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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">IT</font> be  <a href="graph_1.html#V8">oriented</a>  <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">vs</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>assume </b><a NAME="E1:3"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">IT</font> is <a href="graph_4.html#V1">Simple</a> &amp; <font color="Maroon">vs</font> <a href="graph_4.html#R3">is_oriented_vertex_seq_of</a> <font color="Maroon">IT</font> )
 ;<br/>

<b>then 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="E3:3"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">vs'</font> <a href="graph_4.html#R3">is_oriented_vertex_seq_of</a> <font color="Maroon">IT</font> &amp; ( for <font color="Olive">n</font>, <font color="Olive">m</font> being   <a href="subset_1.html#M2">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="ref" href="graph_4.html#D7">GRAPH_4:def 7</a></i>;<br/>
<a NAME="E4:3"/><i><font color="Green">E36</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">IT</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">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#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">IT</font> holds <br/><font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="graph_4.html#R1">orientedly_joins</a> <font color="Maroon">vs</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">n</font>,<font color="Maroon">vs</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) )
 
<b>by </b><i>, <a class="ref" href="graph_4.html#D5">GRAPH_4:def 5</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:3_1"/>
( <font color="Maroon">IT</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">IT</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1_1"><b>suppose </b></a><a NAME="E1:3_1_1"/>
<font color="Maroon">IT</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:3_1_1"/><b>then </b>
<font color="Maroon">vs</font> <a href="hidden.html#R1">=</a> <font color="Maroon">vs'</font>
 <b>by </b><i>, , <a class="ref" href="graph_4.html#T11">GRAPH_4:11</a></i>;<br/><b>hence </b><a NAME="E3:3_1_1"/>
for <font color="Olive">n</font>, <font color="Olive">m</font> being   <a href="subset_1.html#M2">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/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1_2"><b>suppose </b></a><a NAME="E1:3_1_2"/>
<font color="Maroon">IT</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:3_1_2"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">IT</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/><b>hence </b><a NAME="E3:3_1_2"/>
for <font color="Olive">n</font>, <font color="Olive">m</font> being   <a href="subset_1.html#M2">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="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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