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

<b>let </b><font color="Maroon" title="c1">G</font> be   <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>;<br/><b>let </b><font color="Maroon" title="c2">P</font> be   <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c1">G</font>;<br/>



<b>assume </b><a NAME="E1:29"/><i><font color="Green" title="E25">A1</font></i>: 
not <font color="Maroon" title="c2">P</font> is <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">e</font>, <font color="Maroon" title="c5">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E2:29"/><i><font color="Green" title="E26">A2</font></i>: 
not <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <b>and </b><br/><a NAME="E3:29"/><i><font color="Green" title="E27">A3</font></i>: 
( <font color="Maroon" title="c5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  &amp; <font color="Maroon" title="c4">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c1">G</font> )
 ;<br/>

<b>set </b><font color="Maroon" title="c6">T</font> = <font color="Maroon" title="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c5">b</font>;<br/>
<b>set </b><font color="Maroon" title="c7">J</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c5">b</font></span>)</span> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c2">P</font>;<br/>
<a NAME="E4:29"/><i><font color="Green" title="E28">A4</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c5">b</font></span>)</span> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> 
 
<b>by </b><i><a class="txt" href="chord.html#E3:29"><i><font color="Green" title="E27">A3</font></i></a>, <a class="ref" href="glib_001.html#T16" title="GLIB_001:th.16">GLIB_001:16</a></i>;<br/>
<a NAME="E5:29"/>
3 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 3
 
<b>by </b><i><a class="ref" href="finseq_1.html#T5" title="FINSEQ_1:th.5">FINSEQ_1:5</a></i>;<br/>
<a NAME="E6:29"/><b>then </b>
3 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c5">b</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="chord.html#E3:29"><i><font color="Green" title="E27">A3</font></i></a>, <a class="ref" href="glib_001.html#T15" title="GLIB_001:th.15">GLIB_001:15</a></i>;<br/>
<a NAME="E7:29"/><b>then </b>
3 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c5">b</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">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" title="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c5">b</font></span>)</span> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c2">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c5">b</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3
 
<b>by </b><i><a class="ref" href="glib_001.html#T33" title="GLIB_001:th.33">GLIB_001:33</a></i>;<br/>
<a NAME="E9:29"/><b>then </b><i><font color="Green" title="E29">A5</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c5">b</font></span>)</span> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c2">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c5">b</font></span>)</span> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> 
 
<b>by </b><i><a class="txt" href="chord.html#E3:29"><i><font color="Green" title="E27">A3</font></i></a>, <a class="ref" href="glib_001.html#T15" title="GLIB_001:th.15">GLIB_001:15</a></i>;<br/>
<a NAME="E10:29"/><b>then </b><i><font color="Green" title="E30">A6</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c5">b</font></span>)</span> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c2">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font>
 
<b>by </b><i><a class="txt" href="chord.html#E3:29"><i><font color="Green" title="E27">A3</font></i></a>, <a class="ref" href="glib_001.html#T16" title="GLIB_001:th.16">GLIB_001:16</a></i>;<br/>
<div><i><font color="Green" title="E31">A7</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" title="c8">m</font>, <font color="Maroon" title="c9">n</font> be  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a> <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:29_1"/><i><font color="Green" title="E32">A8</font></i>: 
( <font color="Maroon" title="c8">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">n</font> &amp; <font color="Maroon" title="c9">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c5">b</font></span>)</span> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c2">P</font></span>)</span> )
 ;<br/><b>assume </b><a NAME="E2:29_1"/><i><font color="Green" title="E33">A9</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c5">b</font></span>)</span> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c2">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c5">b</font></span>)</span> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c2">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font>
 ;<br/><a NAME="E3:29_1"/><i><font color="Green" title="E34">A10</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">m</font> &amp; <font color="Maroon" title="c8">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c5">b</font></span>)</span> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c2">P</font></span>)</span> )
 <b>by </b><i><a class="ref" href="heyting3.html#T1" title="HEYTING3:th.1">HEYTING3:1</a>, <a class="txt" href="chord.html#E1:29_1"><i><font color="Green" title="E32">A8</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E4:29_1"/><b>then </b><i><font color="Green" title="E35">A11</font></i>: 
<font color="Maroon" title="c8">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c5">b</font></span>)</span> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c2">P</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E5:29_1"/>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 0</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">n</font>
 <b>by </b><i><a class="txt" href="chord.html#E3:29_1"><i><font color="Green" title="E34">A10</font></i></a>, <a class="txt" href="chord.html#E1:29_1"><i><font color="Green" title="E32">A8</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E6:29_1"/><b>then </b><i><font color="Green" title="E36">A12</font></i>: 
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n</font>
 <b>by </b><i><a class="ref" href="chord.html#T4" target="_self" title="CHORD:th.4">Th4</a></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" title="c8">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c5">b</font></span>)</span> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c2">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c5">b</font></span>)</span> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c2">P</font></span>)</span> <a href="glib_001.html#K3" title="GLIB_001:func.3">.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" title="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c5">b</font></span>)</span> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c2">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c5">b</font></span>)</span> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> 
 <b>by </b><i><a class="txt" href="chord.html#E4:29"><i><font color="Green" title="E28">A4</font></i></a>, <a class="ref" href="glib_001.html#T31" title="GLIB_001:th.31">GLIB_001:31</a></i>;<br/><a NAME="E4:29_1_1"/><b>then </b><i><font color="Green" title="E37">A13</font></i>: 
not <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c5">b</font></span>)</span> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c2">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <b>by </b><i><a class="txt" href="chord.html#E2:29"><i><font color="Green" title="E26">A2</font></i></a>, <a class="txt" href="chord.html#E3:29"><i><font color="Green" title="E27">A3</font></i></a>, <a class="ref" href="glib_001.html#T16" title="GLIB_001:th.16">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" title="c9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 or <font color="Maroon" title="c9">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 3 )
 </span><b>by </b><i><a class="txt" href="chord.html#E6:29_1"><i><font color="Green" title="E36">A12</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">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" title="c9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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_1"><i><font color="Green" title="E33">A9</font></i></a>, <a class="txt" href="chord.html#E4:29_1_1"><i><font color="Green" title="E37">A13</font></i></a>, <a class="txt" href="chord.html#E3:29"><i><font color="Green" title="E27">A3</font></i></a>, <a class="txt" href="chord.html#E10:29"><i><font color="Green" title="E30">A6</font></i></a>, <a class="ref" href="glib_001.html#T89" title="GLIB_001:th.89">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" title="E38">A14</font></i>: 
<font color="Maroon" title="c9">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 3
 ;<br/><div class="add"><a NAME="E2:29_1_1_1_2"/><b>then </b>
not <font color="Maroon" title="c9">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 3
 <b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E3:29_1_1_1_2"/><b>then </b>
not <font color="Maroon" title="c9">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c5">b</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="chord.html#E3:29"><i><font color="Green" title="E27">A3</font></i></a>, <a class="ref" href="glib_001.html#T15" title="GLIB_001:th.15">GLIB_001:15</a></i>;<br/><a NAME="E4:29_1_1_1_2"/><b>then </b><i><font color="Green" title="E39">A15</font></i>: 
not <font color="Maroon" title="c9">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c5">b</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E5:29_1_1_1_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n</font>
 <b>by </b><i><a class="txt" href="chord.html#E1:29_1_1_1_2"><i><font color="Green" title="E38">A14</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E6:29_1_1_1_2"/><b>then </b><i><font color="Green" title="E40">A16</font></i>: 
<font color="Maroon" title="c9">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c5">b</font></span>)</span> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c2">P</font></span>)</span>
 <b>by </b><i><a class="txt" href="chord.html#E1:29_1"><i><font color="Green" title="E32">A8</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>consider </b><font color="Maroon" title="c10">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E8:29_1_1_1_2"/><i><font color="Green" title="E41">A17</font></i>: 
( <font color="Maroon" title="c10">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Maroon" title="c9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c5">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c10">j</font> )
 <b>by </b><i><a class="txt" href="chord.html#E6:29_1_1_1_2"><i><font color="Green" title="E40">A16</font></i></a>, <a class="txt" href="chord.html#E4:29_1_1_1_2"><i><font color="Green" title="E39">A15</font></i></a>, <a class="ref" href="glib_001.html#T35" title="GLIB_001:th.35">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" title="c10">j</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c11">jj</font> = <font color="Maroon" title="c10">j</font> as  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a> <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> ;<br/><a NAME="E3:29_1_1_1_2_1"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c5">b</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c11">jj</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a>
 ;<br/><b>hence </b><a NAME="E4:29_1_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E8:29_1_1_1_2"><i><font color="Green" title="E41">A17</font></i></a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c11">jj</font> = <font color="Maroon" title="c10">j</font> as  <a href="abian.html#V1" title="ABIAN:attr.1">even</a> <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> ;<br/><b>reconsider </b><font color="Maroon" title="c12">j1</font> = <font color="Maroon" title="c11">jj</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 as  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a> <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> ;<br/><a NAME="E12:29_1_1_1_2"/>
<font color="Maroon" title="c10">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">P</font>
 <b>by </b><i><a class="txt" href="chord.html#E8:29_1_1_1_2"><i><font color="Green" title="E41">A17</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E13:29_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <b>by </b><i><a class="ref" href="glib_001.html#T88" title="GLIB_001:th.88">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_1"><i><font color="Green" title="E33">A9</font></i></a>, <a class="txt" href="chord.html#E4:29_1_1"><i><font color="Green" title="E37">A13</font></i></a>, <a class="txt" href="chord.html#E8:29_1_1_1_2"><i><font color="Green" title="E41">A17</font></i></a>, <a class="txt" href="chord.html#E4:29"><i><font color="Green" title="E28">A4</font></i></a>, <a class="ref" href="glib_001.html#T34" title="GLIB_001:th.34">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" title="NAT_1:func.2">*</a> 0</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">m</font>
 <b>by </b><i><a class="txt" href="chord.html#E3:29_1"><i><font color="Green" title="E34">A10</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E9:29_1"/><b>then </b><i><font color="Green" title="E37">A18</font></i>: 
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">m</font>
 <b>by </b><i><a class="ref" href="chord.html#T4" target="_self" title="CHORD:th.4">Th4</a></i>;<br/><a NAME="E10:29_1"/><b>then </b>
3 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">n</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a>, <a class="txt" href="chord.html#E1:29_1"><i><font color="Green" title="E32">A8</font></i></a></i>;<br/><a NAME="E11:29_1"/><b>then </b>
not <font color="Maroon" title="c9">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 3
 <b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E12:29_1"/><b>then </b>
not <font color="Maroon" title="c9">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c5">b</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="chord.html#E3:29"><i><font color="Green" title="E27">A3</font></i></a>, <a class="ref" href="glib_001.html#T15" title="GLIB_001:th.15">GLIB_001:15</a></i>;<br/><a NAME="E13:29_1"/><b>then </b><i><font color="Green" title="E38">A19</font></i>: 
not <font color="Maroon" title="c9">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c5">b</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E14:29_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 3 &amp; 3 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n</font> )
 <b>by </b><i><a class="txt" href="chord.html#E1:29_1"><i><font color="Green" title="E32">A8</font></i></a>, <a class="txt" href="chord.html#E9:29_1"><i><font color="Green" title="E37">A18</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E15:29_1"/><b>then </b>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E16:29_1"/><b>then </b>
<font color="Maroon" title="c9">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c5">b</font></span>)</span> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c2">P</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="chord.html#E1:29_1"><i><font color="Green" title="E32">A8</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E17:29_1"/><b>then </b><i><font color="Green" title="E39">A20</font></i>: 
<font color="Maroon" title="c9">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c5">b</font></span>)</span> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c2">P</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>consider </b><font color="Maroon" title="c10">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E19:29_1"/><i><font color="Green" title="E40">A21</font></i>: 
( <font color="Maroon" title="c10">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Maroon" title="c9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c5">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c10">j</font> )
 <b>by </b><i><a class="txt" href="chord.html#E17:29_1"><i><font color="Green" title="E39">A20</font></i></a>, <a class="txt" href="chord.html#E13:29_1"><i><font color="Green" title="E38">A19</font></i></a>, <a class="ref" href="glib_001.html#T35" title="GLIB_001:th.35">GLIB_001:35</a></i>;<br/><a NAME="E20:29_1"/><i><font color="Green" title="E41">A22</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c5">b</font></span>)</span> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c2">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="chord.html#E19:29_1"><i><font color="Green" title="E40">A21</font></i></a>, <a class="txt" href="chord.html#E4:29"><i><font color="Green" title="E28">A4</font></i></a>, <a class="ref" href="glib_001.html#T34" title="GLIB_001:th.34">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" title="c10">j</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c11">jj</font> = <font color="Maroon" title="c10">j</font> as  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a> <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> ;<br/><a NAME="E3:29_1_2"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c5">b</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c11">jj</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a>
 ;<br/><b>hence </b><a NAME="E4:29_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E19:29_1"><i><font color="Green" title="E40">A21</font></i></a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c11">jj</font> = <font color="Maroon" title="c10">j</font> as  <a href="abian.html#V1" title="ABIAN:attr.1">even</a> <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> ;<br/><b>reconsider </b><font color="Maroon" title="c12">j1</font> = <font color="Maroon" title="c11">jj</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 as  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a> <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> ;<br/><a NAME="E24:29_1"/><i><font color="Green" title="E42">A23</font></i>: 
<font color="Maroon" title="c12">j1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">P</font>
 <b>by </b><i><a class="txt" href="chord.html#E19:29_1"><i><font color="Green" title="E40">A21</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</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" title="c8">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 ;<br/><a NAME="E2:29_1_3"/><b>then </b><i><font color="Green" title="E43">A24</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c5">b</font></span>)</span> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c2">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 <b>by </b><i><a class="txt" href="chord.html#E3:29"><i><font color="Green" title="E27">A3</font></i></a>, <a class="ref" href="glib_001.html#T16" title="GLIB_001:th.16">GLIB_001:16</a>, <a class="txt" href="chord.html#E9:29"><i><font color="Green" title="E29">A5</font></i></a></i>;<br/><a NAME="E3:29_1_3"/>
( 0 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">j</font> &amp; 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">j</font> )
 <b>by </b><i><a class="txt" href="chord.html#E1:29_1"><i><font color="Green" title="E32">A8</font></i></a>, <a class="txt" href="chord.html#E9:29_1"><i><font color="Green" title="E37">A18</font></i></a>, <a class="txt" href="chord.html#E3:29"><i><font color="Green" title="E27">A3</font></i></a>, <a class="txt" href="chord.html#E19:29_1"><i><font color="Green" title="E40">A21</font></i></a>, <a class="ref" href="glib_001.html#T15" title="GLIB_001:th.15">GLIB_001:15</a>, <a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a></i>;<br/><a NAME="E4:29_1_3"/><b>then </b>
0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">j</font>
 <b>by </b><i><a class="ref" href="real_1.html#D5" title="REAL_1:def.5">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" title="NAT_1:func.2">*</a> 0</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">j1</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><b>hence </b><a NAME="E6:29_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E2:29_1_3"><i><font color="Green" title="E43">A24</font></i></a>, <a class="txt" href="chord.html#E20:29_1"><i><font color="Green" title="E41">A22</font></i></a>, <a class="txt" href="chord.html#E2:29_1"><i><font color="Green" title="E33">A9</font></i></a>, <a class="txt" href="chord.html#E1:29"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="chord.html#E24:29_1"><i><font color="Green" title="E42">A23</font></i></a>, <a class="ref" href="glib_001.html#T148" title="GLIB_001:th.148">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" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">m</font>
 <b>by </b><i><a class="txt" href="chord.html#E9:29_1"><i><font color="Green" title="E37">A18</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E27:29_1"/><b>then </b>
not <font color="Maroon" title="c8">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 3
 <b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E28:29_1"/><b>then </b>
not <font color="Maroon" title="c8">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c5">b</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="chord.html#E3:29"><i><font color="Green" title="E27">A3</font></i></a>, <a class="ref" href="glib_001.html#T15" title="GLIB_001:th.15">GLIB_001:15</a></i>;<br/><a NAME="E29:29_1"/><b>then </b><i><font color="Green" title="E43">A25</font></i>: 
not <font color="Maroon" title="c8">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c5">b</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>consider </b><font color="Maroon" title="c13">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E31:29_1"/><i><font color="Green" title="E44">A26</font></i>: 
( <font color="Maroon" title="c13">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Maroon" title="c8">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c5">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c13">k</font> )
 <b>by </b><i><a class="txt" href="chord.html#E29:29_1"><i><font color="Green" title="E43">A25</font></i></a>, <a class="txt" href="chord.html#E4:29_1"><i><font color="Green" title="E35">A11</font></i></a>, <a class="ref" href="glib_001.html#T35" title="GLIB_001:th.35">GLIB_001:35</a></i>;<br/><a NAME="E32:29_1"/><i><font color="Green" title="E45">A27</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c5">b</font></span>)</span> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c2">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="chord.html#E31:29_1"><i><font color="Green" title="E44">A26</font></i></a>, <a class="txt" href="chord.html#E4:29"><i><font color="Green" title="E28">A4</font></i></a>, <a class="ref" href="glib_001.html#T34" title="GLIB_001:th.34">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" title="c13">k</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c14">kk</font> = <font color="Maroon" title="c13">k</font> as  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a> <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> ;<br/><a NAME="E3:29_1_4"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c5">b</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c14">kk</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a>
 ;<br/><b>hence </b><a NAME="E4:29_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E31:29_1"><i><font color="Green" title="E44">A26</font></i></a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c14">kk</font> = <font color="Maroon" title="c13">k</font> as  <a href="abian.html#V1" title="ABIAN:attr.1">even</a> <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> ;<br/><b>reconsider </b><font color="Maroon" title="c15">k1</font> = <font color="Maroon" title="c14">kk</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 as  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a> <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> ;<br/><a NAME="E36:29_1"/>
<font color="Maroon" title="c13">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">j</font>
 <b>by </b><i><a class="txt" href="chord.html#E31:29_1"><i><font color="Green" title="E44">A26</font></i></a>, <a class="txt" href="chord.html#E19:29_1"><i><font color="Green" title="E40">A21</font></i></a>, <a class="txt" href="chord.html#E1:29_1"><i><font color="Green" title="E32">A8</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><a NAME="E37:29_1"/><b>then </b>
<font color="Maroon" title="c15">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">j1</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><b>hence </b><a NAME="E38:29_1"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E32:29_1"><i><font color="Green" title="E45">A27</font></i></a>, <a class="txt" href="chord.html#E20:29_1"><i><font color="Green" title="E41">A22</font></i></a>, <a class="txt" href="chord.html#E2:29_1"><i><font color="Green" title="E33">A9</font></i></a>, <a class="txt" href="chord.html#E1:29"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="chord.html#E24:29_1"><i><font color="Green" title="E42">A23</font></i></a>, <a class="ref" href="glib_001.html#T148" title="GLIB_001:th.148">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" title="c8">m</font>, <font color="Maroon" title="c9">n</font> be  <a href="abian.html#NV3" title="ABIAN:NV.3">odd</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:29_2"/><i><font color="Green" title="E32">A28</font></i>: 
( <font color="Maroon" title="c8">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c5">b</font></span>)</span> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c2">P</font></span>)</span> &amp; <font color="Maroon" title="c9">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c5">b</font></span>)</span> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c2">P</font></span>)</span> )
 ;<br/><b>assume </b><a NAME="E2:29_2"/><i><font color="Green" title="E33">A29</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c5">b</font></span>)</span> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c2">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c5">b</font></span>)</span> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c2">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font>
 ;<br/><a NAME="E3:29_2"/><i><font color="Green" title="E34">A30</font></i>: 
<font color="Maroon" title="c8">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c9">n</font>
 <b>by </b><i><a class="txt" href="chord.html#E1:29_2"><i><font color="Green" title="E32">A28</font></i></a>, <a class="txt" href="chord.html#E2:29_2"><i><font color="Green" title="E33">A29</font></i></a>, <a class="txt" href="chord.html#E11:29"><i><font color="Green" title="E31">A7</font></i></a></i>;<br/><a NAME="E4:29_2"/>
not <font color="Maroon" title="c9">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">m</font>
 <b>by </b><i><a class="txt" href="chord.html#E1:29_2"><i><font color="Green" title="E32">A28</font></i></a>, <a class="txt" href="chord.html#E11:29"><i><font color="Green" title="E31">A7</font></i></a>, <a class="txt" href="chord.html#E2:29_2"><i><font color="Green" title="E33">A29</font></i></a></i>;<br/><b>hence </b><a NAME="E5:29_2"/>
<font color="Maroon" title="c9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">m</font>
 <b>by </b><i><a class="txt" href="chord.html#E3:29_2"><i><font color="Green" title="E34">A30</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">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" title="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c5">b</font></span>)</span> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c2">P</font> is <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a>
 <b>by </b><i><a class="ref" href="glib_001.html#T147" title="GLIB_001:th.147">GLIB_001:147</a></i>;<br/>


</div>
