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

<b>let </b><font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c2">i</font>, <font color="Maroon" title="c3">k</font> be    <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>let </b><font color="Maroon" title="c4">g</font>, <font color="Maroon" title="c5">f</font>, <font color="Maroon" title="c6">h</font> be    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> ;<br/>











<b>set </b><font color="Maroon" title="c7">R</font> =  <a href="graphsp.html#K19" title="GRAPHSP:func.19">Relax</a> <font color="Maroon" title="c1">n</font>;<br/>
<b>set </b><font color="Maroon" title="c8">M</font> =  <a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font>;<br/>
<b>set </b><font color="Maroon" title="c9">ff</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K5" title="GRAPHSP:func.5">repeat</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="graphsp.html#K19" title="GRAPHSP:func.19">Relax</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K3" title="GRAPHSP:func.3">*</a> <span class="p4">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">i</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c5">f</font>;<br/>
<b>set </b><font color="Maroon" title="c10">Fi1</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K5" title="GRAPHSP:func.5">repeat</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="graphsp.html#K19" title="GRAPHSP:func.19">Relax</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K3" title="GRAPHSP:func.3">*</a> <span class="p4">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c5">f</font>;<br/>
<b>set </b><font color="Maroon" title="c11">mi</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default">3 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1;<br/>
<b>assume </b><a NAME="E1:69"/><i><font color="Green" title="E57">A1</font></i>: 
( <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K5" title="GRAPHSP:func.5">repeat</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="graphsp.html#K19" title="GRAPHSP:func.19">Relax</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K3" title="GRAPHSP:func.3">*</a> <span class="p4">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">i</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c5">f</font> &amp; <font color="Maroon" title="c6">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K5" title="GRAPHSP:func.5">repeat</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="graphsp.html#K19" title="GRAPHSP:func.19">Relax</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K3" title="GRAPHSP:func.3">*</a> <span class="p4">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c5">f</font> &amp; <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p1">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font> &amp;  <a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 ;<br/>

<a NAME="E2:69"/><b>then </b><i><font color="Green" title="E58">A2</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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"><span class="p3">(<span class="default"><a href="graphsp.html#K5" title="GRAPHSP:func.5">repeat</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="graphsp.html#K19" title="GRAPHSP:func.19">Relax</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K3" title="GRAPHSP:func.3">*</a> <span class="p5">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">i</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c5">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="graphsp.html#T38" target="_self" title="GRAPHSP:th.38">Th38</a></i>;<br/>
<a NAME="E3:69"/><b>then </b><i><font color="Green" title="E59">A3</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="graphsp.html#K5" title="GRAPHSP:func.5">repeat</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="graphsp.html#K19" title="GRAPHSP:func.19">Relax</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K3" title="GRAPHSP:func.3">*</a> <span class="p0">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">i</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c5">f</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="graphsp.html#T34" target="_self" title="GRAPHSP:th.34">Th34</a></i>;<br/>
<a NAME="E4:69"/><i><font color="Green" title="E60">A4</font></i>: 
<span class="p1">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="graphsp.html#K5" title="GRAPHSP:func.5">repeat</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="graphsp.html#K19" title="GRAPHSP:func.19">Relax</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K3" title="GRAPHSP:func.3">*</a> <span class="p5">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">i</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="graphsp.html#K5" title="GRAPHSP:func.5">repeat</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="graphsp.html#K19" title="GRAPHSP:func.19">Relax</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K3" title="GRAPHSP:func.3">*</a> <span class="p5">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">i</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c5">f</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default">3 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="graphsp.html#K2" title="GRAPHSP:func.2">:=</a> <font color="Maroon" title="c3">k</font>,<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="graphsp.html#E1:69"><i><font color="Green" title="E57">A1</font></i></a>, <a class="ref" href="graphsp.html#D11" target="_self" title="GRAPHSP:def.11">Def11</a></i>;<br/>
<a NAME="E5:69"/><i><font color="Green" title="E61">A5</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="graphsp.html#K5" title="GRAPHSP:func.5">repeat</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="graphsp.html#K19" title="GRAPHSP:func.19">Relax</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K3" title="GRAPHSP:func.3">*</a> <span class="p0">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">i</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c5">f</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="graphsp.html#E1:69"><i><font color="Green" title="E57">A1</font></i></a>, <a class="ref" href="graphsp.html#T34" target="_self" title="GRAPHSP:th.34">Th34</a></i>;<br/>
<div><i><font color="Green" title="E62">A6</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c12">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:69_1"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c1">n</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c13">m</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="E3:69_1"/><i><font color="Green" title="E63">A7</font></i>: 
( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">m</font> &amp; <font color="Maroon" title="c13">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> <font color="Maroon" title="c6">h</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">m</font> &amp; <font color="Maroon" title="c13">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c6">h</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c13">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 )
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:69_1_1"/>
( <font color="Maroon" title="c13">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> or <font color="Maroon" title="c13">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">k</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_1_1"><b>suppose </b></a><a NAME="E1:69_1_1_1"/>
<font color="Maroon" title="c13">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font>
 ;<br/><div class="add"><a NAME="E2:69_1_1_1"/><b>then </b>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">k</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="graphsp.html#E3:69_1"><i><font color="Green" title="E63">A7</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:69_1_1_1"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">k</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_1_2"><b>suppose </b></a><a NAME="E1:69_1_1_2"/><i><font color="Green" title="E64">A8</font></i>: 
<font color="Maroon" title="c13">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">k</font>
 ;<br/><div class="add"><a NAME="E2:69_1_1_2"/><i><font color="Green" title="E65">A9</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default">3 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="graphsp.html#E52"><i><font color="Green" title="E46">Lm7</font></i></a></i>;<br/><a NAME="E3:69_1_1_2"/><i><font color="Green" title="E66">A10</font></i>: 
<font color="Maroon" title="c13">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"><span class="p3">(<span class="default"><a href="graphsp.html#K5" title="GRAPHSP:func.5">repeat</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="graphsp.html#K19" title="GRAPHSP:func.19">Relax</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K3" title="GRAPHSP:func.3">*</a> <span class="p5">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">i</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c5">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="graphsp.html#E1:69"><i><font color="Green" title="E57">A1</font></i></a>, <a class="txt" href="graphsp.html#E3:69_1"><i><font color="Green" title="E63">A7</font></i></a>, <a class="ref" href="graphsp.html#T38" target="_self" title="GRAPHSP:th.38">Th38</a></i>;<br/> <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K19" title="GRAPHSP:func.19">Relax</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="graphsp.html#K5" title="GRAPHSP:func.5">repeat</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="graphsp.html#K19" title="GRAPHSP:func.19">Relax</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K3" title="GRAPHSP:func.3">*</a> <span class="p1">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">i</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c5">f</font></span>)</span></span>)</span></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c13">m</font>
<b>by </b><i><a class="txt" href="graphsp.html#E1:69"><i><font color="Green" title="E57">A1</font></i></a>, <a class="txt" href="graphsp.html#E3:69_1"><i><font color="Green" title="E63">A7</font></i></a>, <a class="ref" href="graphsp.html#T23" target="_self" title="GRAPHSP:th.23">Th23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="graphsp.html#K5" title="GRAPHSP:func.5">repeat</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="graphsp.html#K19" title="GRAPHSP:func.19">Relax</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K3" title="GRAPHSP:func.3">*</a> <span class="p0">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">i</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c5">f</font></span>)</span></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c13">m</font>
<b>by </b><i><a class="txt" href="graphsp.html#E3:69"><i><font color="Green" title="E59">A3</font></i></a>, <a class="txt" href="graphsp.html#E3:69_1"><i><font color="Green" title="E63">A7</font></i></a>, <a class="ref" href="graphsp.html#T37" target="_self" title="GRAPHSP:th.37">Th37</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="graphsp.html#K5" title="GRAPHSP:func.5">repeat</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="graphsp.html#K19" title="GRAPHSP:func.19">Relax</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K3" title="GRAPHSP:func.3">*</a> <span class="p5">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">i</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c13">m</font>
<b>by </b><i><a class="txt" href="graphsp.html#E4:69"><i><font color="Green" title="E60">A4</font></i></a>, <a class="txt" href="graphsp.html#E3:69_1"><i><font color="Green" title="E63">A7</font></i></a>, <a class="txt" href="graphsp.html#E1:69_1_1_2"><i><font color="Green" title="E64">A8</font></i></a>, <a class="txt" href="graphsp.html#E2:69_1_1_2"><i><font color="Green" title="E65">A9</font></i></a>, <a class="txt" href="graphsp.html#E3:69_1_1_2"><i><font color="Green" title="E66">A10</font></i></a>, <a class="ref" href="graphsp.html#T19" target="_self" title="GRAPHSP:th.19">Th19</a></i>
;<br/><a NAME="E5:69_1_1_2"/><b>then </b>
<font color="Maroon" title="c13">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">j</font> where <font color="Olive" title="b1">j</font> is    <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>  : ( <font color="Olive" title="b1">j</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"><span class="p3">(<span class="default"><a href="graphsp.html#K5" title="GRAPHSP:func.5">repeat</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="graphsp.html#K19" title="GRAPHSP:func.19">Relax</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K3" title="GRAPHSP:func.3">*</a> <span class="p5">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">i</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c5">f</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="graphsp.html#K5" title="GRAPHSP:func.5">repeat</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="graphsp.html#K19" title="GRAPHSP:func.19">Relax</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K3" title="GRAPHSP:func.3">*</a> <span class="p5">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">i</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 ) </span>}</span> 
 <b>by </b><i><a class="txt" href="graphsp.html#E2:69"><i><font color="Green" title="E58">A2</font></i></a>, <a class="txt" href="graphsp.html#E3:69_1"><i><font color="Green" title="E63">A7</font></i></a></i>;<br/><b>hence </b><a NAME="E6:69_1_1_2"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">k</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="graphsp.html#E1:69"><i><font color="Green" title="E57">A1</font></i></a>, <a class="txt" href="graphsp.html#E3:69_1"><i><font color="Green" title="E63">A7</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c12">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:69_2"/><i><font color="Green" title="E63">A11</font></i>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">k</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:69_2_1"/>
( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font> or <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">k</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 </span><b>by </b><i><a class="txt" href="graphsp.html#E1:69_2"><i><font color="Green" title="E63">A11</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_2_1_1"><b>suppose </b></a><a NAME="E1:69_2_1_1"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c13">m</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="E3:69_2_1_1"/><i><font color="Green" title="E64">A12</font></i>: 
( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">m</font> &amp; <font color="Maroon" title="c13">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> <font color="Maroon" title="c4">g</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">m</font> &amp; <font color="Maroon" title="c13">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c4">g</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c13">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 )
 ;<br/><a NAME="E4:69_2_1_1"/><i><font color="Green" title="E65">A13</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default">3 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="graphsp.html#E52"><i><font color="Green" title="E46">Lm7</font></i></a></i>;<br/><font color="Maroon" title="c6">h</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c13">m</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K19" title="GRAPHSP:func.19">Relax</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="graphsp.html#K5" title="GRAPHSP:func.5">repeat</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="graphsp.html#K19" title="GRAPHSP:func.19">Relax</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K3" title="GRAPHSP:func.3">*</a> <span class="p1">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">i</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c5">f</font></span>)</span></span>)</span></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c13">m</font>
<b>by </b><i><a class="txt" href="graphsp.html#E1:69"><i><font color="Green" title="E57">A1</font></i></a>, <a class="ref" href="graphsp.html#T23" target="_self" title="GRAPHSP:th.23">Th23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="graphsp.html#K5" title="GRAPHSP:func.5">repeat</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="graphsp.html#K19" title="GRAPHSP:func.19">Relax</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K3" title="GRAPHSP:func.3">*</a> <span class="p0">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">i</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c5">f</font></span>)</span></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c13">m</font>
<b>by </b><i><a class="txt" href="graphsp.html#E5:69"><i><font color="Green" title="E61">A5</font></i></a>, <a class="txt" href="graphsp.html#E3:69_2_1_1"><i><font color="Green" title="E64">A12</font></i></a>, <a class="ref" href="graphsp.html#T37" target="_self" title="GRAPHSP:th.37">Th37</a></i>
<br/>.= 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
<b>by </b><i><a class="txt" href="graphsp.html#E1:69"><i><font color="Green" title="E57">A1</font></i></a>, <a class="txt" href="graphsp.html#E3:69_2_1_1"><i><font color="Green" title="E64">A12</font></i></a>, <a class="txt" href="graphsp.html#E4:69_2_1_1"><i><font color="Green" title="E65">A13</font></i></a>, <a class="ref" href="graphsp.html#T33" target="_self" title="GRAPHSP:th.33">Th33</a></i>
;<br/><b>hence </b><a NAME="E6:69_2_1_1"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E1:69"><i><font color="Green" title="E57">A1</font></i></a>, <a class="txt" href="graphsp.html#E2:69"><i><font color="Green" title="E58">A2</font></i></a>, <a class="txt" href="graphsp.html#E3:69_2_1_1"><i><font color="Green" title="E64">A12</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_2_1_2"><b>suppose </b></a><a NAME="E1:69_2_1_2"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">k</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ;<br/><div class="add"><a NAME="E2:69_2_1_2"/><b>then </b><i><font color="Green" title="E64">A14</font></i>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E3:69_2_1_2"/><i><font color="Green" title="E65">A15</font></i>: 
( <font color="Maroon" title="c3">k</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> <font color="Maroon" title="c4">g</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> &amp; <font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> )
 <b>by </b><i><a class="txt" href="graphsp.html#E1:69"><i><font color="Green" title="E57">A1</font></i></a>, <a class="ref" href="graphsp.html#T30" target="_self" title="GRAPHSP:th.30">Th30</a></i>;<br/><font color="Maroon" title="c6">h</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c3">k</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K19" title="GRAPHSP:func.19">Relax</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="graphsp.html#K5" title="GRAPHSP:func.5">repeat</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="graphsp.html#K19" title="GRAPHSP:func.19">Relax</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K3" title="GRAPHSP:func.3">*</a> <span class="p1">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">i</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c5">f</font></span>)</span></span>)</span></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c3">k</font>
<b>by </b><i><a class="txt" href="graphsp.html#E1:69"><i><font color="Green" title="E57">A1</font></i></a>, <a class="ref" href="graphsp.html#T23" target="_self" title="GRAPHSP:th.23">Th23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="graphsp.html#K5" title="GRAPHSP:func.5">repeat</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="graphsp.html#K19" title="GRAPHSP:func.19">Relax</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K3" title="GRAPHSP:func.3">*</a> <span class="p0">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">i</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c5">f</font></span>)</span></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c3">k</font>
<b>by </b><i><a class="txt" href="graphsp.html#E5:69"><i><font color="Green" title="E61">A5</font></i></a>, <a class="txt" href="graphsp.html#E3:69_2_1_2"><i><font color="Green" title="E65">A15</font></i></a>, <a class="ref" href="graphsp.html#T37" target="_self" title="GRAPHSP:th.37">Th37</a></i>
<br/>.= 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
<b>by </b><i><a class="txt" href="graphsp.html#E1:69"><i><font color="Green" title="E57">A1</font></i></a>, <a class="txt" href="graphsp.html#E4:69"><i><font color="Green" title="E60">A4</font></i></a>, <a class="txt" href="graphsp.html#E3:69_2_1_2"><i><font color="Green" title="E65">A15</font></i></a>, <a class="ref" href="graphsp.html#T20" target="_self" title="GRAPHSP:th.20">Th20</a></i>
;<br/><b>hence </b><a NAME="E5:69_2_1_2"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E1:69"><i><font color="Green" title="E57">A1</font></i></a>, <a class="txt" href="graphsp.html#E2:69"><i><font color="Green" title="E58">A2</font></i></a>, <a class="txt" href="graphsp.html#E2:69_2_1_2"><i><font color="Green" title="E64">A14</font></i></a>, <a class="txt" href="graphsp.html#E3:69_2_1_2"><i><font color="Green" title="E65">A15</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E8:69"/>
 <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">k</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="graphsp.html#E6:69"><i><font color="Green" title="E62">A6</font></i></a>, <a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>

<b>assume </b><a NAME="E9:69"/>
<font color="Maroon" title="c3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c12">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="E11:69"/><i><font color="Green" title="E63">A16</font></i>: 
( <font color="Maroon" title="c12">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> &amp; <font color="Maroon" title="c12">j</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> <font color="Maroon" title="c4">g</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">j</font> &amp; <font color="Maroon" title="c12">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c4">g</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c12">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 )
 ;<br/>
<b>thus </b><a NAME="E12:69"/>
contradiction
 <b>by </b><i><a class="txt" href="graphsp.html#E1:69"><i><font color="Green" title="E57">A1</font></i></a>, <a class="txt" href="graphsp.html#E11:69"><i><font color="Green" title="E63">A16</font></i></a>, <a class="ref" href="graphsp.html#T30" target="_self" title="GRAPHSP:th.30">Th30</a></i>;<br/>


</div>
