<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">G</font> be   <a href="glib_000.html#NM2">_Graph</a>;<br/><b>let </b><font color="Maroon">P</font> be   <a href="glib_001.html#NM5">Path</a> of <font color="Maroon">G</font>;<br/>



<b>assume </b><a NAME="E1:29"/><i><font color="Green">E30</font></i>: 
not <font color="Maroon">P</font> is <a href="glib_001.html#V1">closed</a>
 ;<br/>

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">e</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">b</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E2:29"/><i><font color="Green">E33</font></i>: 
not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="glib_001.html#K13">.vertices()</a> 
 <b>and </b><br/><a NAME="E3:29"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="glib_001.html#K3">.first()</a>  &amp; <font color="Maroon">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">G</font> )
 ;<br/>

<b>set </b><font color="Maroon">T</font> = <font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">a</font>,<font color="Maroon">e</font>,<font color="Maroon">b</font>;<br/>
<b>set </b><font color="Maroon">J</font> = <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">a</font>,<font color="Maroon">e</font>,<font color="Maroon">b</font></span>)</span> <a href="glib_001.html#K7">.append</a> <font color="Maroon">P</font>;<br/>
<a NAME="E4:29"/><i><font color="Green">E56</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">a</font>,<font color="Maroon">e</font>,<font color="Maroon">b</font></span>)</span> <a href="glib_001.html#K4">.last()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="glib_001.html#K3">.first()</a> 
 
<b>by </b><i>, <a class="ref" href="glib_001.html#T16">GLIB_001:16</a></i>;<br/>
<a NAME="E5:29"/>
3 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 3
 
<b>by </b><i><a class="ref" href="finseq_1.html#T5">FINSEQ_1:5</a></i>;<br/>
<a NAME="E6:29"/><b>then </b>
3 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">a</font>,<font color="Maroon">e</font>,<font color="Maroon">b</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="glib_001.html#T15">GLIB_001:15</a></i>;<br/>
<a NAME="E7:29"/><b>then </b>
3 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">a</font>,<font color="Maroon">e</font>,<font color="Maroon">b</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E8:29"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">a</font>,<font color="Maroon">e</font>,<font color="Maroon">b</font></span>)</span> <a href="glib_001.html#K7">.append</a> <font color="Maroon">P</font></span>)</span> <a href="funct_1.html#K1">.</a> 3 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">a</font>,<font color="Maroon">e</font>,<font color="Maroon">b</font></span>)</span> <a href="funct_1.html#K1">.</a> 3
 
<b>by </b><i><a class="ref" href="glib_001.html#T33">GLIB_001:33</a></i>;<br/>
<a NAME="E9:29"/><b>then </b><i><font color="Green">E59</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">a</font>,<font color="Maroon">e</font>,<font color="Maroon">b</font></span>)</span> <a href="glib_001.html#K7">.append</a> <font color="Maroon">P</font></span>)</span> <a href="funct_1.html#K1">.</a> 3 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">a</font>,<font color="Maroon">e</font>,<font color="Maroon">b</font></span>)</span> <a href="glib_001.html#K4">.last()</a> 
 
<b>by </b><i>, <a class="ref" href="glib_001.html#T15">GLIB_001:15</a></i>;<br/>
<a NAME="E10:29"/><b>then </b><i><font color="Green">E66</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">a</font>,<font color="Maroon">e</font>,<font color="Maroon">b</font></span>)</span> <a href="glib_001.html#K7">.append</a> <font color="Maroon">P</font></span>)</span> <a href="funct_1.html#K1">.</a> 3 <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 
<b>by </b><i>, <a class="ref" href="glib_001.html#T16">GLIB_001:16</a></i>;<br/>
<div><i><font color="Green">E67</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">m</font> be  <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">n</font> be  <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:29_1"/><i><font color="Green">E68</font></i>: 
( <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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">a</font>,<font color="Maroon">e</font>,<font color="Maroon">b</font></span>)</span> <a href="glib_001.html#K7">.append</a> <font color="Maroon">P</font></span>)</span> )
 ;<br/><b>assume </b><a NAME="E2:29_1"/><i><font color="Green">E69</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">a</font>,<font color="Maroon">e</font>,<font color="Maroon">b</font></span>)</span> <a href="glib_001.html#K7">.append</a> <font color="Maroon">P</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">a</font>,<font color="Maroon">e</font>,<font color="Maroon">b</font></span>)</span> <a href="glib_001.html#K7">.append</a> <font color="Maroon">P</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 ;<br/><a NAME="E3:29_1"/><i><font color="Green">E70</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>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">a</font>,<font color="Maroon">e</font>,<font color="Maroon">b</font></span>)</span> <a href="glib_001.html#K7">.append</a> <font color="Maroon">P</font></span>)</span> )
 <b>by </b><i><a class="ref" href="heyting3.html#T1">HEYTING3:1</a>, , <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E4:29_1"/><b>then </b><i><font color="Green">E71</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">a</font>,<font color="Maroon">e</font>,<font color="Maroon">b</font></span>)</span> <a href="glib_001.html#K7">.append</a> <font color="Maroon">P</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E5:29_1"/>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="chord.html#E3"><i><font color="Green">Lemma34</font></i></a>, , <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E6:29_1"/><b>then </b><i><font color="Green">E72</font></i>: 
1 <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:29_1_1"/>
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><a NAME="E2:29_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">a</font>,<font color="Maroon">e</font>,<font color="Maroon">b</font></span>)</span> <a href="glib_001.html#K7">.append</a> <font color="Maroon">P</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">a</font>,<font color="Maroon">e</font>,<font color="Maroon">b</font></span>)</span> <a href="glib_001.html#K7">.append</a> <font color="Maroon">P</font></span>)</span> <a href="glib_001.html#K3">.first()</a> 
 ;<br/><a NAME="E3:29_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">a</font>,<font color="Maroon">e</font>,<font color="Maroon">b</font></span>)</span> <a href="glib_001.html#K7">.append</a> <font color="Maroon">P</font></span>)</span> <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">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">a</font>,<font color="Maroon">e</font>,<font color="Maroon">b</font></span>)</span> <a href="glib_001.html#K3">.first()</a> 
 <b>by </b><i><a class="ref" href="chord.html#T1" target="_self">Th1</a>, <a class="ref" href="glib_001.html#T31">GLIB_001:31</a></i>;<br/><a NAME="E4:29_1_1"/><b>then </b><i><font color="Green">E73</font></i>: 
not <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">a</font>,<font color="Maroon">e</font>,<font color="Maroon">b</font></span>)</span> <a href="glib_001.html#K7">.append</a> <font color="Maroon">P</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="glib_001.html#K13">.vertices()</a> 
 <b>by </b><i>, , <a class="ref" href="glib_001.html#T16">GLIB_001:16</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_1_1_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 3 or <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 3 )
 </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="29_1_1_1_1"><b>suppose </b></a><a NAME="E1:29_1_1_1_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 3
 ;<br/><div class="add"><b>hence </b><a NAME="E2:29_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E2:29"><i><font color="Green">E33</font></i></a>, , , <a class="txt" href="chord.html#E1:29"><i><font color="Green">E30</font></i></a>, <a class="ref" href="glib_001.html#T89">GLIB_001:89</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_1_2"><b>suppose </b></a><a NAME="E1:29_1_1_1_2"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 3
 ;<br/><div class="add"><a NAME="E2:29_1_1_1_2"/><b>then </b>
not <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 3
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E3:29_1_1_1_2"/><b>then </b>
not <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">a</font>,<font color="Maroon">e</font>,<font color="Maroon">b</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="glib_001.html#T15">GLIB_001:15</a></i>;<br/><a NAME="E4:29_1_1_1_2"/><b>then </b><i><font color="Green">E75</font></i>: 
not <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">a</font>,<font color="Maroon">e</font>,<font color="Maroon">b</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E5:29_1_1_1_2"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="chord.html#T3" target="_self">Th3</a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E6:29_1_1_1_2"/><b>then </b><i><font color="Green">E76</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">a</font>,<font color="Maroon">e</font>,<font color="Maroon">b</font></span>)</span> <a href="glib_001.html#K7">.append</a> <font color="Maroon">P</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>consider </b><font color="Maroon">j</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="E8:29_1_1_1_2"/><i><font color="Green">E80</font></i>: 
( <font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> &amp; <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">a</font>,<font color="Maroon">e</font>,<font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font> )
 <b>by </b><i><a class="ref" href="chord.html#T4" target="_self">Th4</a>, , <a class="ref" href="glib_001.html#T35">GLIB_001:35</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:29_1_1_1_2_1"/>
not <font color="Maroon">j</font> is <a href="abian.html#V1">even</a>
 ;<br/><b>then reconsider </b><font color="Maroon">jj</font> = <font color="Maroon">j</font> as  <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a> ;<br/><a NAME="E3:29_1_1_1_2_1"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">a</font>,<font color="Maroon">e</font>,<font color="Maroon">b</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">jj</font> is <a href="abian.html#V1">even</a>
 ;<br/><b>hence </b><a NAME="E4:29_1_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="ref" href="chord.html#T5" target="_self">Th5</a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">jj</font> = <font color="Maroon">j</font> as  <a href="abian.html#V1">even</a> <a href="nat_1.html#NM1">Nat</a> ;<br/><b>reconsider </b><font color="Maroon">j1</font> = <font color="Maroon">jj</font> <a href="xcmplx_0.html#K2">+</a> 1 as  <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a> ;<br/><a NAME="E12:29_1_1_1_2"/>
<font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="chord.html#T5" target="_self">Th5</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E13:29_1_1_1_2"/><b>then </b>
<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="glib_001.html#K13">.vertices()</a> 
 <b>by </b><i><a class="ref" href="glib_001.html#T88">GLIB_001:88</a></i>;<br/><b>hence </b><a NAME="E14:29_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E2:29"><i><font color="Green">E33</font></i></a>, , <a class="ref" href="chord.html#T5" target="_self">Th5</a>, <a class="ref" href="chord.html#T1" target="_self">Th1</a>, <a class="ref" href="glib_001.html#T34">GLIB_001:34</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E8:29_1"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="txt" href="chord.html#E3"><i><font color="Green">Lemma34</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E9:29_1"/><b>then </b><i><font color="Green">E81</font></i>: 
1 <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 <b>by </b><i/>;<br/><a NAME="E10:29_1"/><b>then </b>
3 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T2">XREAL_1:2</a>, </i>;<br/><a NAME="E11:29_1"/><b>then </b>
not <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 3
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E12:29_1"/><b>then </b>
not <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">a</font>,<font color="Maroon">e</font>,<font color="Maroon">b</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="glib_001.html#T15">GLIB_001:15</a></i>;<br/><a NAME="E13:29_1"/><b>then </b><i><font color="Green">E82</font></i>: 
not <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">a</font>,<font color="Maroon">e</font>,<font color="Maroon">b</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E14:29_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> 3 &amp; 3 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> )
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E15:29_1"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E16:29_1"/><b>then </b>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">a</font>,<font color="Maroon">e</font>,<font color="Maroon">b</font></span>)</span> <a href="glib_001.html#K7">.append</a> <font color="Maroon">P</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E17:29_1"/><b>then </b><i><font color="Green">E83</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">a</font>,<font color="Maroon">e</font>,<font color="Maroon">b</font></span>)</span> <a href="glib_001.html#K7">.append</a> <font color="Maroon">P</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>consider </b><font color="Maroon">j</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="E19:29_1"/><i><font color="Green">E84</font></i>: 
( <font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> &amp; <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">a</font>,<font color="Maroon">e</font>,<font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font> )
 <b>by </b><i>, <a class="ref" href="chord.html#T3" target="_self">Th3</a>, <a class="ref" href="glib_001.html#T35">GLIB_001:35</a></i>;<br/><a NAME="E20:29_1"/><i><font color="Green">E85</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">a</font>,<font color="Maroon">e</font>,<font color="Maroon">b</font></span>)</span> <a href="glib_001.html#K7">.append</a> <font color="Maroon">P</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="chord.html#T4" target="_self">Th4</a>, <a class="ref" href="chord.html#T1" target="_self">Th1</a>, <a class="ref" href="glib_001.html#T34">GLIB_001:34</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:29_1_2"/>
not <font color="Maroon">j</font> is <a href="abian.html#V1">even</a>
 ;<br/><b>then reconsider </b><font color="Maroon">jj</font> = <font color="Maroon">j</font> as  <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a> ;<br/><a NAME="E3:29_1_2"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">a</font>,<font color="Maroon">e</font>,<font color="Maroon">b</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">jj</font> is <a href="abian.html#V1">even</a>
 ;<br/><b>hence </b><a NAME="E4:29_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="chord.html#T4" target="_self">Th4</a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">jj</font> = <font color="Maroon">j</font> as  <a href="abian.html#V1">even</a> <a href="nat_1.html#NM1">Nat</a> ;<br/><b>reconsider </b><font color="Maroon">j1</font> = <font color="Maroon">jj</font> <a href="xcmplx_0.html#K2">+</a> 1 as  <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a> ;<br/><a NAME="E24:29_1"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">j1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="chord.html#T4" target="_self">Th4</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:29_1_3"/>
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 3
 ;<br/><a NAME="E2:29_1_3"/><b>then </b><i><font color="Green">E112</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">a</font>,<font color="Maroon">e</font>,<font color="Maroon">b</font></span>)</span> <a href="glib_001.html#K7">.append</a> <font color="Maroon">P</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> 1
 <b>by </b><i>, <a class="ref" href="glib_001.html#T16">GLIB_001:16</a>, </i>;<br/><a NAME="E3:29_1_3"/>
( 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">j</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> )
 <b>by </b><i>, , , <a class="ref" href="chord.html#T4" target="_self">Th4</a>, <a class="ref" href="glib_001.html#T15">GLIB_001:15</a>, <a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/><a NAME="E4:29_1_3"/><b>then </b>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E5:29_1_3"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j1</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><b>hence </b><a NAME="E6:29_1_3"/>
contradiction
 <b>by </b><i><a class="ref" href="chord.html#T7" target="_self">Th7</a>, <a class="ref" href="chord.html#T5" target="_self">Th5</a>, <a class="txt" href="chord.html#E2:29"><i><font color="Green">E33</font></i></a>, , <a class="ref" href="chord.html#T6" target="_self">Th6</a>, <a class="ref" href="glib_001.html#T148">GLIB_001:148</a></i>;<br/></div><b>end;</b></div><a NAME="E26:29_1"/><b>then </b>
3 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E27:29_1"/><b>then </b>
not <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 3
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E28:29_1"/><b>then </b>
not <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">a</font>,<font color="Maroon">e</font>,<font color="Maroon">b</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="glib_001.html#T15">GLIB_001:15</a></i>;<br/><a NAME="E29:29_1"/><b>then </b><i><font color="Green">E113</font></i>: 
not <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">a</font>,<font color="Maroon">e</font>,<font color="Maroon">b</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>consider </b><font color="Maroon">k</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="E31:29_1"/><i><font color="Green">E114</font></i>: 
( <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> &amp; <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">a</font>,<font color="Maroon">e</font>,<font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> )
 <b>by </b><i><a class="ref" href="chord.html#T7" target="_self">Th7</a>, , <a class="ref" href="glib_001.html#T35">GLIB_001:35</a></i>;<br/><a NAME="E32:29_1"/><i><font color="Green">E115</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">a</font>,<font color="Maroon">e</font>,<font color="Maroon">b</font></span>)</span> <a href="glib_001.html#K7">.append</a> <font color="Maroon">P</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="chord.html#T8" target="_self">Th8</a>, <a class="ref" href="chord.html#T1" target="_self">Th1</a>, <a class="ref" href="glib_001.html#T34">GLIB_001:34</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:29_1_4"/>
not <font color="Maroon">k</font> is <a href="abian.html#V1">even</a>
 ;<br/><b>then reconsider </b><font color="Maroon">kk</font> = <font color="Maroon">k</font> as  <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a> ;<br/><a NAME="E3:29_1_4"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">a</font>,<font color="Maroon">e</font>,<font color="Maroon">b</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">kk</font> is <a href="abian.html#V1">even</a>
 ;<br/><b>hence </b><a NAME="E4:29_1_4"/>
contradiction
 <b>by </b><i><a class="ref" href="chord.html#T8" target="_self">Th8</a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">kk</font> = <font color="Maroon">k</font> as  <a href="abian.html#V1">even</a> <a href="nat_1.html#NM1">Nat</a> ;<br/><b>reconsider </b><font color="Maroon">k1</font> = <font color="Maroon">kk</font> <a href="xcmplx_0.html#K2">+</a> 1 as  <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a> ;<br/><a NAME="E36:29_1"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="chord.html#T8" target="_self">Th8</a>, <a class="ref" href="chord.html#T4" target="_self">Th4</a>, , <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E37:29_1"/><b>then </b>
<font color="Maroon">k1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j1</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><b>hence </b><a NAME="E38:29_1"/>
contradiction
 <b>by </b><i><a class="ref" href="chord.html#T10" target="_self">Th10</a>, <a class="ref" href="chord.html#T5" target="_self">Th5</a>, <a class="txt" href="chord.html#E2:29"><i><font color="Green">E33</font></i></a>, , <a class="ref" href="chord.html#T6" target="_self">Th6</a>, <a class="ref" href="glib_001.html#T148">GLIB_001:148</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2"><b>now </b></a><div class="add"><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> , <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="E1:29_2"/><i><font color="Green">E117</font></i>: 
( <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"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">a</font>,<font color="Maroon">e</font>,<font color="Maroon">b</font></span>)</span> <a href="glib_001.html#K7">.append</a> <font color="Maroon">P</font></span>)</span> &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"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">a</font>,<font color="Maroon">e</font>,<font color="Maroon">b</font></span>)</span> <a href="glib_001.html#K7">.append</a> <font color="Maroon">P</font></span>)</span> )
 ;<br/><b>assume </b><a NAME="E2:29_2"/><i><font color="Green">E118</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">a</font>,<font color="Maroon">e</font>,<font color="Maroon">b</font></span>)</span> <a href="glib_001.html#K7">.append</a> <font color="Maroon">P</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">a</font>,<font color="Maroon">e</font>,<font color="Maroon">b</font></span>)</span> <a href="glib_001.html#K7">.append</a> <font color="Maroon">P</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 ;<br/><a NAME="E3:29_2"/><i><font color="Green">E119</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="txt" href="chord.html#E2:29"><i><font color="Green">E33</font></i></a>, <a class="ref" href="chord.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E4:29_2"/>
not <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="chord.html#T2" target="_self">Th2</a>, <a class="txt" href="chord.html#E2:29"><i><font color="Green">E33</font></i></a></i>;<br/><b>hence </b><a NAME="E5:29_2"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="txt" href="chord.html#E3"><i><font color="Green">Lemma34</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E13:29"/>
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">a</font>,<font color="Maroon">e</font>,<font color="Maroon">b</font></span>)</span> <a href="glib_001.html#K7">.append</a> <font color="Maroon">P</font> is <a href="glib_001.html#V5">Path-like</a>
 <b>by </b><i><a class="ref" href="glib_001.html#T147">GLIB_001:147</a></i>;<br/>


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