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<span class="kw">let </span><font color="Maroon" title="c1">MR</font> be   <a href="matrix_1.html#NM1" title="MATRIX_1:NM.1">Matrix</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="matrprob.html#K5" title="MATRPROB:func.5">SumAll</a> <font color="Maroon" title="c1">MR</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="entropy1.html#K2" title="ENTROPY1:func.2">Mx2FinS</a> <font color="Maroon" title="c1">MR</font></span>)</span></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:52_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">MR</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">MR</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1"><span class="kw">suppose </span></a><a NAME="E1:52_1_1"/><span class="lab"><font color="Green" title="E46">A1</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">MR</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="matrprob.html#K5" title="MATRPROB:func.5">SumAll</a> <font color="Maroon" title="c1">MR</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="entropy1.html#K2" title="ENTROPY1:func.2">Mx2FinS</a> <font color="Maroon" title="c1">MR</font></span>)</span></span><br/><div class="add"><span class="kw">hence </span> <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="entropy1.html#K2" title="ENTROPY1:func.2">Mx2FinS</a> <font color="Maroon" title="c1">MR</font></span>)</span> = 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="entropy1.html#D5" target="_self" title="ENTROPY1:def.5">Def5</a>, <a class="ref" href="rvsum_1.html#T102" title="RVSUM_1:th.102">RVSUM_1:102</a></span>
<br/>.= 
 <a href="matrprob.html#K5" title="MATRPROB:func.5">SumAll</a> <font color="Maroon" title="c1">MR</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:52_1_1"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="ref" href="matrprob.html#T23" title="MATRPROB:th.23">MATRPROB:23</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_2"><span class="kw">suppose </span></a><a NAME="E1:52_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">MR</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="matrprob.html#K5" title="MATRPROB:func.5">SumAll</a> <font color="Maroon" title="c1">MR</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="entropy1.html#K2" title="ENTROPY1:func.2">Mx2FinS</a> <font color="Maroon" title="c1">MR</font></span>)</span></span><br/><div class="add"><a NAME="E2:52_1_2"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">p</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</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>  st <br/>(  <a href="entropy1.html#K2" title="ENTROPY1:func.2">Mx2FinS</a> <font color="Maroon" title="c1">MR</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> <a href="matrprob.html#K1" title="MATRPROB:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">MR</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">MR</font> &amp; <font color="Olive" title="b1">p</font> <a href="matrprob.html#K1" title="MATRPROB:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">MR</font> <a href="matrprob.html#K1" title="MATRPROB:func.1">.</a> 1 &amp; ( for <font color="Olive" title="b2">k</font> being   <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>   st <font color="Olive" title="b2">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Olive" title="b2">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">MR</font> holds <br/><font color="Olive" title="b1">p</font> <a href="matrprob.html#K1" title="MATRPROB:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="matrprob.html#K1" title="MATRPROB:func.1">.</a> <font color="Olive" title="b2">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">MR</font> <a href="matrprob.html#K1" title="MATRPROB:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="entropy1.html#D5" target="_self" title="ENTROPY1:def.5">Def5</a></span>;<br/><span class="kw">hence </span><a NAME="E3:52_1_2"/>
 <a href="matrprob.html#K5" title="MATRPROB:func.5">SumAll</a> <font color="Maroon" title="c1">MR</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="entropy1.html#K2" title="ENTROPY1:func.2">Mx2FinS</a> <font color="Maroon" title="c1">MR</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="entropy1.html#T38" target="_self" title="ENTROPY1:th.38">Th38</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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