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



<b>assume </b><a NAME="E1:25"/><i><font color="Green" title="E22">A1</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c4">W</font> be    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G</font>;<br/>

<b>assume </b><a NAME="E2:25"/><i><font color="Green" title="E23">A2</font></i>: 
<font color="Maroon" title="c4">W</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 ;<br/>

<a NAME="E3:25"/><i><font color="Green" title="E24">A3</font></i>: 
<font color="Maroon" title="c4">W</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">W</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#T89" title="GLIB_001:th.89">GLIB_001:89</a></i>;<br/>
<div><i><font color="Green" title="E25">A4</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:25_1"/><i><font color="Green" title="E26">A5</font></i>: 
<font color="Maroon" title="c4">W</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font>
 ;<br/><b>let </b><font color="Maroon" title="c6">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E2:25_1"/><i><font color="Green" title="E27">A6</font></i>: 
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/><a NAME="E3:25_1"/><i><font color="Green" title="E28">A7</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>set </b><font color="Maroon" title="c7">k</font> = <font color="Maroon" title="c4">W</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <font color="Maroon" title="c6">y</font>;<br/><a NAME="E4:25_1"/><i><font color="Green" title="E29">A8</font></i>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> or <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> )
 <b>by </b><i><a class="txt" href="chord.html#E2:25"><i><font color="Green" title="E23">A2</font></i></a>, <a class="txt" href="chord.html#E3:25"><i><font color="Green" title="E24">A3</font></i></a>, <a class="txt" href="chord.html#E1:25_1"><i><font color="Green" title="E26">A5</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><a NAME="E5:25_1"/><i><font color="Green" title="E30">A9</font></i>: 
<font color="Maroon" title="c4">W</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <font color="Maroon" title="c6">y</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="c4">W</font>
 <b>by </b><i><a class="txt" href="chord.html#E2:25"><i><font color="Green" title="E23">A2</font></i></a>, <a class="txt" href="chord.html#E2:25_1"><i><font color="Green" title="E27">A6</font></i></a>, <a class="ref" href="glib_001.html#D19" title="GLIB_001:def.19">GLIB_001:def 19</a></i>;<br/><a NAME="E6:25_1"/><i><font color="Green" title="E31">A10</font></i>: 
<font color="Maroon" title="c4">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">W</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <font color="Maroon" title="c6">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y</font>
 <b>by </b><i><a class="txt" href="chord.html#E2:25"><i><font color="Green" title="E23">A2</font></i></a>, <a class="txt" href="chord.html#E2:25_1"><i><font color="Green" title="E27">A6</font></i></a>, <a class="ref" href="glib_001.html#D19" title="GLIB_001:def.19">GLIB_001:def 19</a></i>;<br/><a NAME="E7:25_1"/><b>then </b>
<font color="Maroon" title="c4">W</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <font color="Maroon" title="c6">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 <b>by </b><i><a class="txt" href="chord.html#E1:25_1"><i><font color="Green" title="E26">A5</font></i></a>, <a class="txt" href="chord.html#E2:25_1"><i><font color="Green" title="E27">A6</font></i></a>, <a class="txt" href="chord.html#E3:25_1"><i><font color="Green" title="E28">A7</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><b>then consider </b><font color="Maroon" title="c8">m</font> being  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a> <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E9:25_1"/><i><font color="Green" title="E32">A11</font></i>: 
<font color="Maroon" title="c8">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">W</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <font color="Maroon" title="c6">y</font>
 <b>by </b><i><a class="ref" href="chord.html#T5" target="_self" title="CHORD:th.5">Th5</a></i>;<br/><a NAME="E10:25_1"/><i><font color="Green" title="E33">A12</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="c4">W</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <font color="Maroon" title="c6">y</font>
 <b>by </b><i><a class="txt" href="chord.html#E9:25_1"><i><font color="Green" title="E32">A11</font></i></a>, <a class="ref" href="nat_1.html#T16" title="NAT_1:th.16">NAT_1:16</a></i>;<br/><a NAME="E11:25_1"/><b>then </b><i><font color="Green" title="E34">A13</font></i>: 
<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> <font color="Maroon" title="c4">W</font>
 <b>by </b><i><a class="txt" href="chord.html#E5:25_1"><i><font color="Green" title="E30">A9</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E12:25_1"/><i><font color="Green" title="E35">A14</font></i>: 
<font color="Maroon" title="c8">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E13:25_1"/><b>then </b>
<font color="Maroon" title="c4">W</font> <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> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <b>by </b><i><a class="txt" href="chord.html#E2:25"><i><font color="Green" title="E23">A2</font></i></a>, <a class="txt" href="chord.html#E11:25_1"><i><font color="Green" title="E34">A13</font></i></a>, <a class="ref" href="glib_001.html#T88" title="GLIB_001:th.88">GLIB_001:88</a></i>;<br/><a NAME="E14:25_1"/><b>then </b><i><font color="Green" title="E36">A15</font></i>: 
( <font color="Maroon" title="c4">W</font> <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">a</font> or <font color="Maroon" title="c4">W</font> <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="c3">b</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><a NAME="E15:25_1"/><i><font color="Green" title="E37">A16</font></i>: 
( <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> or <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> )
 <b>by </b><i><a class="txt" href="chord.html#E2:25_1"><i><font color="Green" title="E27">A6</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><a NAME="E16:25_1"/>
<font color="Maroon" title="c4">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c4">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">m</font>,<font color="Maroon" title="c4">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">W</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <font color="Maroon" title="c6">y</font></span>)</span>,<font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="chord.html#E9:25_1"><i><font color="Green" title="E32">A11</font></i></a>, <a class="txt" href="chord.html#E11:25_1"><i><font color="Green" title="E34">A13</font></i></a>, <a class="txt" href="chord.html#E12:25_1"><i><font color="Green" title="E35">A14</font></i></a>, <a class="ref" href="glib_001.html#D3" title="GLIB_001:def.3">GLIB_001:def 3</a></i>;<br/><b>hence </b><a NAME="E17:25_1"/>
 ex <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Olive" title="b1">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="chord.html#E2:25"><i><font color="Green" title="E23">A2</font></i></a>, <a class="txt" href="chord.html#E2:25_1"><i><font color="Green" title="E27">A6</font></i></a>, <a class="txt" href="chord.html#E3:25_1"><i><font color="Green" title="E28">A7</font></i></a>, <a class="txt" href="chord.html#E4:25_1"><i><font color="Green" title="E29">A8</font></i></a>, <a class="txt" href="chord.html#E6:25_1"><i><font color="Green" title="E31">A10</font></i></a>, <a class="txt" href="chord.html#E10:25_1"><i><font color="Green" title="E33">A12</font></i></a>, <a class="txt" href="chord.html#E11:25_1"><i><font color="Green" title="E34">A13</font></i></a>, <a class="txt" href="chord.html#E12:25_1"><i><font color="Green" title="E35">A14</font></i></a>, <a class="txt" href="chord.html#E14:25_1"><i><font color="Green" title="E36">A15</font></i></a>, <a class="txt" href="chord.html#E15:25_1"><i><font color="Green" title="E37">A16</font></i></a>, <a class="ref" href="glib_001.html#D19" title="GLIB_001:def.19">GLIB_001:def 19</a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:25_2"/>
( <font color="Maroon" title="c4">W</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> or <font color="Maroon" title="c4">W</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> )
 </span><b>by </b><i><a class="txt" href="chord.html#E2:25"><i><font color="Green" title="E23">A2</font></i></a>, <a class="txt" href="chord.html#E3:25"><i><font color="Green" title="E24">A3</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_2_1"><b>suppose </b></a><a NAME="E1:25_2_1"/><i><font color="Green" title="E26">A17</font></i>: 
<font color="Maroon" title="c4">W</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font>
 ;<br/><div class="add"><a NAME="E2:25_2_1"/>
<font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E3:25_2_1"/><b>then </b>
<font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="chord.html#E1:25"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="zfmisc_1.html#T23" title="ZFMISC_1:th.23">ZFMISC_1:23</a></i>;<br/><b>hence </b><a NAME="E4:25_2_1"/>
 ex <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Olive" title="b1">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="chord.html#E4:25"><i><font color="Green" title="E25">A4</font></i></a>, <a class="txt" href="chord.html#E1:25_2_1"><i><font color="Green" title="E26">A17</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_2_2"><b>suppose </b></a><a NAME="E1:25_2_2"/><i><font color="Green" title="E26">A18</font></i>: 
<font color="Maroon" title="c4">W</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font>
 ;<br/><div class="add"><a NAME="E2:25_2_2"/>
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E3:25_2_2"/><b>then </b>
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="chord.html#E1:25"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="zfmisc_1.html#T23" title="ZFMISC_1:th.23">ZFMISC_1:23</a></i>;<br/><b>then consider </b><font color="Maroon" title="c5">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:25_2_2"/><i><font color="Green" title="E27">A19</font></i>: 
<font color="Maroon" title="c5">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="chord.html#E4:25"><i><font color="Green" title="E25">A4</font></i></a>, <a class="txt" href="chord.html#E1:25_2_2"><i><font color="Green" title="E26">A18</font></i></a></i>;<br/><a NAME="E6:25_2_2"/>
<font color="Maroon" title="c5">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="chord.html#E5:25_2_2"><i><font color="Green" title="E27">A19</font></i></a>, <a class="ref" href="glib_000.html#T17" title="GLIB_000:th.17">GLIB_000:17</a></i>;<br/><b>hence </b><a NAME="E7:25_2_2"/>
 ex <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Olive" title="b1">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c1">G</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

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