<?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">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>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>assume </b><a NAME="E1:72"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   holds <br/>(  not <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> or  not  <a href="graphsp.html#K8" target="_self">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" 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="Olive">i</font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> implies  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="graphsp.html#K14" target="_self">UnusedVx</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">a<sub>1</sub></font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="real_1.html#K5">-</a> <font color="Maroon">a<sub>1</sub></font> );<br/>
<a NAME="E2:72"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:72_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 ;<br/>

<b>set </b><font color="Maroon">f0</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> 0</span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">f</font>;<br/>
<a NAME="E2:72_1"/>
 <a href="graphsp.html#K14" target="_self">UnusedVx</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> 0</span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">n</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="graphsp.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E3:72_1"/><b>then </b>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="graphsp.html#K14" target="_self">UnusedVx</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> 0</span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="card_1.html#T80">CARD_1:80</a></i>;<br/>
<b>hence </b><a NAME="E4:72_1"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="graphsp.html#K14" target="_self">UnusedVx</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> 0</span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="real_1.html#K5">-</a> 0
 <b>by </b><i><a class="ref" href="finseq_1.html#T78">FINSEQ_1:78</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:72"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_2"><b>proof </b></a><div class="add">

<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>assume </b><a NAME="E1:72_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:72_2_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 ;<br/><a NAME="E2:72_2_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E3:72_2_1"/><b>then </b><i><font color="Green">E64</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E2:72_2_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>set </b><font color="Maroon">fk</font> =  <a href="graphsp.html#K14" target="_self">UnusedVx</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">k</font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">n</font>;<br/><b>set </b><font color="Maroon">fk1</font> =  <a href="graphsp.html#K14" target="_self">UnusedVx</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> <span class="p3">(<span class="default"><font color="Maroon">k</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></span>)</span>,<font color="Maroon">n</font>;<br/><a NAME="E4:72_2_1"/>
 <a href="graphsp.html#K8" target="_self">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" 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">k</font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="txt" href="graphsp.html#E5:72"><i><font color="Green">E65</font></i></a></i>;<br/><a NAME="E5:72_2_1"/><b>then </b>
 <a href="graphsp.html#K14" target="_self">UnusedVx</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> <span class="p3">(<span class="default"><font color="Maroon">k</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></span>)</span>,<font color="Maroon">n</font> <a href="xboole_0.html#R2">c&lt;</a>  <a href="graphsp.html#K14" target="_self">UnusedVx</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">k</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#E6"><i><font color="Green">Lemma56</font></i></a></i>;<br/><a NAME="E6:72_2_1"/><b>then </b>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="graphsp.html#K14" target="_self">UnusedVx</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> <span class="p4">(<span class="default"><font color="Maroon">k</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></span>)</span>,<font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="graphsp.html#K14" target="_self">UnusedVx</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">k</font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="card_2.html#T67">CARD_2:67</a></i>;<br/><a NAME="E7:72_2_1"/><b>then </b>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="graphsp.html#K14" target="_self">UnusedVx</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> <span class="p4">(<span class="default"><font color="Maroon">k</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></span>)</span>,<font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font> <a href="real_1.html#K5">-</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E1:72_2_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="graphsp.html#E2:72_2_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="graphsp.html#E3:72_2_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E8:72_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="graphsp.html#K14" target="_self">UnusedVx</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> <span class="p5">(<span class="default"><font color="Maroon">k</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></span>)</span>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="real_1.html#K5">-</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="int_1.html#T20">INT_1:20</a></i>;<br/><a NAME="E9:72_2_1"/><b>then </b>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="graphsp.html#K14" target="_self">UnusedVx</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> <span class="p4">(<span class="default"><font color="Maroon">k</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></span>)</span>,<font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K5">-</a> <font color="Maroon">k</font></span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><b>hence </b><a NAME="E10:72_2_1"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="graphsp.html#K14" target="_self">UnusedVx</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> <span class="p4">(<span class="default"><font color="Maroon">k</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></span>)</span>,<font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:72_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:72"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<a NAME="E5:72"/><b>then </b><i><font color="Green">E65</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>]
 
;<br/>
<b>set </b><font color="Maroon">nf</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">n</font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">f</font>;<br/>
<a NAME="E6:72"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="graphsp.html#K14" target="_self">UnusedVx</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">n</font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E7:72"/><b>then </b><i><font color="Green">E66</font></i>: 
 <a href="graphsp.html#K14" target="_self">UnusedVx</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">n</font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="graph_5.html#T5">GRAPH_5:5</a></i>;<br/>
<a NAME="E8:72"/>
 <a href="graphsp.html#K8" target="_self">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" 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">n</font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">n</font> <a href="tarski.html#R1">c=</a>  <a href="graphsp.html#K14" target="_self">UnusedVx</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">n</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/>;<br/>
<a NAME="E9:72"/><b>then </b>
 <a href="graphsp.html#K8" target="_self">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" 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">n</font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/>
<b>hence </b><a NAME="E10:72"/>
contradiction
 <b>by </b><i/>;<br/>


</div>
