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





<b>assume </b><a NAME="E1:236"/>
<font color="Maroon">W1</font> is    <a href="glib_001.html#M4" target="_self">Subwalk</a> of <font color="Maroon">W2</font>
 ;<br/>

<b>then consider </b><font color="Maroon">es</font> being    <a href="graph_2.html#M1">FinSubsequence</a> of <font color="Maroon">W2</font> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> <b> such that </b><br/><a NAME="E3:236"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">W1</font> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a>  <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">es</font>
 <b>by </b><i/>;<br/>
<b>let </b><font color="Maroon">m</font> be  <a href="abian.html#V1">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E4:236"/><i><font color="Green">E40</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</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">W1</font> )
 ;<br/>

<a NAME="E5:236"/>
( <font color="Maroon">m</font> <a href="nat_1.html#K5">div</a> 2 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span> &amp; <font color="Maroon">W1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K5">div</a> 2</span>)</span> )
 
<b>by </b><i><a class="txt" href="glib_001.html#E52"><i><font color="Green">Lemma155</font></i></a>, </i>;<br/>
<b>then consider </b><font color="Maroon">ndiv2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E7:236"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">ndiv2</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">W2</font> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span> &amp; <font color="Maroon">m</font> <a href="nat_1.html#K5">div</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ndiv2</font> &amp; <font color="Maroon">W1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">W2</font> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">ndiv2</font> )
 <b>by </b><i><a class="txt" href="glib_001.html#E51"><i><font color="Green">Lemma154</font></i></a>, </i>;<br/>
<a NAME="E8:236"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ndiv2</font> &amp; <font color="Maroon">ndiv2</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">W2</font> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span> )
 
<b>by </b><i><a class="txt" href="glib_001.html#E53"><i><font color="Green">Lemma156</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E9:236"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">W1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">ndiv2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="glib_001.html#E53"><i><font color="Green">Lemma156</font></i></a>, </i>;<br/>
<a NAME="E10:236"/>
2 <a href="nat_1.html#R1">divides</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="ref" href="pepin.html#T22">PEPIN:22</a></i>;<br/>
<a NAME="E11:236"/><b>then </b>
2 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K5">div</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T49">NAT_1:49</a></i>;<br/>
<a NAME="E12:236"/><b>then </b><i><font color="Green">E44</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">ndiv2</font>
 
<b>by </b><i><a class="txt" href="glib_001.html#E53"><i><font color="Green">Lemma156</font></i></a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E13:236"/>
2 <a href="nat_1.html#K2">*</a> <font color="Maroon">ndiv2</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">W2</font>
 
<b>by </b><i><a class="txt" href="glib_001.html#E53"><i><font color="Green">Lemma156</font></i></a>, </i>;<br/>
<a NAME="E14:236"/><b>then </b>
2 <a href="nat_1.html#K2">*</a> <font color="Maroon">ndiv2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W2</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>hence </b><a NAME="E15:236"/>
 ex <font color="Olive">n</font> being <a href="abian.html#V1">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">m</font> <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">W2</font> &amp; <font color="Maroon">W1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> )
 <b>by </b><i><a class="txt" href="glib_001.html#E54"><i><font color="Green">Lemma157</font></i></a>, <a class="txt" href="glib_001.html#E55"><i><font color="Green">Lemma158</font></i></a></i>;<br/>


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