<?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">m</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">RF</font> = <span class="p1">(<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="p1">(<span class="default"><a href="graphsp.html#K17" title="GRAPHSP:func.17">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c8">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>set </b><font color="Maroon" title="c9">cn</font> =  <a href="graphsp.html#K9" title="GRAPHSP:func.9">LifeSpan</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>,<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c1">n</font>;<br/>
<b>assume </b><a NAME="E1:81"/><i><font color="Green" title="E69">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#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> 1</span>)</span> <a href="graphsp.html#K7" title="GRAPHSP:func.7">.</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#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> <font color="Maroon" title="c2">i</font></span>)</span> <a href="graphsp.html#K7" title="GRAPHSP:func.7">.</a> <font color="Maroon" title="c5">f</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">i</font> &amp; <font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="graphsp.html#K9" title="GRAPHSP:func.9">LifeSpan</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>,<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c3">m</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="c4">g</font>,<font color="Maroon" title="c1">n</font> )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <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>means </b>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> $1 &amp; $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="graphsp.html#K9" title="GRAPHSP:func.9">LifeSpan</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>,<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c1">n</font> implies <font color="Maroon" title="c3">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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="c5">f</font></span>)</span>,<font color="Maroon" title="c1">n</font> );<br/>
<a NAME="E2:81"/><i><font color="Green" title="E70">A2</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<a NAME="E3:81"/><i><font color="Green" title="E71">A3</font></i>: 
for <font color="Olive" title="b1">k</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 <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">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>assume </b><a NAME="E1:81_1"/><i><font color="Green" title="E72">A4</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">k</font>]
 ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:81_1_1"/><i><font color="Green" title="E73">A5</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c10">k</font> <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#K9" title="GRAPHSP:func.9">LifeSpan</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>,<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c1">n</font> )
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:81_1_1_1"/>
( <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Maroon" title="c10">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_1_1_1_1"><b>suppose </b></a><a NAME="E1:81_1_1_1_1"/>
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:81_1_1_1_1"/>
<font color="Maroon" title="c3">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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"><font color="Maroon" title="c10">k</font> <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="c5">f</font></span>)</span>,<font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E1:81"><i><font color="Green" title="E69">A1</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_1_1_1_2"><b>suppose </b></a><a NAME="E1:81_1_1_1_2"/>
<font color="Maroon" title="c10">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:81_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/><a NAME="E3:81_1_1_1_2"/><b>then </b><i><font color="Green" title="E74">A6</font></i>: 
<font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 0
 <b>by </b><i><a class="ref" href="int_1.html#T20" title="INT_1:th.20">INT_1:20</a></i>;<br/><a NAME="E4:81_1_1_1_2"/>
<font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="graphsp.html#K9" title="GRAPHSP:func.9">LifeSpan</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>,<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E1:81_1_1"><i><font color="Green" title="E73">A5</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E5:81_1_1_1_2"/><b>then </b>
 <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> <font color="Maroon" title="c10">k</font></span>)</span> <a href="graphsp.html#K7" title="GRAPHSP:func.7">.</a> <font color="Maroon" title="c5">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="ref" href="graphsp.html#D4" target="_self" title="GRAPHSP:def.4">Def4</a></i>;<br/><a NAME="E6:81_1_1_1_2"/><b>then </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> <span class="p3">(<span class="default"><font color="Maroon" title="c10">k</font> <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="c5">f</font></span>)</span>,<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#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> <font color="Maroon" title="c10">k</font></span>)</span> <a href="graphsp.html#K7" title="GRAPHSP:func.7">.</a> <font color="Maroon" title="c5">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"><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> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><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></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="graphsp.html#K7" title="GRAPHSP:func.7">.</a> <font color="Maroon" title="c5">f</font></span>)</span>,<font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="graphsp.html#K6" title="GRAPHSP:func.6">repeat</a> <span class="p0">(<span class="default"><span class="p1">(<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="p1">(<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> <font color="Maroon" title="c10">k</font></span>)</span> <a href="graphsp.html#K7" title="GRAPHSP:func.7">.</a> <font color="Maroon" title="c5">f</font></span>)</span>,<font color="Maroon" title="c1">n</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="graphsp.html#T40" target="_self" title="GRAPHSP:th.40">Th40</a></i>;<br/><a NAME="E7:81_1_1_1_2"/><b>then </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> <font color="Maroon" title="c10">k</font></span>)</span> <a href="graphsp.html#K7" title="GRAPHSP:func.7">.</a> <font color="Maroon" title="c5">f</font></span>)</span>,<font color="Maroon" title="c1">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <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"><font color="Maroon" title="c10">k</font> <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="c5">f</font></span>)</span>,<font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E8:81_1_1_1_2"/>
<font color="Maroon" title="c3">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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"><font color="Maroon" title="c10">k</font> <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="c5">f</font></span>)</span>,<font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E1:81_1"><i><font color="Green" title="E72">A4</font></i></a>, <a class="txt" href="graphsp.html#E1:81_1_1"><i><font color="Green" title="E73">A5</font></i></a>, <a class="txt" href="graphsp.html#E3:81_1_1_1_2"><i><font color="Green" title="E74">A6</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div>


</div><b>end;</b></div>
<a NAME="E4:81"/>
for <font color="Olive" title="b1">k</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>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="graphsp.html#E2:81"><i><font color="Green" title="E70">A2</font></i></a>, <a class="txt" href="graphsp.html#E3:81"><i><font color="Green" title="E71">A3</font></i></a>);<br/></i>
<b>hence </b><a NAME="E5:81"/>
<font color="Maroon" title="c3">m</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="c6">h</font>,<font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E1:81"><i><font color="Green" title="E69">A1</font></i></a></i>;<br/>


</div>
