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

<span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font> being    <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>   ex <font color="Olive" title="b2">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 <br/>( <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">i</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Olive" title="b1">f</font></span>)</span>,<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>  )</span><br/><span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <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 <br/>( <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> &amp;  <a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c2">f</font></span>)</span>,<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>  )</span><br/>



<span class="kw">set </span><font color="Maroon" title="c3">R</font> =  <a href="graphsp.html#K19" title="GRAPHSP:func.19">Relax</a> <font color="Maroon" title="c1">n</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">M</font> =  <a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font>;<br/>
<span class="kw">defpred </span><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> ] <span class="kw">means </span>( $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> implies  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="graphsp.html#K13" title="GRAPHSP:func.13">UnusedVx</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="nat_1.html#K6" title="NAT_1:func.6">.</a> $1</span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> $1 );<br/>
<span class="kw">set </span><font color="Maroon" title="c5">nf</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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c2">f</font>;<br/>
<span class="kw">assume </span><a NAME="E1:70"/><span class="lab"><font color="Green" title="E58">A1</font></span>: 
 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>   holds <br/>(  not <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> or  not  <a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c2">f</font></span>)</span>,<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>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:70"/><span class="lab"><font color="Green" title="E59">A2</font></span>: 
 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#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:70_1"/><span class="lab"><font color="Green" title="E60">A3</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">set </span><font color="Maroon" title="c7">fk</font> =  <a href="graphsp.html#K13" title="GRAPHSP:func.13">UnusedVx</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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c1">n</font>);<br/><span class="kw">set </span><font color="Maroon" title="c8">fk1</font> =  <a href="graphsp.html#K13" title="GRAPHSP:func.13">UnusedVx</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="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c1">n</font>);<br/><a NAME="E1:70_1_1"/><span class="lab"><font color="Green" title="E61">A4</font></span>: 
<font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/><span class="kw">assume </span><a NAME="E2:70_1_1"/><span class="lab"><font color="Green" title="E62">A5</font></span>: 
<font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="graphsp.html#K13" title="GRAPHSP:func.13">UnusedVx</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="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><br/><a NAME="E3:70_1_1"/><span class="kw">then </span>
 <a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70"><span class="lab"><font color="Green" title="E58">A1</font></span></a>, <a class="txt" href="#E1:70_1_1"><span class="lab"><font color="Green" title="E61">A4</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E4:70_1_1"/><span class="kw">then </span>
 <a href="graphsp.html#K13" title="GRAPHSP:func.13">UnusedVx</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="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c1">n</font>) <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a>  <a href="graphsp.html#K13" title="GRAPHSP:func.13">UnusedVx</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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c1">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="graphsp.html#T38" target="_self" title="GRAPHSP:th.38">Th39</a></span>;<br/><a NAME="E5:70_1_1"/><span class="kw">then </span>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="graphsp.html#K13" title="GRAPHSP:func.13">UnusedVx</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="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c6">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70_1"><span class="lab"><font color="Green" title="E60">A3</font></span></a>, <a class="txt" href="#E2:70_1_1"><span class="lab"><font color="Green" title="E62">A5</font></span></a>, <a class="txt" href="#E1:70_1_1"><span class="lab"><font color="Green" title="E61">A4</font></span></a>, <a class="ref" href="card_2.html#T48" title="CARD_2:th.48">CARD_2:48</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E6:70_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><a href="graphsp.html#K13" title="GRAPHSP:func.13">UnusedVx</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="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c6">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T7" title="INT_1:th.7">INT_1:7</a></span>;<br/><a NAME="E7:70_1_1"/><span class="kw">then </span>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="graphsp.html#K13" title="GRAPHSP:func.13">UnusedVx</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="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T19" title="XREAL_1:th.19">XREAL_1:19</a></span>;<br/><span class="kw">hence </span><a NAME="E8:70_1_1"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="graphsp.html#K13" title="GRAPHSP:func.13">UnusedVx</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="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:70_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:70"/><span class="lab"><font color="Green" title="E60">A6</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c6">f0</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="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c2">f</font>;<br/>
<span class="kw">assume </span><a NAME="E1:70_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="graphsp.html#K13" title="GRAPHSP:func.13">UnusedVx</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="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><br/>

<a NAME="E2:70_2"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="graphsp.html#K13" title="GRAPHSP:func.13">UnusedVx</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="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="graphsp.html#T26" target="_self" title="GRAPHSP:th.26">Th27</a>, <a class="ref" href="nat_1.html#T43" title="NAT_1:th.43">NAT_1:43</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:70_2"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="graphsp.html#K13" title="GRAPHSP:func.13">UnusedVx</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="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E4:70"/>
 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>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E3:70"><span class="lab"><font color="Green" title="E60">A6</font></span></a>, <a class="txt" href="#E2:70"><span class="lab"><font color="Green" title="E59">A2</font></span></a>);<br/></span>
<a NAME="E5:70"/><span class="kw">then </span>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">n</font>]
 
;<br/>
<a NAME="E6:70"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A7</font></span>: 
 <a href="graphsp.html#K13" title="GRAPHSP:func.13">UnusedVx</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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c2">f</font></span>)</span>,<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> 
 
;<br/>
<a NAME="E7:70"/>
 <a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c2">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#K13" title="GRAPHSP:func.13">UnusedVx</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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c1">n</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="graphsp.html#T27" target="_self" title="GRAPHSP:th.27">Th28</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:70"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70"><span class="lab"><font color="Green" title="E58">A1</font></span></a>, <a class="txt" href="#E6:70"><span class="lab"><font color="Green" title="E61">A7</font></span></a>, <a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
