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<b>let </b><font color="Maroon" title="c1">i</font>, <font color="Maroon" title="c2">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c3">f</font> be    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> ;<br/>





<b>set </b><font color="Maroon" title="c4">k</font> =  <a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p1">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">n</font>;<br/>
<b>set </b><font color="Maroon" title="c5">mi</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default">3 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1;<br/>
<b>assume </b><a NAME="E1:54"/><i><font color="Green" title="E43">A1</font></i>: 
( <font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> &amp; <font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">n</font> &amp; <font color="Maroon" title="c1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default">3 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 ;<br/>

<a NAME="E2:54"/><i><font color="Green" title="E44">A2</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p4">(<span class="default">3 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="graphsp.html#K2" title="GRAPHSP:func.2">:=</a> <span class="p2">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p3">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c1">i</font>
 
<b>by </b><i><a class="ref" href="graphsp.html#D11" target="_self" title="GRAPHSP:def.11">Def11</a></i>;<br/>
<a NAME="E3:54"/>
 <a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p1">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font>
 
<b>by </b><i><a class="ref" href="graphsp.html#T31" target="_self" title="GRAPHSP:th.31">Th31</a></i>;<br/>
<b>hence </b><a NAME="E4:54"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c1">i</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E1:54"><i><font color="Green" title="E43">A1</font></i></a>, <a class="txt" href="graphsp.html#E2:54"><i><font color="Green" title="E44">A2</font></i></a>, <a class="ref" href="graphsp.html#T19" target="_self" title="GRAPHSP:th.19">Th19</a></i>;<br/>


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