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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">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">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>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> <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</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">a<sub>1</sub></font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">f</font></span>)</span>;<br/>
<a NAME="E1:75"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1"><b>proof </b></a><div class="add">

<b>thus </b> <a href="finseq_1.html#K4">dom</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> = 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K3" target="_self">id</a> <span class="p3">(<span class="default"><a href="numbers.html#K1">REAL</a>  <a href="finseq_2.html#K3">*</a> </span>)</span></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font>
<b>by </b><i><a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E2:75"/><i><font color="Green">E52</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]
 
<b>by </b><i><a class="ref" href="graphsp.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E3:75"/>
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>
<b>hence </b><a NAME="E4:75"/>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</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">i</font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">f</font></span>)</span>
 ;<br/>


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