i><font color="Green" title="E59">A5</font></i></a>, <a class="txt" href="graph_2.html#E7:66_1"><i><font color="Green" title="E60">A6</font></i></a>, <a class="txt" href="graph_2.html#E9:66_1"><i><font color="Green" title="E62">A8</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:66"/>
<font color="Maroon" title="c3">sc</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span> is  <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="c2">G</font>
 <b>by </b><i><a class="ref" href="graph_2.html#D10" target="_self" title="GRAPH_2:def.10">Def10</a></i>;<br/>


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