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

<b>set </b><font color="Maroon">V</font> = <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span>;<br/>
<b>set </b><font color="Maroon">E</font> = <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>;<br/>
<b>set </b><font color="Maroon">S</font> = 0 <a href="cqc_lang.html#K3">.--&gt;</a> 0;<br/>
<b>set </b><font color="Maroon">T</font> = 0 <a href="cqc_lang.html#K3">.--&gt;</a> 1;<br/>
<a NAME="E1:13_2_1"/><i><font color="Green">E41</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">0 <a href="cqc_lang.html#K3">.--&gt;</a> 0</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">0 <a href="cqc_lang.html#K3">.--&gt;</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:13_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>
 ;<br/><a NAME="E2:13_2_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:13_2_1_1"/><b>then </b>
<span class="p1">(<span class="default">0 <a href="cqc_lang.html#K3">.--&gt;</a> 0</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>;<br/><b>hence </b><a NAME="E4:13_2_1_1"/>
<span class="p1">(<span class="default">0 <a href="cqc_lang.html#K3">.--&gt;</a> 0</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">S</font> = 0 <a href="cqc_lang.html#K3">.--&gt;</a> 0 as   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>,<span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span> <b>by </b><i>, <a class="ref" href="funct_2.html#T5">FUNCT_2:5</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:13_2_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>
 ;<br/><a NAME="E2:13_2_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:13_2_1_2"/><b>then </b>
<span class="p1">(<span class="default">0 <a href="cqc_lang.html#K3">.--&gt;</a> 1</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>;<br/><b>hence </b><a NAME="E4:13_2_1_2"/>
<span class="p1">(<span class="default">0 <a href="cqc_lang.html#K3">.--&gt;</a> 1</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">T</font> = 0 <a href="cqc_lang.html#K3">.--&gt;</a> 1 as   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>,<span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span> <b>by </b><i>, <a class="ref" href="funct_2.html#T5">FUNCT_2:5</a></i>;<br/>
<b>set </b><font color="Maroon">G</font> =  <a href="glib_000.html#K12">createGraph</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span>,<span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">S</font>,<font color="Maroon">T</font>;<br/>
<a NAME="E6:13_2_1"/><i><font color="Green">E42</font></i>: 
(  <a href="glib_000.html#K6">the_Vertices_of</a> <span class="p1">(<span class="default"><a href="glib_000.html#K12">createGraph</a> <span class="p2"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span> &amp;  <a href="glib_000.html#K7">the_Edges_of</a> <span class="p1">(<span class="default"><a href="glib_000.html#K12">createGraph</a> <span class="p2"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> &amp;  <a href="glib_000.html#K10">the_Source_of</a> <span class="p1">(<span class="default"><a href="glib_000.html#K12">createGraph</a> <span class="p2"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="funct_2.html#R1">=</a> <font color="Maroon">S</font> &amp;  <a href="glib_000.html#K11">the_Target_of</a> <span class="p1">(<span class="default"><a href="glib_000.html#K12">createGraph</a> <span class="p2"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="funct_2.html#R1">=</a> <font color="Maroon">T</font> )
 
<b>by </b><i><a class="ref" href="glib_000.html#T8">GLIB_000:8</a></i>;<br/>
<b>take </b>
 <a href="glib_000.html#K12">createGraph</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span>,<span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">S</font>,<font color="Maroon">T</font>
;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_2_1_3"/>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <span class="p2">(<span class="default"><a href="glib_000.html#K12">createGraph</a> <span class="p3"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span>,<span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 ;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:13_2_1_3"/><i><font color="Green">E43</font></i>: 
 <a href="glib_000.html#K6">the_Vertices_of</a> <span class="p1">(<span class="default"><a href="glib_000.html#K12">createGraph</a> <span class="p2"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="card_2.html#T60">CARD_2:60</a></i>;<br/><b>thus </b><a NAME="E4:13_2_1_3"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T9">ZFMISC_1:9</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:13_2_1"/>
( not  <a href="glib_000.html#K12">createGraph</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span>,<span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">S</font>,<font color="Maroon">T</font> is <a href="glib_000.html#V4">trivial</a> &amp;  <a href="glib_000.html#K12">createGraph</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span>,<span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">S</font>,<font color="Maroon">T</font> is <a href="glib_000.html#V2">finite</a> )
 <b>by </b><i><a class="ref" href="glib_000.html#D21">GLIB_000:def 21</a></i>;<br/>

<b>reconsider </b><font color="Maroon">a0</font> = 0, <font color="Maroon">a1</font> = 1 as   <a href="glib_000.html#NM10">Vertex</a> of <span class="p1">(<span class="default"><a href="glib_000.html#K12">createGraph</a> <span class="p2"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E10:13_2_1"/><i><font color="Green">E44</font></i>: 
0 <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K7">the_Edges_of</a> <span class="p1">(<span class="default"><a href="glib_000.html#K12">createGraph</a> <span class="p2"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E11:13_2_1"/><i><font color="Green">E59</font></i>: 
( <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p2">(<span class="default"><a href="glib_000.html#K12">createGraph</a> <span class="p3"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span>,<span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> 0 &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p2">(<span class="default"><a href="glib_000.html#K12">createGraph</a> <span class="p3"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span>,<span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> 1 )
 
<b>by </b><i>, <a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>;<br/>
<a NAME="E12:13_2_1"/><b>then </b><i><font color="Green">E60</font></i>: 
0 <a href="glib_000.html#R1">Joins</a> 0,1, <a href="glib_000.html#K12">createGraph</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span>,<span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">S</font>,<font color="Maroon">T</font>
 
<b>by </b><i>, <a class="ref" href="glib_000.html#D15">GLIB_000:def 15</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_1_4"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">W</font> being    <a href="glib_001.html#M3">Walk</a> of  <a href="glib_000.html#K12">createGraph</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span>,<span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">S</font>,<font color="Maroon">T</font><b> such that </b><a NAME="E1:13_2_1_4"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">W</font> is <a href="glib_001.html#V8">Cycle-like</a>
 ;<br/><a NAME="E2:13_2_1_4"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">W</font> is <a href="glib_001.html#V1">closed</a> &amp; not <font color="Maroon">W</font> is <a href="glib_001.html#V3">trivial</a> &amp; <font color="Maroon">W</font> is <a href="glib_001.html#V5">Path-like</a> )
 <b>by </b><i>, <a class="ref" href="glib_001.html#D31">GLIB_001:def 31</a></i>;<br/><a NAME="E3:13_2_1_4"/><b>then </b><i><font color="Green">E64</font></i>: 
<font color="Maroon">W</font> is <a href="glib_001.html#V4">Trail-like</a>
 <b>by </b><i><a class="ref" href="glib_001.html#D28">GLIB_001:def 28</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_1_4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_2_1_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:13_2_1_4_1_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font> <a href="hidden.html#R1">=</a> 3 or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 3 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_2_1_4_1_1_1"><b>suppose </b></a><a NAME="E1:13_2_1_4_1_1_1"/><i><font color="Green">E65</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font> <a href="hidden.html#R1">=</a> 3
 ;<br/><div class="add"><a NAME="E2:13_2_1_4_1_1_1"/><b>then </b><i><font color="Green">E66</font></i>: 
<font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> 1,<font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 2</span>)</span>, <a href="glib_000.html#K12">createGraph</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span>,<span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">S</font>,<font color="Maroon">T</font>
 <b>by </b><i><a class="ref" href="glib_001.html#D3">GLIB_001:def 3</a>, <a class="ref" href="jordan12.html#T3">JORDAN12:3</a></i>;<br/><b>set </b><font color="Maroon">e</font> = <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><b>set </b><font color="Maroon">v1</font> = <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> 1;<br/><b>set </b><font color="Maroon">v2</font> = <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 2</span>)</span>;<br/><a NAME="E3:13_2_1_4_1_1_1"/>
<font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 2</span>)</span>
 <b>by </b><i>, , <a class="ref" href="glib_001.html#T119">GLIB_001:119</a></i>;<br/><a NAME="E4:13_2_1_4_1_1_1"/><b>then </b>
( <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K7">the_Edges_of</a> <span class="p1">(<span class="default"><a href="glib_000.html#K12">createGraph</a> <span class="p2"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> &amp; ( ( <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p2">(<span class="default"><a href="glib_000.html#K12">createGraph</a> <span class="p3"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span>,<span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> 1 &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p2">(<span class="default"><a href="glib_000.html#K12">createGraph</a> <span class="p3"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span>,<span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> 1 ) or ( <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p2">(<span class="default"><a href="glib_000.html#K12">createGraph</a> <span class="p3"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span>,<span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> 1 &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p2">(<span class="default"><a href="glib_000.html#K12">createGraph</a> <span class="p3"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span>,<span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> 1 ) ) )
 <b>by </b><i>, <a class="ref" href="glib_000.html#D15">GLIB_000:def 15</a></i>;<br/><a NAME="E5:13_2_1_4_1_1_1"/><b>then </b>
( ( <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> 1 ) or ( <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> 0 ) )
 <b>by </b><i>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E6:13_2_1_4_1_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_2_1_4_1_1_2"><b>suppose </b></a><a NAME="E1:13_2_1_4_1_1_2"/><i><font color="Green">E67</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 3
 ;<br/><div class="add"><a NAME="E2:13_2_1_4_1_1_2"/><i><font color="Green">E68</font></i>: 
3 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font>
 <b>by </b><i>, <a class="ref" href="glib_001.html#T126">GLIB_001:126</a></i>;<br/><a NAME="E3:13_2_1_4_1_1_2"/><b>then </b>
3 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font>
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E4:13_2_1_4_1_1_2"/><b>then </b><i><font color="Green">E69</font></i>: 
3 <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">W</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E5:13_2_1_4_1_1_2"/><b>then </b><i><font color="Green">E70</font></i>: 
2 <a href="nat_1.html#K2">*</a> 2 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">W</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E6:13_2_1_4_1_1_2"/>
3 <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 0
 <b>by </b><i>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E7:13_2_1_4_1_1_2"/><b>then </b><i><font color="Green">E72</font></i>: 
2 <a href="nat_1.html#K2">*</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">W</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E8:13_2_1_4_1_1_2"/>
( <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 2</span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i>, , , <a class="ref" href="glib_001.html#T9">GLIB_001:9</a></i>;<br/><a NAME="E9:13_2_1_4_1_1_2"/><b>then </b>
( <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 2</span>)</span> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E10:13_2_1_4_1_1_2"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="glib_001.html#T139">GLIB_001:139</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:13_2_1_4"/>
contradiction
 ;<br/></div><b>end;</b></div>
<a NAME="E14:13_2_1"/><b>then </b><i><font color="Green">E73</font></i>: 
 <a href="glib_000.html#K12">createGraph</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span>,<span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">S</font>,<font color="Maroon">T</font> is <a href="glib_002.html#V2" target="_self">acyclic</a>
 
<b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">W1</font> = <span class="p1">(<span class="default"><a href="glib_000.html#K12">createGraph</a> <span class="p2"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="glib_001.html#K2">.walkOf</a> 0,0,1;<br/>
<b>set </b><font color="Maroon">W2</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K12">createGraph</a> <span class="p3"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span>,<span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="glib_001.html#K2">.walkOf</a> 0,0,1</span>)</span> <a href="glib_001.html#K6">.reverse()</a> ;<br/>
<a NAME="E15:13_2_1"/><i><font color="Green">E82</font></i>: 
<span class="p1">(<span class="default"><a href="glib_000.html#K12">createGraph</a> <span class="p2"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="glib_001.html#K2">.walkOf</a> 0,0,1 <a href="glib_001.html#R1">is_Walk_from</a> 0,1
 
<b>by </b><i>, <a class="ref" href="glib_001.html#T16">GLIB_001:16</a></i>;<br/>
<a NAME="E16:13_2_1"/><b>then </b><i><font color="Green">E83</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K12">createGraph</a> <span class="p3"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span>,<span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="glib_001.html#K2">.walkOf</a> 0,0,1</span>)</span> <a href="glib_001.html#K6">.reverse()</a>  <a href="glib_001.html#R1">is_Walk_from</a> 1,0
 
<b>by </b><i><a class="ref" href="glib_001.html#T24">GLIB_001:24</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_1_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">u</font> be   <a href="glib_000.html#NM10">Vertex</a> of <span class="p1">(<span class="default"><a href="glib_000.html#K12">createGraph</a> <span class="p2"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span>, <font color="Maroon">v</font> be   <a href="glib_000.html#NM10">Vertex</a> of <span class="p1">(<span class="default"><a href="glib_000.html#K12">createGraph</a> <span class="p2"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_1_5_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_2_1_5_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:13_2_1_5_1_1"/>
( ( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> 0 ) or ( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> 1 ) or ( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> 0 ) or ( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> 1 ) )
 </span><b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_2_1_5_1_1_1"><b>suppose </b></a><a NAME="E1:13_2_1_5_1_1_1"/>
( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/><div class="add"><a NAME="E2:13_2_1_5_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="glib_000.html#K12">createGraph</a> <span class="p2"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="glib_001.html#K1">.walkOf</a> <font color="Maroon">a0</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">u</font>,<font color="Maroon">v</font>
 <b>by </b><i><a class="ref" href="glib_001.html#T14">GLIB_001:14</a></i>;<br/><b>hence </b><a NAME="E3:13_2_1_5_1_1_1"/>
 ex <font color="Olive">W</font> being   <a href="glib_001.html#M3">Walk</a> of  <a href="glib_000.html#K12">createGraph</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span>,<span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">S</font>,<font color="Maroon">T</font> st <font color="Olive">W</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">u</font>,<font color="Maroon">v</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_2_1_5_1_1_2"><b>suppose </b></a><a NAME="E1:13_2_1_5_1_1_2"/>
( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> 1 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:13_2_1_5_1_1_2"/>
 ex <font color="Olive">W</font> being   <a href="glib_001.html#M3">Walk</a> of  <a href="glib_000.html#K12">createGraph</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span>,<span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">S</font>,<font color="Maroon">T</font> st <font color="Olive">W</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">u</font>,<font color="Maroon">v</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_2_1_5_1_1_3"><b>suppose </b></a><a NAME="E1:13_2_1_5_1_1_3"/>
( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:13_2_1_5_1_1_3"/>
 ex <font color="Olive">W</font> being   <a href="glib_001.html#M3">Walk</a> of  <a href="glib_000.html#K12">createGraph</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span>,<span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">S</font>,<font color="Maroon">T</font> st <font color="Olive">W</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">u</font>,<font color="Maroon">v</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_2_1_5_1_1_4"><b>suppose </b></a><a NAME="E1:13_2_1_5_1_1_4"/>
( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> 1 )
 ;<br/><div class="add"><a NAME="E2:13_2_1_5_1_1_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="glib_000.html#K12">createGraph</a> <span class="p2"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="glib_001.html#K1">.walkOf</a> <font color="Maroon">a1</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">u</font>,<font color="Maroon">v</font>
 <b>by </b><i><a class="ref" href="glib_001.html#T14">GLIB_001:14</a></i>;<br/><b>hence </b><a NAME="E3:13_2_1_5_1_1_4"/>
 ex <font color="Olive">W</font> being   <a href="glib_001.html#M3">Walk</a> of  <a href="glib_000.html#K12">createGraph</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span>,<span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">S</font>,<font color="Maroon">T</font> st <font color="Olive">W</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">u</font>,<font color="Maroon">v</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E2:13_2_1_5"/>
 ex <font color="Olive">W</font> being   <a href="glib_001.html#M3">Walk</a> of  <a href="glib_000.html#K12">createGraph</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span>,<span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">S</font>,<font color="Maroon">T</font> st <font color="Olive">W</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">u</font>,<font color="Maroon">v</font>
 ;<br/></div><b>end;</b></div>
<a NAME="E18:13_2_1"/><b>then </b>
 <a href="glib_000.html#K12">createGraph</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span>,<span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">S</font>,<font color="Maroon">T</font> is <a href="glib_002.html#V1" target="_self">connected</a>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E19:13_2_1"/>
 <a href="glib_000.html#K12">createGraph</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span>,<span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">S</font>,<font color="Maroon">T</font> is <a href="glib_002.html#V3" target="_self">Tree-like</a>
 <b>by </b><i>, </i>;<br/>


</div>
