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

<b>let </b><font color="Maroon" title="c1">n</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="c2">f</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="c3">R</font> =  <a href="graphsp.html#K20" title="GRAPHSP:func.20">Relax</a> <font color="Maroon" title="c1">n</font>;<br/>
<b>set </b><font color="Maroon" title="c4">M</font> =  <a href="graphsp.html#K17" title="GRAPHSP:func.17">findmin</a> <font color="Maroon" title="c1">n</font>;<br/>
<b>set </b><font color="Maroon" title="c5">f0</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K6" title="GRAPHSP:func.6">repeat</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="graphsp.html#K20" title="GRAPHSP:func.20">Relax</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K4" title="GRAPHSP:func.4">*</a> <span class="p4">(<span class="default"><a href="graphsp.html#K17" title="GRAPHSP:func.17">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> 0</span>)</span> <a href="graphsp.html#K7" title="GRAPHSP:func.7">.</a> <font color="Maroon" title="c2">f</font>;<br/>
<b>set </b><font color="Maroon" title="c6">RT</font> =  <a href="graphsp.html#K6" title="GRAPHSP:func.6">repeat</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K20" title="GRAPHSP:func.20">Relax</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K4" title="GRAPHSP:func.4">*</a> <span class="p2">(<span class="default"><a href="graphsp.html#K17" title="GRAPHSP:func.17">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>;<br/>
<b>assume </b><a NAME="E1:80"/><i><font color="Green" title="E68">A1</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; 1 <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="c2">f</font> &amp; <font color="Maroon" title="c2">f</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp; ( for <font color="Olive" title="b1">i</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>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> holds <br/><font color="Maroon" title="c2">f</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) &amp; ( for <font color="Olive" title="b1">i</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>   st 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> holds <br/><font color="Maroon" title="c2">f</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Olive" title="b1">i</font></span>)</span> <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="E2:80"/><i><font color="Green" title="E69">A2</font></i>: 
 <a href="graphsp.html#K8" title="GRAPHSP:func.8">OuterVx</a> <font color="Maroon" title="c2">f</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> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1"><b>proof </b></a><div class="add">

<a NAME="E1:80_1"/>
<font color="Maroon" title="c2">f</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="graphsp.html#E1:80"><i><font color="Green" title="E68">A1</font></i></a></i>;<br/>
<a NAME="E2:80_1"/><b>then </b>
1 <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> <font color="Maroon" title="c2">f</font> &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; <font color="Maroon" title="c2">f</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp; <font color="Maroon" title="c2">f</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Olive" title="b1">j</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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#E1:80"><i><font color="Green" title="E68">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:80_1"/>
 <a href="graphsp.html#K8" title="GRAPHSP:func.8">OuterVx</a> <font color="Maroon" title="c2">f</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/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon" title="c7">k</font> =  <a href="graphsp.html#K16" title="GRAPHSP:func.16">Argmin</a> <span class="p1">(<span class="default"><a href="graphsp.html#K8" title="GRAPHSP:func.8">OuterVx</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">n</font>;<br/>
<b>thus </b><a NAME="E3:80"/><i><font color="Green" title="E70">A3</font></i>: 
 <a href="graphsp.html#K16" title="GRAPHSP:func.16">Argmin</a> <span class="p1">(<span class="default"><a href="graphsp.html#K8" title="GRAPHSP:func.8">OuterVx</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:80_2"/><i><font color="Green" title="E71">A4</font></i>: 
 <a href="graphsp.html#K16" title="GRAPHSP:func.16">Argmin</a> <span class="p1">(<span class="default"><a href="graphsp.html#K8" title="GRAPHSP:func.8">OuterVx</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ;<br/>

<a NAME="E2:80_2"/><i><font color="Green" title="E72">A5</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="graphsp.html#K16" title="GRAPHSP:func.16">Argmin</a> <span class="p1">(<span class="default"><a href="graphsp.html#K8" title="GRAPHSP:func.8">OuterVx</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">n</font> &amp;  <a href="graphsp.html#K16" title="GRAPHSP:func.16">Argmin</a> <span class="p1">(<span class="default"><a href="graphsp.html#K8" title="GRAPHSP:func.8">OuterVx</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">n</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="c2">f</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">Argmin</a> <span class="p2">(<span class="default"><a href="graphsp.html#K8" title="GRAPHSP:func.8">OuterVx</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp; <font color="Maroon" title="c2">f</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">Argmin</a> <span class="p3">(<span class="default"><a href="graphsp.html#K8" title="GRAPHSP:func.8">OuterVx</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 )
 
<b>by </b><i><a class="txt" href="graphsp.html#E2:80"><i><font color="Green" title="E69">A2</font></i></a>, <a class="ref" href="graphsp.html#T30" target="_self" title="GRAPHSP:th.30">Th30</a></i>;<br/>
<a NAME="E3:80_2"/><b>then </b>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="graphsp.html#K16" title="GRAPHSP:func.16">Argmin</a> <span class="p1">(<span class="default"><a href="graphsp.html#K8" title="GRAPHSP:func.8">OuterVx</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="txt" href="graphsp.html#E1:80_2"><i><font color="Green" title="E71">A4</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/>
<a NAME="E4:80_2"/><b>then </b>
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="graphsp.html#K16" title="GRAPHSP:func.16">Argmin</a> <span class="p1">(<span class="default"><a href="graphsp.html#K8" title="GRAPHSP:func.8">OuterVx</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="int_1.html#T20" title="INT_1:th.20">INT_1:20</a></i>;<br/>
<b>hence </b><a NAME="E5:80_2"/>
contradiction
 <b>by </b><i><a class="txt" href="graphsp.html#E1:80"><i><font color="Green" title="E68">A1</font></i></a>, <a class="txt" href="graphsp.html#E2:80_2"><i><font color="Green" title="E72">A5</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E4:80"/><i><font color="Green" title="E71">A6</font></i>: 
 <a href="graphsp.html#K15" title="GRAPHSP:func.15">UsedVx</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:80_3"/>
 <a href="graphsp.html#K15" title="GRAPHSP:func.15">UsedVx</a> <font color="Maroon" title="c2">f</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/>

<b>then consider </b><font color="Maroon" title="c8">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:80_3"/><i><font color="Green" title="E72">A7</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K15" title="GRAPHSP:func.15">UsedVx</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E4:80_3"/>
<font color="Maroon" title="c8">x</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> <font color="Maroon" title="c2">f</font> &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; <font color="Maroon" title="c2">f</font> <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#E3:80_3"><i><font color="Green" title="E72">A7</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">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="E6:80_3"/><i><font color="Green" title="E73">A8</font></i>: 
( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">j</font> &amp; <font color="Maroon" title="c9">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="c2">f</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">j</font> &amp; <font color="Maroon" title="c9">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="c2">f</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c9">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="E7:80_3"/>
contradiction
 <b>by </b><i><a class="txt" href="graphsp.html#E1:80"><i><font color="Green" title="E68">A1</font></i></a>, <a class="txt" href="graphsp.html#E6:80_3"><i><font color="Green" title="E73">A8</font></i></a></i>;<br/>


</div><b>end;</b></div>

<a NAME="E5:80"/><i><font color="Green" title="E72">A9</font></i>: 
 <a href="graphsp.html#K8" title="GRAPHSP:func.8">OuterVx</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="graphsp.html#K6" title="GRAPHSP:func.6">repeat</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="graphsp.html#K20" title="GRAPHSP:func.20">Relax</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K4" title="GRAPHSP:func.4">*</a> <span class="p5">(<span class="default"><a href="graphsp.html#K17" title="GRAPHSP:func.17">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> 0</span>)</span> <a href="graphsp.html#K7" title="GRAPHSP:func.7">.</a> <font color="Maroon" title="c2">f</font></span>)</span>,<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> 
 
<b>by </b><i><a class="txt" href="graphsp.html#E2:80"><i><font color="Green" title="E69">A2</font></i></a>, <a class="ref" href="graphsp.html#T22" target="_self" title="GRAPHSP:th.22">Th22</a></i>;<br/>
<i><font color="Green" title="E73">A10</font></i>:  <a href="graphsp.html#K16" title="GRAPHSP:func.16">Argmin</a> <span class="p1">(<span class="default"><a href="graphsp.html#K8" title="GRAPHSP:func.8">OuterVx</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="graphsp.html#K6" title="GRAPHSP:func.6">repeat</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="graphsp.html#K20" title="GRAPHSP:func.20">Relax</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K4" title="GRAPHSP:func.4">*</a> <span class="p0">(<span class="default"><a href="graphsp.html#K17" title="GRAPHSP:func.17">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> 0</span>)</span> <a href="graphsp.html#K7" title="GRAPHSP:func.7">.</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="graphsp.html#K6" title="GRAPHSP:func.6">repeat</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="graphsp.html#K20" title="GRAPHSP:func.20">Relax</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K4" title="GRAPHSP:func.4">*</a> <span class="p5">(<span class="default"><a href="graphsp.html#K17" title="GRAPHSP:func.17">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> 0</span>)</span> <a href="graphsp.html#K7" title="GRAPHSP:func.7">.</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c1">n</font> = 
 <a href="graphsp.html#K16" title="GRAPHSP:func.16">Argmin</a> <span class="p1">(<span class="default"><a href="graphsp.html#K8" title="GRAPHSP:func.8">OuterVx</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="graphsp.html#K6" title="GRAPHSP:func.6">repeat</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="graphsp.html#K20" title="GRAPHSP:func.20">Relax</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K4" title="GRAPHSP:func.4">*</a> <span class="p5">(<span class="default"><a href="graphsp.html#K17" title="GRAPHSP:func.17">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> 0</span>)</span> <a href="graphsp.html#K7" title="GRAPHSP:func.7">.</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="ref" href="graphsp.html#T22" target="_self" title="GRAPHSP:th.22">Th22</a></i>
<br/>.= 
1
<b>by </b><i><a class="txt" href="graphsp.html#E3:80"><i><font color="Green" title="E70">A3</font></i></a>, <a class="ref" href="graphsp.html#T22" target="_self" title="GRAPHSP:th.22">Th22</a></i>
;<br/>
<b>thus </b> <a href="graphsp.html#K15" title="GRAPHSP:func.15">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" title="GRAPHSP:func.6">repeat</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="graphsp.html#K20" title="GRAPHSP:func.20">Relax</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K4" title="GRAPHSP:func.4">*</a> <span class="p5">(<span class="default"><a href="graphsp.html#K17" title="GRAPHSP:func.17">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> 1</span>)</span> <a href="graphsp.html#K7" title="GRAPHSP:func.7">.</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c1">n</font> = 
 <a href="graphsp.html#K15" title="GRAPHSP:func.15">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" title="GRAPHSP:func.6">repeat</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="graphsp.html#K20" title="GRAPHSP:func.20">Relax</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K4" title="GRAPHSP:func.4">*</a> <span class="p5">(<span class="default"><a href="graphsp.html#K17" title="GRAPHSP:func.17">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="p3">(<span class="default">0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="graphsp.html#K7" title="GRAPHSP:func.7">.</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c1">n</font>

<br/>.= 
<span class="p1">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">UsedVx</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="graphsp.html#K6" title="GRAPHSP:func.6">repeat</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="graphsp.html#K20" title="GRAPHSP:func.20">Relax</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K4" title="GRAPHSP:func.4">*</a> <span class="p0">(<span class="default"><a href="graphsp.html#K17" title="GRAPHSP:func.17">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> 0</span>)</span> <a href="graphsp.html#K7" title="GRAPHSP:func.7">.</a> <font color="Maroon" title="c2">f</font></span>)</span>,<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">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
<b>by </b><i><a class="txt" href="graphsp.html#E5:80"><i><font color="Green" title="E72">A9</font></i></a>, <a class="txt" href="graphsp.html#E6:80"><i><font color="Green" title="E73">A10</font></i></a>, <a class="ref" href="graphsp.html#T40" target="_self" title="GRAPHSP:th.40">Th40</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">UsedVx</a> <font color="Maroon" title="c2">f</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">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
<b>by </b><i><a class="ref" href="graphsp.html#T22" target="_self" title="GRAPHSP:th.22">Th22</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
<b>by </b><i><a class="txt" href="graphsp.html#E4:80"><i><font color="Green" title="E71">A6</font></i></a></i>
;<br/>


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