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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">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">IX</font> =  <a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">f</font>,<font color="Maroon">n</font>;<br/>
<b>assume </b><a NAME="E1:62"/>
 <a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">f</font>,<font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>then consider </b><font color="Maroon">i</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="E3:62"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <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">f</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">f</font>,<font color="Maroon">n</font> &amp; <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">f</font>,<font color="Maroon">n</font> &amp; ( 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="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">f</font>,<font color="Maroon">n</font> holds <br/><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Olive">k</font></span>)</span> ) &amp; ( 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="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">f</font>,<font color="Maroon">n</font> &amp; <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Olive">k</font></span>)</span> holds <br/><font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> ) )
 <b>by </b><i/>;<br/>
<a NAME="E4:62"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">f</font> <a href="goboard1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="real_1.html#K1">-</a> 1 &amp; <font color="Maroon">f</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="Olive">k</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="real_1.html#K1">-</a> 1 ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="graphsp.html#T4" target="_self">Th4</a></i>;<br/>
<b>then consider </b><font color="Maroon">k</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="E6:62"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">f</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="real_1.html#K1">-</a> 1 &amp; <font color="Maroon">f</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">k</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="real_1.html#K1">-</a> 1 )
 ;<br/>
<b>take </b>
<font color="Maroon">i</font>
;<br/>

<b>thus </b><a NAME="E7:62"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">f</font>,<font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="graphsp.html#T4" target="_self">Th4</a></i>;<br/>

<b>thus </b><a NAME="E8:62"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> )
 <b>by </b><i/>;<br/>

<b>thus </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">f</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<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="p4">(<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> <font color="Maroon">i</font>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i><a class="ref" href="graphsp.html#T4" target="_self">Th4</a>, </i>
<br/>.= 
 <a href="real_1.html#K1">-</a> 1
<b>by </b><i>, </i>
;<br/>


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