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<div class="add">

<a NAME="E1:12_1_1"/><i><font color="Green" title="E11">A1</font></i>: 
<span class="p1"><a href="matrprob.html#K2" title="MATRPROB:func.2">&lt;*</a><span class="default">1</span><a href="matrprob.html#K2" title="MATRPROB:func.2">*&gt;</a></span> is <a href="partfun3.html#V4" title="PARTFUN3:attr.4">nonnegative</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon" title="c1">p</font> = <span class="p1"><a href="matrprob.html#K2" title="MATRPROB:func.2">&lt;*</a><span class="default">1</span><a href="matrprob.html#K2" title="MATRPROB:func.2">*&gt;</a></span>;<br/>
<a NAME="E1:12_1_1_1"/>
for <font color="Olive" title="b1">i</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="b1">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> <span class="p1"><a href="matrprob.html#K2" title="MATRPROB:func.2">&lt;*</a><span class="default">1</span><a href="matrprob.html#K2" title="MATRPROB:func.2">*&gt;</a></span> holds <br/><span class="p1"><a href="matrprob.html#K2" title="MATRPROB:func.2">&lt;*</a><span class="default">1</span><a href="matrprob.html#K2" title="MATRPROB:func.2">*&gt;</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">i</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>assume </b><a NAME="E1:12_1_1_1_1"/><i><font color="Green" title="E12">A2</font></i>: 
<font color="Maroon" title="c2">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> <span class="p1"><a href="matrprob.html#K2" title="MATRPROB:func.2">&lt;*</a><span class="default">1</span><a href="matrprob.html#K2" title="MATRPROB:func.2">*&gt;</a></span>
 ;<br/>

<a NAME="E2:12_1_1_1_1"/>
<font color="Maroon" title="c2">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1
 
<b>by </b><i><a class="txt" href="entropy1.html#E1:12_1_1_1_1"><i><font color="Green" title="E12">A2</font></i></a>, <a class="ref" href="finseq_1.html#D8" title="FINSEQ_1:def.8">FINSEQ_1:def 8</a></i>;<br/>
<a NAME="E3:12_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E4:12_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>
<b>hence </b><a NAME="E5:12_1_1_1_1"/>
<span class="p1"><a href="matrprob.html#K2" title="MATRPROB:func.2">&lt;*</a><span class="default">1</span><a href="matrprob.html#K2" title="MATRPROB:func.2">*&gt;</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:12_1_1_1"/>
<span class="p1"><a href="matrprob.html#K2" title="MATRPROB:func.2">&lt;*</a><span class="default">1</span><a href="matrprob.html#K2" title="MATRPROB:func.2">*&gt;</a></span> is <a href="partfun3.html#V4" title="PARTFUN3:attr.4">nonnegative</a>
 <b>by </b><i><a class="ref" href="entropy1.html#D1" target="_self" title="ENTROPY1:def.1">Def1</a></i>;<br/>


</div><b>end;</b></div>
<b>take </b>
<span class="p1"><a href="matrprob.html#K2" title="MATRPROB:func.2">&lt;*</a><span class="default">1</span><a href="matrprob.html#K2" title="MATRPROB:func.2">*&gt;</a></span>
;<br/>

<b>thus </b><a NAME="E2:12_1_1"/>
<span class="p1"><a href="matrprob.html#K2" title="MATRPROB:func.2">&lt;*</a><span class="default">1</span><a href="matrprob.html#K2" title="MATRPROB:func.2">*&gt;</a></span> is <a href="partfun3.html#V4" title="PARTFUN3:attr.4">nonnegative</a>
 <b>by </b><i><a class="txt" href="entropy1.html#E1:12_1_1"><i><font color="Green" title="E11">A1</font></i></a></i>;<br/>


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