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<b>let </b><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> ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:25_1"/><i><font color="Green" title="E20">A1</font></i>: 
<font color="Maroon" title="c1">MR</font> is <a href="matrprob.html#V2" title="MATRPROB:attr.2">m-nonnegative</a>
 ;<br/><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="E2:25_1"/><i><font color="Green" title="E21">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> <font color="Maroon" title="c1">MR</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">k</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="E1:25_1_1"/><i><font color="Green" title="E22">A3</font></i>: 
<font color="Maroon" title="c3">k</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">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c1">MR</font>,<font color="Maroon" title="c2">i</font></span>)</span>
 ;<br/><a NAME="E2:25_1_1"/><i><font color="Green" title="E23">A4</font></i>: 
<font color="Maroon" title="c3">k</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">(<span class="default"><font color="Maroon" title="c1">MR</font> <a href="matrprob.html#K1" title="MATRPROB:func.1">.</a> <font color="Maroon" title="c2">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="entropy1.html#E2:25_1"><i><font color="Green" title="E21">A2</font></i></a>, <a class="txt" href="entropy1.html#E1:25_1_1"><i><font color="Green" title="E22">A3</font></i></a>, <a class="ref" href="matrix_2.html#T18" title="MATRIX_2:th.18">MATRIX_2:18</a></i>;<br/><a NAME="E3:25_1_1"/><b>then </b><i><font color="Green" title="E24">A5</font></i>: 
<font color="Maroon" title="c3">k</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> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c1">MR</font></span>)</span>
 <b>by </b><i><a class="txt" href="entropy1.html#E2:25_1"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="entropy1.html#T18" target="_self" title="ENTROPY1:th.18">Th18</a></i>;<br/><a NAME="E4:25_1_1"/><i><font color="Green" title="E25">A6</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">k</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c1">MR</font>
 <b>by </b><i><a class="txt" href="entropy1.html#E2:25_1"><i><font color="Green" title="E21">A2</font></i></a>, <a class="txt" href="entropy1.html#E2:25_1_1"><i><font color="Green" title="E23">A4</font></i></a>, <a class="ref" href="matrprob.html#T13" title="MATRPROB:th.13">MATRPROB:13</a></i>;<br/><a NAME="E5:25_1_1"/>
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c1">MR</font>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">MR</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">k</font>
 <b>by </b><i><a class="txt" href="entropy1.html#E3:25_1_1"><i><font color="Green" title="E24">A5</font></i></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></i>;<br/><b>hence </b><a NAME="E6:25_1_1"/>
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c1">MR</font>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="entropy1.html#E1:25_1"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="entropy1.html#E4:25_1_1"><i><font color="Green" title="E25">A6</font></i></a>, <a class="ref" href="matrprob.html#D6" title="MATRPROB:def.6">MATRPROB:def 6</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:25_1"/>
 <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c1">MR</font>,<font color="Maroon" title="c2">i</font> 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>assume </b><a NAME="E2:25"/><i><font color="Green" title="E20">A7</font></i>: 
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> <font color="Maroon" title="c1">MR</font> holds <br/> <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c1">MR</font>,<font color="Olive" title="b1">i</font> is <a href="partfun3.html#V4" title="PARTFUN3:attr.4">nonnegative</a>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c2">i</font>, <font color="Maroon" title="c3">j</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="E1:25_2"/><i><font color="Green" title="E21">A8</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c1">MR</font>
 ;<br/><a NAME="E2:25_2"/><i><font color="Green" title="E22">A9</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#K2" title="FINSEQ_1:func.2">Seg</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; <font color="Maroon" title="c3">j</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> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c1">MR</font></span>)</span> )
 <b>by </b><i><a class="txt" href="entropy1.html#E1:25_2"><i><font color="Green" title="E21">A8</font></i></a>, <a class="ref" href="matrprob.html#T12" title="MATRPROB:th.12">MATRPROB:12</a></i>;<br/><a NAME="E3:25_2"/><b>then </b>
<font color="Maroon" title="c3">j</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> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c1">MR</font>,<font color="Maroon" title="c2">i</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></i>;<br/><a NAME="E4:25_2"/><b>then </b><i><font color="Green" title="E23">A10</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> <font color="Maroon" title="c1">MR</font> &amp; <font color="Maroon" title="c3">j</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">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c1">MR</font>,<font color="Maroon" title="c2">i</font></span>)</span> )
 <b>by </b><i><a class="txt" href="entropy1.html#E2:25_2"><i><font color="Green" title="E22">A9</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E5:25_2"/><b>then </b><i><font color="Green" title="E24">A11</font></i>: 
 <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c1">MR</font>,<font color="Maroon" title="c2">i</font> is <a href="partfun3.html#V4" title="PARTFUN3:attr.4">nonnegative</a>
 <b>by </b><i><a class="txt" href="entropy1.html#E2:25"><i><font color="Green" title="E20">A7</font></i></a></i>;<br/><a NAME="E6:25_2"/>
<font color="Maroon" title="c1">MR</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c1">MR</font>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">j</font>
 <b>by </b><i><a class="txt" href="entropy1.html#E2:25_2"><i><font color="Green" title="E22">A9</font></i></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></i>;<br/><b>hence </b><a NAME="E7:25_2"/>
<font color="Maroon" title="c1">MR</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="entropy1.html#E4:25_2"><i><font color="Green" title="E23">A10</font></i></a>, <a class="txt" href="entropy1.html#E5:25_2"><i><font color="Green" title="E24">A11</font></i></a>, <a class="ref" href="entropy1.html#D1" target="_self" title="ENTROPY1:def.1">Def1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:25"/>
<font color="Maroon" title="c1">MR</font> is <a href="matrprob.html#V2" title="MATRPROB:attr.2">m-nonnegative</a>
 <b>by </b><i><a class="ref" href="matrprob.html#D6" title="MATRPROB:def.6">MATRPROB:def 6</a></i>;<br/>


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