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<span class="kw">let </span><font color="Maroon" title="c1">M</font> be   <a href="matrprob.html#V2" title="MATRPROB:attr.2">m-nonnegative</a>  <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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">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="b1">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> <font color="Maroon" title="c1">M</font> holds <br/> <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<span class="p1">(<span class="default"><a href="entropy1.html#K5" title="ENTROPY1:func.5">Infor_FinSeq_of</a> <font color="Maroon" title="c1">M</font></span>)</span>,<font color="Olive" title="b1">k</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="entropy1.html#K4" title="ENTROPY1:func.4">Infor_FinSeq_of</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">M</font>,<font color="Olive" title="b1">k</font>)</span>)</span></span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">M1</font> =  <a href="entropy1.html#K5" title="ENTROPY1:func.5">Infor_FinSeq_of</a> <font color="Maroon" title="c1">M</font>;<br/>
<a NAME="E1:66"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="entropy1.html#K5" title="ENTROPY1:func.5">Infor_FinSeq_of</a> <font color="Maroon" title="c1">M</font></span>)</span> <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">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="entropy1.html#D8" target="_self" title="ENTROPY1:def.8">Def8</a></span>;<br/>
<a NAME="E2:66"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A1</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="entropy1.html#K5" title="ENTROPY1:func.5">Infor_FinSeq_of</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T29" title="FINSEQ_3:th.29">FINSEQ_3:29</a></span>;<br/>
<span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> <font color="Maroon" title="c1">M</font> implies  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<span class="p1">(<span class="default"><a href="entropy1.html#K5" title="ENTROPY1:func.5">Infor_FinSeq_of</a> <font color="Maroon" title="c1">M</font></span>)</span>,<font color="Maroon" title="c3">k</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="entropy1.html#K4" title="ENTROPY1:func.4">Infor_FinSeq_of</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">M</font>,<font color="Maroon" title="c3">k</font>)</span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E3:66"/><span class="lab"><font color="Green" title="E58">A2</font></span>: 
<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> <font color="Maroon" title="c1">M</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<span class="p1">(<span class="default"><a href="entropy1.html#K5" title="ENTROPY1:func.5">Infor_FinSeq_of</a> <font color="Maroon" title="c1">M</font></span>)</span>,<font color="Maroon" title="c3">k</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="entropy1.html#K4" title="ENTROPY1:func.4">Infor_FinSeq_of</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">M</font>,<font color="Maroon" title="c3">k</font>)</span>)</span></span><br/>

<span class="kw">thus </span> <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<span class="p1">(<span class="default"><a href="entropy1.html#K5" title="ENTROPY1:func.5">Infor_FinSeq_of</a> <font color="Maroon" title="c1">M</font></span>)</span>,<font color="Maroon" title="c3">k</font>) = 
<span class="p1">(<span class="default"><a href="entropy1.html#K5" title="ENTROPY1:func.5">Infor_FinSeq_of</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="matrprob.html#K1" title="MATRPROB:func.1">.</a> <font color="Maroon" title="c3">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:66"><span class="lab"><font color="Green" title="E58">A2</font></span></a>, <a class="txt" href="#E2:66"><span class="lab"><font color="Green" title="E57">A1</font></span></a>, <a class="ref" href="matrix_2.html#T16" title="MATRIX_2:th.16">MATRIX_2:16</a></span>
<br/>.= 
 <a href="entropy1.html#K4" title="ENTROPY1:func.4">Infor_FinSeq_of</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">M</font>,<font color="Maroon" title="c3">k</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:66"><span class="lab"><font color="Green" title="E58">A2</font></span></a>, <a class="txt" href="#E2:66"><span class="lab"><font color="Green" title="E57">A1</font></span></a>, <a class="ref" href="entropy1.html#D8" target="_self" title="ENTROPY1:def.8">Def8</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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