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<b>let </b><font color="Maroon">G</font> be   <a href="glib_000.html#NM2">_Graph</a>;<br/><b>let </b><font color="Maroon">W2</font> be    <a href="glib_001.html#M3" target="_self">Walk</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">W1</font> be    <a href="glib_001.html#M3" target="_self">Walk</a> of <font color="Maroon">G</font>;<br/>





<b>assume </b><a NAME="E1:215"/>
<font color="Maroon">W2</font> is <a href="glib_001.html#V3" target="_self">trivial</a>
 ;<br/>

<a NAME="E2:215"/><b>then </b><i><font color="Green">E39</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">W2</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i/>;<br/>
<a NAME="E3:215"/><b>then </b><i><font color="Green">E40</font></i>: 
<font color="Maroon">W2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="215_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="215_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:215_1_1"/>
( <font color="Maroon">W1</font> <a href="glib_001.html#K4" target="_self">.last()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">W2</font> <a href="glib_001.html#K3" target="_self">.first()</a>  or <font color="Maroon">W1</font> <a href="glib_001.html#K4" target="_self">.last()</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">W2</font> <a href="glib_001.html#K3" target="_self">.first()</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="215_1_1_1"><b>suppose </b></a><a NAME="E1:215_1_1_1"/>
<font color="Maroon">W1</font> <a href="glib_001.html#K4" target="_self">.last()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">W2</font> <a href="glib_001.html#K3" target="_self">.first()</a> 
 ;<br/><div class="add"><a NAME="E2:215_1_1_1"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">W1</font> <a href="glib_001.html#K7" target="_self">.append</a> <font color="Maroon">W2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W1</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">W2</font>
 <b>by </b><i/>;<br/><a NAME="E3:215_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">W1</font> <a href="glib_001.html#K7" target="_self">.append</a> <font color="Maroon">W2</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="glib_001.html#E44"><i><font color="Green">Lemma147</font></i></a>, <a class="txt" href="glib_001.html#E45"><i><font color="Green">Lemma148</font></i></a>, <a class="ref" href="graph_2.html#T13">GRAPH_2:13</a></i>;<br/><a NAME="E4:215_1_1_1"/><b>then </b><i><font color="Green">E43</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="glib_001.html#K7" target="_self">.append</a> <font color="Maroon">W2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W1</font>
 ;<br/><a NAME="E5:215_1_1_1"/>
for <font color="Olive">k</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">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="glib_001.html#K7" target="_self">.append</a> <font color="Maroon">W2</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="glib_001.html#K7" target="_self">.append</a> <font color="Maroon">W2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E46"><i><font color="Green">Lemma149</font></i></a>, <a class="txt" href="glib_001.html#E47"><i><font color="Green">Lemma150</font></i></a>, <a class="ref" href="graph_2.html#T14">GRAPH_2:14</a></i>;<br/><b>hence </b><a NAME="E6:215_1_1_1"/>
<font color="Maroon">W1</font> <a href="glib_001.html#K7" target="_self">.append</a> <font color="Maroon">W2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W1</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E47"><i><font color="Green">Lemma150</font></i></a>, <a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="215_1_1_2"><b>suppose </b></a><a NAME="E1:215_1_1_2"/>
<font color="Maroon">W1</font> <a href="glib_001.html#K4" target="_self">.last()</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">W2</font> <a href="glib_001.html#K3" target="_self">.first()</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:215_1_1_2"/>
<font color="Maroon">W1</font> <a href="glib_001.html#K7" target="_self">.append</a> <font color="Maroon">W2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W1</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:215"/>
<font color="Maroon">W1</font> <a href="glib_001.html#K7" target="_self">.append</a> <font color="Maroon">W2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W1</font>
 ;<br/>


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