<?xml version="1.0"?>
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<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">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">g</font> be    <a href="finseq_2.html#M2">Element</a> of <a href="numbers.html#K1">REAL</a>  <a href="finseq_2.html#K3">*</a> ;<br/><b>let </b><font color="Maroon">f</font> be    <a href="finseq_2.html#M2">Element</a> of <a href="numbers.html#K1">REAL</a>  <a href="finseq_2.html#K3">*</a> ;<br/><b>let </b><font color="Maroon">h</font> be    <a href="finseq_2.html#M2">Element</a> of <a href="numbers.html#K1">REAL</a>  <a href="finseq_2.html#K3">*</a> ;<br/>











<b>set </b><font color="Maroon">R</font> =  <a href="graphsp.html#K20" target="_self">Relax</a> <font color="Maroon">n</font>;<br/>
<b>set </b><font color="Maroon">M</font> =  <a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">n</font>;<br/>
<b>set </b><font color="Maroon">RF</font> =  <a href="graphsp.html#K6" target="_self">repeat</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K20" target="_self">Relax</a> <font color="Maroon">n</font></span>)</span> <a href="graphsp.html#K4" target="_self">*</a> <span class="p2">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">n</font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">mi</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default">3 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1;<br/>
<b>set </b><font color="Maroon">Ak</font> =  <a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p1">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">g</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">g</font>,<font color="Maroon">n</font>;<br/>
<b>assume </b><a NAME="E1:96"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K6" target="_self">repeat</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="graphsp.html#K20" target="_self">Relax</a> <font color="Maroon">n</font></span>)</span> <a href="graphsp.html#K4" target="_self">*</a> <span class="p4">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">f</font> &amp; <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K6" target="_self">repeat</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="graphsp.html#K20" target="_self">Relax</a> <font color="Maroon">n</font></span>)</span> <a href="graphsp.html#K4" target="_self">*</a> <span class="p4">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">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> &amp;  <a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">g</font>,<font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">g</font>,<font color="Maroon">n</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default">3 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/>

<a NAME="E2:96"/><i><font color="Green">E52</font></i>: 
<span class="p1">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">n</font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default">3 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graphsp.html#K2" target="_self">:=</a> <span class="p1">(<span class="default"><a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p2">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">g</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">g</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>
 
<b>by </b><i/>;<br/>
<i><font color="Green">E53</font></i>: <font color="Maroon">h</font> = 
<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#K7" target="_self">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">n</font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">g</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
 <a href="graphsp.html#K19" target="_self">Relax</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">n</font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">g</font></span>)</span>,<font color="Maroon">n</font>
<b>by </b><i><a class="txt" href="graphsp.html#E2:96"><i><font color="Green">E52</font></i></a></i>
;<br/>
<a NAME="E4:96"/><i><font color="Green">E54</font></i>: 
(  <a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p1">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">g</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">g</font>,<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">g</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p2">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">g</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">g</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="real_1.html#K1">-</a> 1 )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:96"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">j</font> where <font color="Olive">j</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">g</font> <a href="goboard1.html#K1">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 ) </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">j</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E7:96"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">g</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 )
 ;<br/>
<b>set </b><font color="Maroon">ni</font> = <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font>;<br/>
<a NAME="E8:96"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E9:96"/><b>then </b><i><font color="Green">E64</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font>
 
;<br/>
<a NAME="E10:96"/><i><font color="Green">E65</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="ref" href="binarith.html#T39">BINARITH:39</a></i>;<br/>
<a NAME="E11:96"/>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E12:96"/><b>then </b><i><font color="Green">E66</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E13:96"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default">3 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i/>;<br/>
<a NAME="E14:96"/><i><font color="Green">E68</font></i>: 
2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default">3 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i/>;<br/>
<a NAME="E15:96"/><b>then </b><i><font color="Green">E69</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default">3 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E16:96"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<a NAME="E17:96"/><b>then </b>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E18:96"/><b>then </b><i><font color="Green">E70</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E19:96"/><b>then </b><i><font color="Green">E71</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">n</font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">g</font></span>)</span>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:96_2"/>
<span class="p1">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">n</font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">g</font> <a href="graphsp.html#R1" target="_self">hasBetterPathAt</a> <font color="Maroon">n</font>,<font color="Maroon">i</font>
 ;<br/><a NAME="E2:96_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">n</font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">g</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="real_1.html#K1">-</a> 1
 <b>by </b><i><a class="ref" href="graphsp.html#T4" target="_self">Th4</a></i>;<br/><b>hence </b><a NAME="E3:96_2"/>
contradiction
 <b>by </b><i><a class="txt" href="graphsp.html#E13"><i><font color="Green">Lemma80</font></i></a>, , , <a class="ref" href="graphsp.html#T12" target="_self">Th12</a>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><font color="Maroon">h</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">n</font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">g</font></span>)</span> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="ref" href="graphsp.html#T11" target="_self">Th11</a>, , , <a class="txt" href="graphsp.html#E15"><i><font color="Green">Lemma87</font></i></a>, <a class="ref" href="graphsp.html#T13" target="_self">Th13</a>, </i>
<br/>.= 
<font color="Maroon">g</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="txt" href="graphsp.html#E13"><i><font color="Green">Lemma80</font></i></a>, , , <a class="txt" href="graphsp.html#E15"><i><font color="Green">Lemma87</font></i></a>, , , </i>
;<br/><br/>


</div>
