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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="E1:235"/><i><font color="Green">E39</font></i>: 
<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="E3:235"/><i><font color="Green">E40</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">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E4:235"/><i><font color="Green">E42</font></i>: 
<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/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="235_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="235_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:235_1_1"/>
( <font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W1</font> or <font color="Maroon">m</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W1</font> )
 </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="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="235_1_1_1"><b>suppose </b></a><a NAME="E1:235_1_1_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W1</font>
 ;<br/><div class="add"><a NAME="E2:235_1_1_1"/><b>then </b><i><font color="Green">E44</font></i>: 
<font color="Maroon">W1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">W1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>,<font color="Maroon">W1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">G</font>
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">m1</font> = <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 as  <a href="abian.html#V1">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><a NAME="E4:235_1_1_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m1</font> &amp; <font color="Maroon">m1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W1</font> )
 <b>by </b><i><a class="txt" href="glib_001.html#E54"><i><font color="Green">Lemma157</font></i></a>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E5:235_1_1_1"/><b>then </b>
( <font color="Maroon">m1</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">m1</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">m1</font> <a href="nat_1.html#K5">div</a> 2</span>)</span> )
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">x</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="E7:235_1_1_1"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">x</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">m1</font> <a href="nat_1.html#K5">div</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">W1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m1</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">x</font> )
 <b>by </b><i><a class="txt" href="glib_001.html#E52"><i><font color="Green">Lemma155</font></i></a>, </i>;<br/><b>set </b><font color="Maroon">n</font> = 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">x</font>;<br/><a NAME="E8:235_1_1_1"/><i><font color="Green">E46</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</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="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E9:235_1_1_1"/>
2 <a href="nat_1.html#R1">divides</a> <font color="Maroon">m1</font>
 <b>by </b><i><a class="ref" href="pepin.html#T22">PEPIN:22</a></i>;<br/><a NAME="E10:235_1_1_1"/><b>then </b><i><font color="Green">E47</font></i>: 
2 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">m1</font> <a href="nat_1.html#K5">div</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T49">NAT_1:49</a></i>;<br/><a NAME="E11:235_1_1_1"/>
2 <a href="nat_1.html#K2">*</a> <font color="Maroon">x</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>, </i>;<br/><a NAME="E12:235_1_1_1"/><b>then </b><i><font color="Green">E48</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">x</font> &amp; 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">x</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>then reconsider </b><font color="Maroon">naa1</font> = <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 as  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/><a NAME="E14:235_1_1_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">naa1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 0
 <b>by </b><i>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E15:235_1_1_1"/><b>then </b>
<font color="Maroon">W2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">naa1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">W2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">naa1</font>,<font color="Maroon">W2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">naa1</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">G</font>
 <b>by </b><i/>;<br/><a NAME="E16:235_1_1_1"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">W1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m1</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">W2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">naa1</font>,<font color="Maroon">W2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">naa1</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">G</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E56"><i><font color="Green">Lemma160</font></i></a>, , </i>;<br/><a NAME="E17:235_1_1_1"/>
<font color="Maroon">m1</font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E57"><i><font color="Green">Lemma161</font></i></a>, , <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E18:235_1_1_1"/><b>then </b><i><font color="Green">E52</font></i>: 
<font color="Maroon">m1</font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E19:235_1_1_1"/><b>then </b><i><font color="Green">E53</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">naa1</font> <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E20:235_1_1_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">naa1</font> <a href="nat_1.html#K1">+</a> 2 <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="txt" href="glib_001.html#E58"><i><font color="Green">Lemma163</font></i></a>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="235_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="235_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:235_1_1_1_1_1"/>
( <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="Maroon">naa1</font> or <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"><font color="Maroon">naa1</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> )
 </span><b>by </b><i><a class="txt" href="glib_001.html#E55"><i><font color="Green">Lemma158</font></i></a>, , <a class="ref" href="glib_000.html#T18">GLIB_000:18</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="235_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:235_1_1_1_1_1_1"/>
<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="Maroon">naa1</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:235_1_1_1_1_1_1"/>
 ex <font color="Olive">n</font> being <a href="abian.html#V1">odd</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#E58"><i><font color="Green">Lemma163</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="235_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:235_1_1_1_1_1_2"/>
<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"><font color="Maroon">naa1</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:235_1_1_1_1_1_2"/>
 ex <font color="Olive">n</font> being <a href="abian.html#V1">odd</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#E59"><i><font color="Green">Lemma167</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E22:235_1_1_1"/>
 ex <font color="Olive">n</font> being <a href="abian.html#V1">odd</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> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="235_1_1_2"><b>suppose </b></a><a NAME="E1:235_1_1_2"/>
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W1</font>
 ;<br/><div class="add"><b>then </b><i><font color="Green">E88</font></i>: <font color="Maroon">W1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> = 
<font color="Maroon">W1</font> <a href="glib_001.html#K4" target="_self">.last()</a> 

<br/>.= 
<font color="Maroon">W2</font> <a href="glib_001.html#K4" target="_self">.last()</a> 
<b>by </b><i><a class="txt" href="glib_001.html#E51"><i><font color="Green">Lemma154</font></i></a>, <a class="txt" href="glib_001.html#E48"><i><font color="Green">Lemma151</font></i></a></i>
<br/>.= 
<font color="Maroon">W2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W2</font></span>)</span>

;<br/><a NAME="E3:235_1_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">W1</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="txt" href="glib_001.html#E51"><i><font color="Green">Lemma154</font></i></a>, <a class="ref" href="glib_001.html#D11" target="_self">Def11</a></i>;<br/><a NAME="E4:235_1_1_2"/><b>then </b>
<font color="Maroon">m</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="txt" href="glib_001.html#E53"><i><font color="Green">Lemma156</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E5:235_1_1_2"/>
 ex <font color="Olive">n</font> being <a href="abian.html#V1">odd</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></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E6:235"/>
 ex <font color="Olive">n</font> being <a href="abian.html#V1">odd</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> )
 ;<br/>


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