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



<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:91_1"/>
<font color="Maroon">W</font> is <a href="glib_001.html#V4" target="_self">Trail-like</a>
 ;<br/><a NAME="E2:91_1"/><b>then </b><i><font color="Green">E39</font></i>: 
<font color="Maroon">W</font> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a>  is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="ref" href="glib_001.html#T1" target="_self">Th1</a></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> , <font color="Maroon">n</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="E3:91_1"/><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#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> &amp; <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/><a NAME="E4:91_1"/><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">W</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E5:91_1"/><b>then </b><i><font color="Green">E42</font></i>: 
( <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">W</font> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span> &amp; <font color="Maroon">W</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">W</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>, </i>;<br/><a NAME="E6:91_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E7:91_1"/><b>then </b><i><font color="Green">E43</font></i>: 
( <font color="Maroon">n</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">W</font> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span> &amp; <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">W</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">n</font> <a href="nat_1.html#K5">div</a> 2</span>)</span> )
 <b>by </b><i>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:91_1_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">m</font> <a href="nat_1.html#K5">div</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K5">div</a> 2
 ;<br/><a NAME="E2:91_1_1"/>
( 2 <a href="nat_1.html#R1">divides</a> <font color="Maroon">m</font> &amp; 2 <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="ref" href="pepin.html#T22">PEPIN:22</a></i>;<br/><a NAME="E3:91_1_1"/><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> &amp; 2 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K5">div</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="ref" href="nat_1.html#T49">NAT_1:49</a></i>;<br/><b>hence </b><a NAME="E4:91_1_1"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E9:91_1"/>
<font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:91"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">m</font>, <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 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> &amp; <font color="Olive">m</font> <a href="xxreal_0.html#NR3" target="_self">&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">W</font> holds <br/><font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Olive">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:91_2"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">x1</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#K12" target="_self">.edgeSeq()</a> </span>)</span> &amp; <font color="Maroon">x2</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#K12" target="_self">.edgeSeq()</a> </span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> )
 ;<br/><b>then reconsider </b><font color="Maroon">m</font> = <font color="Maroon">x1</font>, <font color="Maroon">n</font> = <font color="Maroon">x2</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><a NAME="E3:91_2"/>
( 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> <span class="p1">(<span class="default"><font color="Maroon">W</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="E4:91_2"/><b>then </b><i><font color="Green">E47</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</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">m</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E5:91_2"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</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#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="E6:91_2"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">W</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">m</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">W</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">n</font></span>)</span>
 <b>by </b><i>, , </i>;<br/><a NAME="E7:91_2"/>
( 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">W</font> &amp; 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">W</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E8:91_2"/><b>then </b><i><font color="Green">E49</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font> &amp; 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> &amp; 2 <a href="nat_1.html#K2">*</a> <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> )
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:91_2_1_1"/>
( 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> or 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> or 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> )
 </span><b>by </b><i><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="91_2_1_1_1"><b>suppose </b></a><a NAME="E1:91_2_1_1_1"/>
2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:91_2_1_1_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_2_1_1_2"><b>suppose </b></a><a NAME="E1:91_2_1_1_2"/>
2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:91_2_1_1_2"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_2_1_1_3"><b>suppose </b></a><a NAME="E1:91_2_1_1_3"/>
2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:91_2_1_1_3"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E10:91_2"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 ;<br/></div><b>end;</b></div>
<a NAME="E4:91"/><b>then </b>
<font color="Maroon">W</font> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a>  is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>
<b>hence </b><a NAME="E5:91"/>
<font color="Maroon">W</font> is <a href="glib_001.html#V4" target="_self">Trail-like</a>
 <b>by </b><i><a class="ref" href="glib_001.html#T1" target="_self">Th1</a></i>;<br/>


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