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

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">g</font> be    <a href="finseq_2.html#M2">Element</a> of <a href="numbers.html#K1">REAL</a>  <a href="finseq_2.html#K3">*</a> ;<br/><b>let </b><font color="Maroon">f</font> be    <a href="finseq_2.html#M2">Element</a> of <a href="numbers.html#K1">REAL</a>  <a href="finseq_2.html#K3">*</a> ;<br/><b>let </b><font color="Maroon">h</font> be    <a href="finseq_2.html#M2">Element</a> of <a href="numbers.html#K1">REAL</a>  <a href="finseq_2.html#K3">*</a> ;<br/>











<b>set </b><font color="Maroon">R</font> =  <a href="graphsp.html#K20" target="_self">Relax</a> <font color="Maroon">n</font>;<br/>
<b>set </b><font color="Maroon">M</font> =  <a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">n</font>;<br/>
<b>set </b><font color="Maroon">ff</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K6" target="_self">repeat</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="graphsp.html#K20" target="_self">Relax</a> <font color="Maroon">n</font></span>)</span> <a href="graphsp.html#K4" target="_self">*</a> <span class="p4">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">f</font>;<br/>
<b>set </b><font color="Maroon">Fi1</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K6" target="_self">repeat</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="graphsp.html#K20" target="_self">Relax</a> <font color="Maroon">n</font></span>)</span> <a href="graphsp.html#K4" target="_self">*</a> <span class="p4">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">f</font>;<br/>
<b>set </b><font color="Maroon">mi</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default">3 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1;<br/>
<b>assume </b><a NAME="E1:71"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K6" target="_self">repeat</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="graphsp.html#K20" target="_self">Relax</a> <font color="Maroon">n</font></span>)</span> <a href="graphsp.html#K4" target="_self">*</a> <span class="p4">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">f</font> &amp; <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K6" target="_self">repeat</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="graphsp.html#K20" target="_self">Relax</a> <font color="Maroon">n</font></span>)</span> <a href="graphsp.html#K4" target="_self">*</a> <span class="p4">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">f</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p1">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">g</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">g</font>,<font color="Maroon">n</font> &amp;  <a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">g</font>,<font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/>

<a NAME="E2:71"/><b>then </b><i><font color="Green">E52</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="graphsp.html#K6" target="_self">repeat</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="graphsp.html#K20" target="_self">Relax</a> <font color="Maroon">n</font></span>)</span> <a href="graphsp.html#K4" target="_self">*</a> <span class="p5">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="graphsp.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E3:71"/><b>then </b><i><font color="Green">E53</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">n</font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="graphsp.html#K6" target="_self">repeat</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="graphsp.html#K20" target="_self">Relax</a> <font color="Maroon">n</font></span>)</span> <a href="graphsp.html#K4" target="_self">*</a> <span class="p0">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E4:71"/><i><font color="Green">E54</font></i>: 
<span class="p1">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">n</font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="graphsp.html#K6" target="_self">repeat</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="graphsp.html#K20" target="_self">Relax</a> <font color="Maroon">n</font></span>)</span> <a href="graphsp.html#K4" target="_self">*</a> <span class="p5">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="graphsp.html#K6" target="_self">repeat</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="graphsp.html#K20" target="_self">Relax</a> <font color="Maroon">n</font></span>)</span> <a href="graphsp.html#K4" target="_self">*</a> <span class="p5">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">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">n</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default">3 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graphsp.html#K2" target="_self">:=</a> <font color="Maroon">k</font>,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="graphsp.html#T6" target="_self">Th6</a>, </i>;<br/>
<a NAME="E5:71"/><i><font color="Green">E62</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">n</font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="graphsp.html#K6" target="_self">repeat</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="graphsp.html#K20" target="_self">Relax</a> <font color="Maroon">n</font></span>)</span> <a href="graphsp.html#K4" target="_self">*</a> <span class="p0">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="graphsp.html#T6" target="_self">Th6</a>, </i>;<br/>
<div><i><font color="Green">E63</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="71_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="E1:71_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">h</font>,<font color="Maroon">n</font>
 ;<br/><a NAME="E2:71_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">m</font> where <font color="Olive">m</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Olive">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> &amp; <font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">h</font> <a href="goboard1.html#K1">.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">m</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:71_1"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> &amp; <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">h</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 )
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:71_1_1"/>
( <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> or <font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">k</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_1_1_1"><b>suppose </b></a><a NAME="E1:71_1_1_1"/>
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><a NAME="E2:71_1_1_1"/><b>then </b>
<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"><font color="Maroon">k</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="graphsp.html#E4:71_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:71_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">g</font>,<font color="Maroon">n</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">k</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_1_1_2"><b>suppose </b></a><a NAME="E1:71_1_1_2"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><a NAME="E2:71_1_1_2"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default">3 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i/>;<br/><a NAME="E3:71_1_1_2"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="graphsp.html#K6" target="_self">repeat</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="graphsp.html#K20" target="_self">Relax</a> <font color="Maroon">n</font></span>)</span> <a href="graphsp.html#K4" target="_self">*</a> <span class="p5">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="graphsp.html#T6" target="_self">Th6</a>, <a class="txt" href="graphsp.html#E4:71_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="graphsp.html#T5" target="_self">Th5</a></i>;<br/> <a href="real_1.html#K1">-</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K20" target="_self">Relax</a> <font color="Maroon">n</font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">n</font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="graphsp.html#K6" target="_self">repeat</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="graphsp.html#K20" target="_self">Relax</a> <font color="Maroon">n</font></span>)</span> <a href="graphsp.html#K4" target="_self">*</a> <span class="p1">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">m</font>
<b>by </b><i><a class="ref" href="graphsp.html#T6" target="_self">Th6</a>, <a class="txt" href="graphsp.html#E4:71_1"><i><font color="Green">E64</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">n</font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="graphsp.html#K6" target="_self">repeat</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="graphsp.html#K20" target="_self">Relax</a> <font color="Maroon">n</font></span>)</span> <a href="graphsp.html#K4" target="_self">*</a> <span class="p0">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">m</font>
<b>by </b><i>, <a class="txt" href="graphsp.html#E4:71_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="graphsp.html#E4:71"><i><font color="Green">E54</font></i></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#K6" target="_self">repeat</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="graphsp.html#K20" target="_self">Relax</a> <font color="Maroon">n</font></span>)</span> <a href="graphsp.html#K4" target="_self">*</a> <span class="p5">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">f</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">m</font>
<b>by </b><i>, <a class="txt" href="graphsp.html#E4:71_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="graphsp.html#E1:71_1_1_2"><i><font color="Green">E65</font></i></a>, <a class="txt" href="graphsp.html#E2:71_1_1_2"><i><font color="Green">E66</font></i></a>, <a class="txt" href="graphsp.html#E3:71_1_1_2"><i><font color="Green">E67</font></i></a>, </i>
;<br/><a NAME="E5:71_1_1_2"/><b>then </b>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">j</font> where <font color="Olive">j</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="graphsp.html#K6" target="_self">repeat</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="graphsp.html#K20" target="_self">Relax</a> <font color="Maroon">n</font></span>)</span> <a href="graphsp.html#K4" target="_self">*</a> <span class="p5">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">f</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">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#K6" target="_self">repeat</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="graphsp.html#K20" target="_self">Relax</a> <font color="Maroon">n</font></span>)</span> <a href="graphsp.html#K4" target="_self">*</a> <span class="p5">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">f</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 ) </span>}</span> 
 <b>by </b><i>, <a class="txt" href="graphsp.html#E4:71_1"><i><font color="Green">E64</font></i></a></i>;<br/><a NAME="E6:71_1_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="graphsp.html#K15" target="_self">UsedVx</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="graphsp.html#K6" target="_self">repeat</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="graphsp.html#K20" target="_self">Relax</a> <font color="Maroon">n</font></span>)</span> <a href="graphsp.html#K4" target="_self">*</a> <span class="p5">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E4:71_1"><i><font color="Green">E64</font></i></a></i>;<br/><b>hence </b><a NAME="E7:71_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">g</font>,<font color="Maroon">n</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">k</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="graphsp.html#T6" target="_self">Th6</a>, <a class="ref" href="xboole_0.html#D2">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="71_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="E1:71_2"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">g</font>,<font color="Maroon">n</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">k</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:71_2_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">g</font>,<font color="Maroon">n</font> or <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"><font color="Maroon">k</font></span><a href="domain_1.html#K6">}</a></span> )
 </span><b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_2_1_1"><b>suppose </b></a><a NAME="E1:71_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">g</font>,<font color="Maroon">n</font>
 ;<br/><div class="add"><a NAME="E2:71_2_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">m</font> where <font color="Olive">m</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Olive">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> &amp; <font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">g</font> <a href="goboard1.html#K1">.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">m</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:71_2_1_1"/><i><font color="Green">E69</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> &amp; <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">g</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 )
 ;<br/><a NAME="E5:71_2_1_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default">3 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i/>;<br/><font color="Maroon">h</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">m</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K20" target="_self">Relax</a> <font color="Maroon">n</font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">n</font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="graphsp.html#K6" target="_self">repeat</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="graphsp.html#K20" target="_self">Relax</a> <font color="Maroon">n</font></span>)</span> <a href="graphsp.html#K4" target="_self">*</a> <span class="p1">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">m</font>
<b>by </b><i><a class="ref" href="graphsp.html#T6" target="_self">Th6</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">n</font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="graphsp.html#K6" target="_self">repeat</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="graphsp.html#K20" target="_self">Relax</a> <font color="Maroon">n</font></span>)</span> <a href="graphsp.html#K4" target="_self">*</a> <span class="p0">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">m</font>
<b>by </b><i><a class="txt" href="graphsp.html#E5:71"><i><font color="Green">E62</font></i></a>, , <a class="txt" href="graphsp.html#E4:71"><i><font color="Green">E54</font></i></a></i>
<br/>.= 
 <a href="real_1.html#K1">-</a> 1
<b>by </b><i><a class="ref" href="graphsp.html#T6" target="_self">Th6</a>, , , </i>
;<br/><a NAME="E7:71_2_1_1"/><b>then </b>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">j</font> where <font color="Olive">j</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">h</font> <a href="goboard1.html#K1">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 ) </span>}</span> 
 <b>by </b><i><a class="ref" href="graphsp.html#T6" target="_self">Th6</a>, , </i>;<br/><b>hence </b><a NAME="E8:71_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">h</font>,<font color="Maroon">n</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_2_1_2"><b>suppose </b></a><a NAME="E1:71_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"><font color="Maroon">k</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><div class="add"><a NAME="E2:71_2_1_2"/><b>then </b><i><font color="Green">E71</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:71_2_1_2"/><i><font color="Green">E84</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="ref" href="graphsp.html#T6" target="_self">Th6</a>, </i>;<br/><font color="Maroon">h</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K20" target="_self">Relax</a> <font color="Maroon">n</font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">n</font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="graphsp.html#K6" target="_self">repeat</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="graphsp.html#K20" target="_self">Relax</a> <font color="Maroon">n</font></span>)</span> <a href="graphsp.html#K4" target="_self">*</a> <span class="p1">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i><a class="ref" href="graphsp.html#T6" target="_self">Th6</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">n</font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="graphsp.html#K6" target="_self">repeat</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="graphsp.html#K20" target="_self">Relax</a> <font color="Maroon">n</font></span>)</span> <a href="graphsp.html#K4" target="_self">*</a> <span class="p0">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i><a class="txt" href="graphsp.html#E5:71"><i><font color="Green">E62</font></i></a>, , <a class="txt" href="graphsp.html#E4:71"><i><font color="Green">E54</font></i></a></i>
<br/>.= 
 <a href="real_1.html#K1">-</a> 1
<b>by </b><i><a class="ref" href="graphsp.html#T6" target="_self">Th6</a>, , , </i>
;<br/><a NAME="E5:71_2_1_2"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">j</font> where <font color="Olive">j</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">h</font> <a href="goboard1.html#K1">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 ) </span>}</span> 
 <b>by </b><i><a class="ref" href="graphsp.html#T6" target="_self">Th6</a>, , </i>;<br/><b>hence </b><a NAME="E6:71_2_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">h</font>,<font color="Maroon">n</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E8:71"/>
 <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">h</font>,<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">g</font>,<font color="Maroon">n</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">k</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="graphsp.html#E6:71"><i><font color="Green">E63</font></i></a>, <a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>

<b>assume </b><a NAME="E9:71"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">g</font>,<font color="Maroon">n</font>
 ;<br/>

<a NAME="E10:71"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">j</font> where <font color="Olive">j</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">g</font> <a href="goboard1.html#K1">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 ) </span>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon">j</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E12:71"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">g</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 )
 ;<br/>
<b>thus </b><a NAME="E13:71"/>
contradiction
 <b>by </b><i><a class="ref" href="graphsp.html#T6" target="_self">Th6</a>, , </i>;<br/>


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