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<div class="add">

<b>let </b><font color="Maroon" title="c1">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">MR</font> being  <a href="matrix_1.html#NM3" title="MATRIX_1:NM.3">Matrix</a> of <font color="Maroon" title="c1">n</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  holds <br/> ( <font color="Olive" title="b1">MR</font> is <a href="entropy1.html#V1" title="ENTROPY1:attr.1">diagonal</a> iff for <font color="Olive" title="b2">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="b2">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="Olive" title="b1">MR</font> holds <br/> <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Olive" title="b1">MR</font>,<font color="Olive" title="b2">i</font> <a href="entropy1.html#R1" title="ENTROPY1:pred.1">has_onlyone_value_in</a> <font color="Olive" title="b2">i</font> )</span><br/><b>let </b><font color="Maroon" title="c2">MR</font> be   <a href="matrix_1.html#NM3" title="MATRIX_1:NM.3">Matrix</a> of <font color="Maroon" title="c1">n</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">MR</font> is <a href="entropy1.html#V1" title="ENTROPY1:attr.1">diagonal</a> iff 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="c2">MR</font> holds <br/> <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c2">MR</font>,<font color="Olive" title="b1">i</font> <a href="entropy1.html#R1" title="ENTROPY1:pred.1">has_onlyone_value_in</a> <font color="Olive" title="b1">i</font> )</span><br/>



<a NAME="E1:32"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">MR</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">MR</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> )
 
<b>by </b><i><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></i>;<br/>
<a NAME="E2:32"/><b>then </b><i><font color="Green" title="E26">A1</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">MR</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c2">MR</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( 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="c2">MR</font> holds <br/> <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c2">MR</font>,<font color="Olive" title="b1">i</font> <a href="entropy1.html#R1" title="ENTROPY1:pred.1">has_onlyone_value_in</a> <font color="Olive" title="b1">i</font> ) implies <font color="Maroon" title="c2">MR</font> is <a href="entropy1.html#V1" title="ENTROPY1:attr.1">diagonal</a> )</span>
<div class="add"><b>assume </b><a NAME="E1:32_1"/><i><font color="Green" title="E27">A2</font></i>: 
<font color="Maroon" title="c2">MR</font> is <a href="entropy1.html#V1" title="ENTROPY1:attr.1">diagonal</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> 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="c2">MR</font> holds <br/> <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c2">MR</font>,<font color="Olive" title="b1">i</font> <a href="entropy1.html#R1" title="ENTROPY1:pred.1">has_onlyone_value_in</a> <font color="Olive" title="b1">i</font></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">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="c2">MR</font> implies  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c2">MR</font>,<font color="Maroon" title="c3">i</font> <a href="entropy1.html#R1" title="ENTROPY1:pred.1">has_onlyone_value_in</a> <font color="Maroon" title="c3">i</font> )</span><br/><b>assume </b><a NAME="E1:32_1_1"/><i><font color="Green" title="E28">A3</font></i>: 
<font color="Maroon" title="c3">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="c2">MR</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c2">MR</font>,<font color="Maroon" title="c3">i</font> <a href="entropy1.html#R1" title="ENTROPY1:pred.1">has_onlyone_value_in</a> <font color="Maroon" title="c3">i</font></span><br/><a NAME="E2:32_1_1"/><i><font color="Green" title="E29">A4</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c2">MR</font>,<font color="Maroon" title="c3">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">MR</font>
 <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="E3:32_1_1"/><b>then </b><i><font color="Green" title="E30">A5</font></i>: 
( <font color="Maroon" title="c3">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="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">MR</font></span>)</span> &amp; <font color="Maroon" title="c3">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">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c2">MR</font>,<font color="Maroon" title="c3">i</font></span>)</span> )
 <b>by </b><i><a class="txt" href="entropy1.html#E2:32"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="entropy1.html#E1:32_1_1"><i><font color="Green" title="E28">A3</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">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="c2">MR</font>,<font color="Maroon" title="c3">i</font></span>)</span> &amp; <font color="Maroon" title="c4">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">i</font> implies <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c2">MR</font>,<font color="Maroon" title="c3">i</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/><b>assume </b><a NAME="E1:32_1_1_1"/><i><font color="Green" title="E31">A6</font></i>: 
( <font color="Maroon" title="c4">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="c2">MR</font>,<font color="Maroon" title="c3">i</font></span>)</span> &amp; <font color="Maroon" title="c4">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">i</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c2">MR</font>,<font color="Maroon" title="c3">i</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/><a NAME="E2:32_1_1_1"/><i><font color="Green" title="E32">A7</font></i>: 
( <font color="Maroon" title="c4">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"><font color="Maroon" title="c2">MR</font> <a href="matrprob.html#K1" title="MATRPROB:func.1">.</a> <font color="Maroon" title="c3">i</font></span>)</span> &amp; <font color="Maroon" title="c4">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="c2">MR</font></span>)</span> )
 <b>by </b><i><a class="txt" href="entropy1.html#E1:32_1_1"><i><font color="Green" title="E28">A3</font></i></a>, <a class="txt" href="entropy1.html#E2:32_1_1"><i><font color="Green" title="E29">A4</font></i></a>, <a class="txt" href="entropy1.html#E1:32_1_1_1"><i><font color="Green" title="E31">A6</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a>, <a class="ref" href="matrix_2.html#T18" title="MATRIX_2:th.18">MATRIX_2:18</a></i>;<br/><a NAME="E3:32_1_1_1"/><b>then </b><i><font color="Green" title="E33">A8</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">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="c2">MR</font>
 <b>by </b><i><a class="txt" href="entropy1.html#E1:32_1_1"><i><font color="Green" title="E28">A3</font></i></a>, <a class="ref" href="matrprob.html#T13" title="MATRPROB:th.13">MATRPROB:13</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c2">MR</font>,<font color="Maroon" title="c3">i</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">j</font> = 
<font color="Maroon" title="c2">MR</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>
<b>by </b><i><a class="txt" href="entropy1.html#E2:32_1_1_1"><i><font color="Green" title="E32">A7</font></i></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></i>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="txt" href="entropy1.html#E1:32_1"><i><font color="Green" title="E27">A2</font></i></a>, <a class="txt" href="entropy1.html#E1:32_1_1_1"><i><font color="Green" title="E31">A6</font></i></a>, <a class="txt" href="entropy1.html#E3:32_1_1_1"><i><font color="Green" title="E33">A8</font></i></a>, <a class="ref" href="entropy1.html#D3" target="_self" title="ENTROPY1:def.3">Def3</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E5:32_1_1"/>
 <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c2">MR</font>,<font color="Maroon" title="c3">i</font> <a href="entropy1.html#R1" title="ENTROPY1:pred.1">has_onlyone_value_in</a> <font color="Maroon" title="c3">i</font>
 <b>by </b><i><a class="txt" href="entropy1.html#E3:32_1_1"><i><font color="Green" title="E30">A5</font></i></a>, <a class="ref" href="entropy1.html#D2" target="_self" title="ENTROPY1:def.2">Def2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E3:32_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> <font color="Maroon" title="c2">MR</font> holds <br/> <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c2">MR</font>,<font color="Olive" title="b1">i</font> <a href="entropy1.html#R1" title="ENTROPY1:pred.1">has_onlyone_value_in</a> <font color="Olive" title="b1">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E4:32"/><i><font color="Green" title="E27">A9</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="c2">MR</font> holds <br/> <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c2">MR</font>,<font color="Olive" title="b1">i</font> <a href="entropy1.html#R1" title="ENTROPY1:pred.1">has_onlyone_value_in</a> <font color="Olive" title="b1">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">MR</font> is <a href="entropy1.html#V1" title="ENTROPY1:attr.1">diagonal</a></span><br/>

<a NAME="E5:32"/>
for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</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 <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">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="c2">MR</font> &amp; <font color="Maroon" title="c2">MR</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/><font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">i</font>, <font color="Maroon" title="c4">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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">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="c2">MR</font> &amp; <font color="Maroon" title="c2">MR</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">j</font> )</span><br/>



<b>assume </b><a NAME="E1:32_2"/><i><font color="Green" title="E28">A10</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">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="c2">MR</font> &amp; <font color="Maroon" title="c2">MR</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">j</font></span><br/>

<b>assume </b><a NAME="E2:32_2"/><i><font color="Green" title="E29">A11</font></i>: 
<font color="Maroon" title="c3">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:32_2"/><i><font color="Green" title="E30">A12</font></i>: 
( <font color="Maroon" title="c3">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="c2">MR</font> &amp; <font color="Maroon" title="c4">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"><font color="Maroon" title="c2">MR</font> <a href="matrprob.html#K1" title="MATRPROB:func.1">.</a> <font color="Maroon" title="c3">i</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="entropy1.html#E1:32_2"><i><font color="Green" title="E28">A10</font></i></a>, <a class="ref" href="matrprob.html#T13" title="MATRPROB:th.13">MATRPROB:13</a></i>;<br/>
<a NAME="E4:32_2"/><b>then </b><i><font color="Green" title="E31">A13</font></i>: 
<font color="Maroon" title="c4">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="c2">MR</font>,<font color="Maroon" title="c3">i</font></span>)</span>
 
<b>by </b><i><a class="ref" href="matrix_2.html#T18" title="MATRIX_2:th.18">MATRIX_2:18</a></i>;<br/>
<a NAME="E5:32_2"/><i><font color="Green" title="E32">A14</font></i>: 
 <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c2">MR</font>,<font color="Maroon" title="c3">i</font> <a href="entropy1.html#R1" title="ENTROPY1:pred.1">has_onlyone_value_in</a> <font color="Maroon" title="c3">i</font>
 
<b>by </b><i><a class="txt" href="entropy1.html#E4:32"><i><font color="Green" title="E27">A9</font></i></a>, <a class="txt" href="entropy1.html#E3:32_2"><i><font color="Green" title="E30">A12</font></i></a></i>;<br/>
<a NAME="E6:32_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c2">MR</font>,<font color="Maroon" title="c3">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">MR</font>
 
<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="E7:32_2"/><b>then </b>
 <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="c2">MR</font>,<font color="Maroon" title="c3">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c2">MR</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>then </b><font color="Maroon" title="c2">MR</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font> = 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c2">MR</font>,<font color="Maroon" title="c3">i</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">j</font>
<b>by </b><i><a class="txt" href="entropy1.html#E4:32_2"><i><font color="Green" title="E31">A13</font></i></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></i>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="txt" href="entropy1.html#E2:32_2"><i><font color="Green" title="E29">A11</font></i></a>, <a class="txt" href="entropy1.html#E4:32_2"><i><font color="Green" title="E31">A13</font></i></a>, <a class="txt" href="entropy1.html#E5:32_2"><i><font color="Green" title="E32">A14</font></i></a>, <a class="ref" href="entropy1.html#D2" target="_self" title="ENTROPY1:def.2">Def2</a></i>
;<br/>
<b>hence </b><a NAME="E9:32_2"/>
contradiction
 <b>by </b><i><a class="txt" href="entropy1.html#E1:32_2"><i><font color="Green" title="E28">A10</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:32"/>
<font color="Maroon" title="c2">MR</font> is <a href="entropy1.html#V1" title="ENTROPY1:attr.1">diagonal</a>
 <b>by </b><i><a class="ref" href="entropy1.html#D3" target="_self" title="ENTROPY1:def.3">Def3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
