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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="glib_000.html#NM2">_Graph</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="glib_001.html#M3" target="_self">Walk</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>





<b>set </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> ;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:80_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="glib_001.html#K14" target="_self">.edges()</a> 
 ;<br/><a NAME="E2:80_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></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">c<sub>2</sub></font> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:80_1"/><i><font color="Green">E80</font></i>: 
( <font color="Maroon">c<sub>5</sub></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">c<sub>2</sub></font> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></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">c<sub>6</sub></font> = 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font>;<br/><b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font> as  <a href="abian.html#V1">even</a> <a href="nat_1.html#NM1">Nat</a> ;<br/><a NAME="E6:80_1"/><i><font color="Green">E81</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>5</sub></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">c<sub>2</sub></font> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span> )
 <b>by </b><i><a class="txt" href="glib_001.html#E4:80_1"><i><font color="Green">E80</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>take </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font>;<br/><a NAME="E7:80_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="glib_001.html#E6:80_1"><i><font color="Green">E81</font></i></a>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><b>hence </b><a NAME="E8:80_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><a NAME="E9:80_1"/>
<font color="Maroon">c<sub>5</sub></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">c<sub>2</sub></font> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span></span>)</span> <a href="nat_1.html#K2">*</a> 2
 <b>by </b><i><a class="txt" href="glib_001.html#E6:80_1"><i><font color="Green">E81</font></i></a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E10:80_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <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> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span></span>)</span> <a href="nat_1.html#K2">*</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><b>hence </b><a NAME="E11:80_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="glib_001.html#D15" target="_self">Def15</a></i>;<br/><b>thus </b><a NAME="E12:80_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="glib_001.html#E4:80_1"><i><font color="Green">E80</font></i></a>, <a class="txt" href="glib_001.html#E6:80_1"><i><font color="Green">E81</font></i></a>, <a class="ref" href="glib_001.html#D15" target="_self">Def15</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:80"/>
ex <font color="Olive">b<sub>1</sub></font> being <a href="abian.html#V1">even</a> <a href="nat_1.html#NM1">Nat</a> st <br/>( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being  <a href="abian.html#V1">even</a> <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:80"/><i><font color="Green">E80</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K3">div</a> 2;<br/>
<a NAME="E5:80"/>
2 <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="pepin.html#T22">PEPIN:22</a></i>;<br/>
<a NAME="E6:80"/><b>then </b><i><font color="Green">E81</font></i>: 
2 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K3">div</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T49">NAT_1:49</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K3">div</a> 2 as   <a href="nat_1.html#NM1">Nat</a> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_2"><b>now </b></a><div class="add"><a NAME="E1:80_2"/>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="glib_001.html#E4:80"><i><font color="Green">E80</font></i></a>, <a class="ref" href="jordan12.html#T3">JORDAN12:3</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E2:80_2"/><b>then </b>
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E3:80_2"/><b>then </b>
2 <a href="nat_1.html#K2">*</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a>, <a class="txt" href="glib_001.html#E6:80"><i><font color="Green">E81</font></i></a></i>;<br/><a NAME="E4:80_2"/><b>then </b><i><font color="Green">E82</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="glib_001.html#E2"><i><font color="Green">Lemma2</font></i></a></i>;<br/><a NAME="E5:80_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="glib_001.html#E4:80"><i><font color="Green">E80</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E6:80_2"/><b>then </b>
2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="glib_001.html#E6:80"><i><font color="Green">E81</font></i></a>, <a class="ref" href="glib_001.html#D15" target="_self">Def15</a></i>;<br/><a NAME="E7:80_2"/><b>then </b>
2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E8:80_2"/><b>then </b><i><font color="Green">E83</font></i>: 
<font color="Maroon">c<sub>7</sub></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">c<sub>2</sub></font> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span>
 <b>by </b><i><a class="txt" href="glib_001.html#E2"><i><font color="Green">Lemma2</font></i></a></i>;<br/><b>hence </b><a NAME="E9:80_2"/>
<font color="Maroon">c<sub>7</sub></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">c<sub>2</sub></font> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span>
 <b>by </b><i><a class="txt" href="glib_001.html#E4:80_2"><i><font color="Green">E82</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>thus </b><a NAME="E10:80_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="glib_001.html#E4:80"><i><font color="Green">E80</font></i></a>, <a class="txt" href="glib_001.html#E6:80"><i><font color="Green">E81</font></i></a>, <a class="txt" href="glib_001.html#E4:80_2"><i><font color="Green">E82</font></i></a>, <a class="txt" href="glib_001.html#E8:80_2"><i><font color="Green">E83</font></i></a>, <a class="ref" href="glib_001.html#D15" target="_self">Def15</a></i>;<br/></div><b>end;</b></div>
<a NAME="E9:80"/><b>then </b>
<font color="Maroon">c<sub>3</sub></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">c<sub>2</sub></font> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E10:80"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="glib_001.html#K14" target="_self">.edges()</a> 
 ;<br/>


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