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<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:25_1_1_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> <a href="glib_001.html#K13" target="_self">.vertices()</a> 
 ;<br/><b>set </b><font color="Maroon">vs</font> = <font color="Maroon">W</font> <a href="glib_001.html#K11" target="_self">.vertexSeq()</a> ;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( not <font color="Maroon">a<sub>1</sub></font> is <a href="abian.html#V1">even</a> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font> &amp; <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> );<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_1_1"><b>now </b></a><div class="add"><a NAME="E1:25_1_1_1_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K11" target="_self">.vertexSeq()</a> </span>)</span>
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">i</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="E3:25_1_1_1_1"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">i</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">W</font> <a href="glib_001.html#K11" target="_self">.vertexSeq()</a> </span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K11" target="_self">.vertexSeq()</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> )
 <b>by </b><i><a class="ref" href="finseq_2.html#T11">FINSEQ_2:11</a></i>;<br/><b>set </b><font color="Maroon">n1</font> = 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">i</font>;<br/><b>reconsider </b><font color="Maroon">n1</font> = 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">i</font> 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/><b>set </b><font color="Maroon">n</font> = <font color="Maroon">n1</font> <a href="xcmplx_0.html#K6">-</a> 1;<br/><a NAME="E5:25_1_1_1_1"/><i><font color="Green">E42</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</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">W</font> <a href="glib_001.html#K11" target="_self">.vertexSeq()</a> </span>)</span> )
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E6:25_1_1_1_1"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><b>then reconsider </b><font color="Maroon">n</font> = <font color="Maroon">n1</font> <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/><b>take </b><font color="Maroon">n</font> = <font color="Maroon">n</font>;<br/><a NAME="E8:25_1_1_1_1"/>
<font color="Maroon">i</font> <a href="nat_1.html#K2">*</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K11" target="_self">.vertexSeq()</a> </span>)</span></span>)</span> <a href="nat_1.html#K2">*</a> 2
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E9:25_1_1_1_1"/><b>then </b>
<font color="Maroon">i</font> <a href="nat_1.html#K2">*</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i/>;<br/><a NAME="E10:25_1_1_1_1"/><b>then </b>
<font color="Maroon">n1</font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</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/><b>hence </b><a NAME="E11:25_1_1_1_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font>
 ;<br/><b>thus </b><a NAME="E12:25_1_1_1_1"/>
<font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><a NAME="E3:25_1_1_1"/><b>then </b><i><font color="Green">E43</font></i>: 
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 ;<br/><b>consider </b><font color="Maroon">IT</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="E5:25_1_1_1"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">IT</font>] &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 <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">IT</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5">NAT_1:sch 5</a>();<br/></i><b>reconsider </b><font color="Maroon">IT</font> = <font color="Maroon">IT</font> 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/>;<br/><b>take </b><font color="Maroon">IT</font> = <font color="Maroon">IT</font>;<br/><b>thus </b><a NAME="E7:25_1_1_1"/>
( <font color="Maroon">IT</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font> &amp; <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">IT</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> )
 <b>by </b><i/>;<br/><b>let </b><font color="Maroon">n</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="E8:25_1_1_1"/>
( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font> &amp; <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> )
 ;<br/><b>hence </b><a NAME="E9:25_1_1_1"/>
<font color="Maroon">IT</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:25_1_1"/>
not <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> <a href="glib_001.html#K13" target="_self">.vertices()</a> 
 ;<br/>

<b>set </b><font color="Maroon">IT</font> =  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font>;<br/>
<b>take </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font>
;<br/>

<b>thus </b><a NAME="E3:25_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font>
 ;<br/>


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