<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">M</font> be   <a href="matrix_1.html#NM1" title="MATRIX_1:NM.1">Matrix</a> of <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> ;<br/>

<b>assume </b><a NAME="E1:61"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">M</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> 
 ;<br/>

<a NAME="E2:61"/><b>then </b>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">M</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T8" title="NAT_1:th.8">NAT_1:8</a></i>;<br/>
<a NAME="E3:61"/><b>then </b>
<span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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">M</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/>
<a NAME="E4:61"/><b>then </b><i><font color="Green" title="E45">A1</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.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">M</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>for <font color="Olive" title="b1">N</font> being  <a href="matrix_1.html#NM1" title="MATRIX_1:NM.1">Matrix</a> of <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>   st $1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <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="Olive" title="b1">N</font> holds <br/> <a href="rvsum_1.html#K17" title="RVSUM_1:func.17">Sum</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K10" title="MATRIXC1:func.10">LineSum</a> <font color="Olive" title="b1">N</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K17" title="RVSUM_1:func.17">Sum</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K11" title="MATRIXC1:func.11">ColSum</a> <font color="Olive" title="b1">N</font></span>)</span>;<br/>
<a NAME="E5:61"/>
for <font color="Olive" title="b1">N</font> being  <a href="matrix_1.html#NM1" title="MATRIX_1:NM.1">Matrix</a> of <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>   st <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <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="Olive" title="b1">N</font> holds <br/> <a href="rvsum_1.html#K17" title="RVSUM_1:func.17">Sum</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K10" title="MATRIXC1:func.10">LineSum</a> <font color="Olive" title="b1">N</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K17" title="RVSUM_1:func.17">Sum</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K11" title="MATRIXC1:func.11">ColSum</a> <font color="Olive" title="b1">N</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">N</font> be   <a href="matrix_1.html#NM1" title="MATRIX_1:NM.1">Matrix</a> of <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> ;<br/>

<b>assume </b><a NAME="E1:61_1"/><i><font color="Green" title="E46">A2</font></i>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <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="c2">N</font>
 ;<br/>

<a NAME="E2:61_1"/><b>then </b><i><font color="Green" title="E47">A3</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K10" title="MATRIXC1:func.10">LineSum</a> <font color="Maroon" title="c2">N</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="matrixc1.html#D10" target="_self" title="MATRIXC1:def.10">Def10</a></i>;<br/>
<a NAME="E3:61_1"/><i><font color="Green" title="E48">A4</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K11" title="MATRIXC1:func.11">ColSum</a> <font color="Maroon" title="c2">N</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">N</font>
 
<b>by </b><i><a class="ref" href="matrixc1.html#D11" target="_self" title="MATRIXC1:def.11">Def11</a></i>;<br/>
<a NAME="E4:61_1"/><i><font color="Green" title="E49">A5</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">N</font>,1</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">N</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="E5:61_1"/><i><font color="Green" title="E50">A6</font></i>: 
1 <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="ref" href="finseq_1.html#T5" title="FINSEQ_1:th.5">FINSEQ_1:5</a></i>;<br/>
<a NAME="E6:61_1"/><i><font color="Green" title="E51">A7</font></i>: 
for <font color="Olive" title="b1">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">N</font> holds <br/><span class="p1">(<span class="default"><a href="matrixc1.html#K11" title="MATRIXC1:func.11">ColSum</a> <font color="Maroon" title="c2">N</font></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Olive" title="b1">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="c2">N</font>,1</span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Olive" title="b1">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:61_1_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">j</font> &amp; <font color="Maroon" title="c3">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">N</font> )
 ;<br/>

<a NAME="E2:61_1_1"/><b>then </b><i><font color="Green" title="E52">A8</font></i>: 
<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="c2">N</font></span>)</span>
 
<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="E3:61_1_1"/><i><font color="Green" title="E53">A9</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#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c2">N</font>,<font color="Maroon" title="c3">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="matrixc1.html#E1:61_1"><i><font color="Green" title="E46">A2</font></i></a>, <a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></i>;<br/>
<a NAME="E4:61_1_1"/><i><font color="Green" title="E54">A10</font></i>: 
1 <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">N</font>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E1:61_1"><i><font color="Green" title="E46">A2</font></i></a>, <a class="txt" href="matrixc1.html#E5:61_1"><i><font color="Green" title="E50">A6</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<span class="p1">(<span class="default"><a href="matrixc1.html#K11" title="MATRIXC1:func.11">ColSum</a> <font color="Maroon" title="c2">N</font></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c3">j</font> = 
 <a href="rvsum_1.html#K17" title="RVSUM_1:func.17">Sum</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c2">N</font>,<font color="Maroon" title="c3">j</font></span>)</span>
<b>by </b><i><a class="txt" href="matrixc1.html#E2:61_1_1"><i><font color="Green" title="E52">A8</font></i></a>, <a class="ref" href="matrixc1.html#D11" target="_self" title="MATRIXC1:def.11">Def11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c2">N</font>,<font color="Maroon" title="c3">j</font></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> 1
<b>by </b><i><a class="txt" href="matrixc1.html#E3:61_1_1"><i><font color="Green" title="E53">A9</font></i></a>, <a class="ref" href="matrixc1.html#T50" target="_self" title="MATRIXC1:th.50">Th50</a></i>
<br/>.= 
<font color="Maroon" title="c2">N</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c3">j</font>
<b>by </b><i><a class="txt" href="matrixc1.html#E4:61_1_1"><i><font color="Green" title="E54">A10</font></i></a>, <a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c2">N</font>,1</span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c3">j</font>
<b>by </b><i><a class="txt" href="matrixc1.html#E2:61_1_1"><i><font color="Green" title="E52">A8</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="E6:61_1_1"/>
<span class="p1">(<span class="default"><a href="matrixc1.html#K11" title="MATRIXC1:func.11">ColSum</a> <font color="Maroon" title="c2">N</font></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <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="c2">N</font>,1</span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c3">j</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E7:61_1"/><i><font color="Green" title="E52">A11</font></i>: 
1 <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="c2">N</font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E1:61_1"><i><font color="Green" title="E46">A2</font></i></a>, <a class="ref" href="finseq_1.html#T5" title="FINSEQ_1:th.5">FINSEQ_1:5</a></i>;<br/>
 <a href="rvsum_1.html#K17" title="RVSUM_1:func.17">Sum</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K11" title="MATRIXC1:func.11">ColSum</a> <font color="Maroon" title="c2">N</font></span>)</span> = 
 <a href="rvsum_1.html#K17" title="RVSUM_1:func.17">Sum</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">N</font>,1</span>)</span>
<b>by </b><i><a class="txt" href="matrixc1.html#E3:61_1"><i><font color="Green" title="E48">A4</font></i></a>, <a class="txt" href="matrixc1.html#E4:61_1"><i><font color="Green" title="E49">A5</font></i></a>, <a class="txt" href="matrixc1.html#E6:61_1"><i><font color="Green" title="E51">A7</font></i></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrixc1.html#K10" title="MATRIXC1:func.10">LineSum</a> <font color="Maroon" title="c2">N</font></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> 1
<b>by </b><i><a class="txt" href="matrixc1.html#E7:61_1"><i><font color="Green" title="E52">A11</font></i></a>, <a class="ref" href="matrixc1.html#D10" target="_self" title="MATRIXC1:def.10">Def10</a></i>
<br/>.= 
 <a href="rvsum_1.html#K17" title="RVSUM_1:func.17">Sum</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K10" title="MATRIXC1:func.10">LineSum</a> <font color="Maroon" title="c2">N</font></span>)</span>
<b>by </b><i><a class="txt" href="matrixc1.html#E2:61_1"><i><font color="Green" title="E47">A3</font></i></a>, <a class="ref" href="matrixc1.html#T50" target="_self" title="MATRIXC1:th.50">Th50</a></i>
;<br/>
<b>hence </b><a NAME="E9:61_1"/>
 <a href="rvsum_1.html#K17" title="RVSUM_1:func.17">Sum</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K10" title="MATRIXC1:func.10">LineSum</a> <font color="Maroon" title="c2">N</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K17" title="RVSUM_1:func.17">Sum</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K11" title="MATRIXC1:func.11">ColSum</a> <font color="Maroon" title="c2">N</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E6:61"/><b>then </b><i><font color="Green" title="E46">A12</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
;<br/>
<a NAME="E7:61"/><i><font color="Green" title="E47">A13</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:61_2"/><i><font color="Green" title="E48">A14</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">k</font>]
 ;<br/>

<a NAME="E2:61_2"/>
for <font color="Olive" title="b1">N</font> being  <a href="matrix_1.html#NM1" title="MATRIX_1:NM.1">Matrix</a> of <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>   st <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <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="Olive" title="b1">N</font> holds <br/> <a href="rvsum_1.html#K17" title="RVSUM_1:func.17">Sum</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K10" title="MATRIXC1:func.10">LineSum</a> <font color="Olive" title="b1">N</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K17" title="RVSUM_1:func.17">Sum</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K11" title="MATRIXC1:func.11">ColSum</a> <font color="Olive" title="b1">N</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">N</font> be   <a href="matrix_1.html#NM1" title="MATRIX_1:NM.1">Matrix</a> of <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> ;<br/>

<b>assume </b><a NAME="E1:61_2_1"/><i><font color="Green" title="E49">A15</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <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="c3">N</font>
 ;<br/>

<a NAME="E2:61_2_1"/><b>then </b><i><font color="Green" title="E50">A16</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">N</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E3:61_2_1"/><i><font color="Green" title="E51">A17</font></i>: 
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E4:61_2_1"/><i><font color="Green" title="E52">A18</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">N</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<b>by </b><i><a class="txt" href="matrixc1.html#E1:61_2_1"><i><font color="Green" title="E49">A15</font></i></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">k1</font> = <font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 as    <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>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E7:61_2_1"/><i><font color="Green" title="E53">A19</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">N</font> holds <br/> ex <font color="Olive" title="b2">p</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">p</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">n</font> )
 <b>by </b><i><a class="ref" href="matrix_1.html#T9" title="MATRIX_1:th.9">MATRIX_1:9</a></i>;<br/>
<a NAME="E8:61_2_1"/><i><font color="Green" title="E54">A20</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">N</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">k1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">N</font>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T21" title="FINSEQ_5:th.21">FINSEQ_5:21</a></i>;<br/>
<a NAME="E9:61_2_1"/><b>then </b>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">N</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">k1</font></span>)</span> holds <br/> ex <font color="Olive" title="b2">p</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">p</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">n</font> )
 
<b>by </b><i><a class="txt" href="matrixc1.html#E7:61_2_1"><i><font color="Green" title="E53">A19</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">N2</font> = <font color="Maroon" title="c3">N</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">k1</font> as   <a href="matrix_1.html#NM1" title="MATRIX_1:NM.1">Matrix</a> of <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  <b>by </b><i><a class="ref" href="matrix_1.html#T9" title="MATRIX_1:th.9">MATRIX_1:9</a></i>;<br/>
<a NAME="E11:61_2_1"/><i><font color="Green" title="E55">A21</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">N2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="matrixc1.html#E2:61_2_1"><i><font color="Green" title="E50">A16</font></i></a>, <a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E12:61_2_1"/><i><font color="Green" title="E56">A22</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">N2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<b>by </b><i><a class="txt" href="matrixc1.html#E2:61_2_1"><i><font color="Green" title="E50">A16</font></i></a>, <a class="txt" href="matrixc1.html#E3:61_2_1"><i><font color="Green" title="E51">A17</font></i></a>, <a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">n3</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E14:61_2_1"/><i><font color="Green" title="E57">A23</font></i>: 
for <font color="Olive" title="b1">x2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">N</font> holds <br/> ex <font color="Olive" title="b2">s</font> being  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>( <font color="Olive" title="b2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x2</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">n3</font> )
 <b>by </b><i><a class="ref" href="matrix_1.html#D1" title="MATRIX_1:def.1">MATRIX_1:def 1</a></i>;<br/>
<a NAME="E15:61_2_1"/>
1 <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="c3">N</font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E4:61_2_1"><i><font color="Green" title="E52">A18</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E16:61_2_1"/><b>then </b>
1 <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="c3">N</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E17:61_2_1"/><b>then </b><i><font color="Green" title="E58">A24</font></i>: 
<font color="Maroon" title="c3">N</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">N</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">s2</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E19:61_2_1"/><i><font color="Green" title="E59">A25</font></i>: 
( <font color="Maroon" title="c8">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">N</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">n3</font> )
 <b>by </b><i><a class="txt" href="matrixc1.html#E14:61_2_1"><i><font color="Green" title="E57">A23</font></i></a></i>;<br/>
<a NAME="E20:61_2_1"/>
1 <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="c6">N2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E12:61_2_1"><i><font color="Green" title="E56">A22</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E21:61_2_1"/><b>then </b>
1 <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="c6">N2</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E22:61_2_1"/><b>then </b><i><font color="Green" title="E60">A26</font></i>: 
<font color="Maroon" title="c6">N2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">N2</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">s3</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E24:61_2_1"/><i><font color="Green" title="E61">A27</font></i>: 
( <font color="Maroon" title="c9">s3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">N2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">s3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">n3</font> )
 <b>by </b><i><a class="txt" href="matrixc1.html#E8:61_2_1"><i><font color="Green" title="E54">A20</font></i></a>, <a class="txt" href="matrixc1.html#E14:61_2_1"><i><font color="Green" title="E57">A23</font></i></a></i>;<br/>
<a NAME="E25:61_2_1"/><i><font color="Green" title="E62">A28</font></i>: 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c6">N2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">n3</font>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E12:61_2_1"><i><font color="Green" title="E56">A22</font></i></a>, <a class="txt" href="matrixc1.html#E22:61_2_1"><i><font color="Green" title="E60">A26</font></i></a>, <a class="txt" href="matrixc1.html#E24:61_2_1"><i><font color="Green" title="E61">A27</font></i></a>, <a class="ref" href="matrix_1.html#D4" title="MATRIX_1:def.4">MATRIX_1:def 4</a></i>;<br/>
<a NAME="E26:61_2_1"/><b>then </b><i><font color="Green" title="E63">A29</font></i>: 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c6">N2</font> <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="c3">N</font>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E1:61_2_1"><i><font color="Green" title="E49">A15</font></i></a>, <a class="txt" href="matrixc1.html#E17:61_2_1"><i><font color="Green" title="E58">A24</font></i></a>, <a class="txt" href="matrixc1.html#E19:61_2_1"><i><font color="Green" title="E59">A25</font></i></a>, <a class="ref" href="matrix_1.html#D4" title="MATRIX_1:def.4">MATRIX_1:def 4</a></i>;<br/>
<a NAME="E27:61_2_1"/><i><font color="Green" title="E64">A30</font></i>: 
for <font color="Olive" title="b1">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">N</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixc1.html#K11" title="MATRIXC1:func.11">ColSum</a> <font color="Maroon" title="c6">N2</font></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Olive" title="b1">j</font></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">N</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Olive" title="b1">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K11" title="MATRIXC1:func.11">ColSum</a> <font color="Maroon" title="c3">N</font></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Olive" title="b1">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:61_2_1_1"/><i><font color="Green" title="E65">A31</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">j</font> &amp; <font color="Maroon" title="c10">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">N</font> )
 ;<br/>

<a NAME="E2:61_2_1_1"/><b>then </b><i><font color="Green" title="E66">A32</font></i>: 
<font color="Maroon" title="c10">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="c3">N</font></span>)</span>
 
<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="E3:61_2_1_1"/><i><font color="Green" title="E67">A33</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <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="c3">N</font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E1:61_2_1"><i><font color="Green" title="E49">A15</font></i></a>, <a class="ref" href="finseq_1.html#T6" title="FINSEQ_1:th.6">FINSEQ_1:6</a></i>;<br/>
<a NAME="E4:61_2_1_1"/><b>then </b><i><font color="Green" title="E68">A34</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <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="c3">N</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E5:61_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <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#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c3">N</font>,<font color="Maroon" title="c10">j</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E3:61_2_1_1"><i><font color="Green" title="E67">A33</font></i></a>, <a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></i>;<br/>
<a NAME="E6:61_2_1_1"/><b>then </b><i><font color="Green" title="E69">A35</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <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#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c3">N</font>,<font color="Maroon" title="c10">j</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/>
<a NAME="E7:61_2_1_1"/><i><font color="Green" title="E70">A36</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#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c3">N</font>,<font color="Maroon" title="c10">j</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="c3">N</font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></i>;<br/>
<b>set </b><font color="Maroon" title="c11">f</font> =  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c3">N</font>,<font color="Maroon" title="c10">j</font>;<br/>
<b>set </b><font color="Maroon" title="c12">g</font> =  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c6">N2</font>,<font color="Maroon" title="c10">j</font>;<br/>
<i><font color="Green" title="E71">A37</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#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c6">N2</font>,<font color="Maroon" title="c10">j</font></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">N2</font>
<b>by </b><i><a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></i>
<br/>.= 
<font color="Maroon" title="c4">k1</font>
<b>by </b><i><a class="txt" href="matrixc1.html#E2:61_2_1"><i><font color="Green" title="E50">A16</font></i></a>, <a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a></i>
;<br/>
<a NAME="E9:61_2_1_1"/><i><font color="Green" title="E72">A38</font></i>: 
<font color="Maroon" title="c4">k1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c3">N</font>,<font color="Maroon" title="c10">j</font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E2:61_2_1"><i><font color="Green" title="E50">A16</font></i></a>, <a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></i>;<br/>
<a NAME="E10:61_2_1_1"/><i><font color="Green" title="E73">A39</font></i>: 
 <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c6">N2</font>,<font color="Maroon" title="c10">j</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c3">N</font>,<font color="Maroon" title="c10">j</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">k1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:61_2_1_1_2"/><i><font color="Green" title="E74">A40</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c3">N</font>,<font color="Maroon" title="c10">j</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">k1</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> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c6">N2</font>,<font color="Maroon" title="c10">j</font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E8:61_2_1_1"><i><font color="Green" title="E71">A37</font></i></a>, <a class="txt" href="matrixc1.html#E9:61_2_1_1"><i><font color="Green" title="E72">A38</font></i></a>, <a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E2:61_2_1_1_2"/>
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c3">N</font>,<font color="Maroon" title="c10">j</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">k1</font></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c3">N</font>,<font color="Maroon" title="c10">j</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">k1</font></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c6">N2</font>,<font color="Maroon" title="c10">j</font></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Olive" title="b1">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:61_2_1_1_2_1"/><i><font color="Green" title="E75">A41</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">n</font> &amp; <font color="Maroon" title="c13">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c3">N</font>,<font color="Maroon" title="c10">j</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">k1</font></span>)</span> )
 ;<br/>

<a NAME="E2:61_2_1_1_2_1"/><b>then </b><i><font color="Green" title="E76">A42</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">n</font> &amp; <font color="Maroon" title="c13">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">k1</font> )
 
<b>by </b><i><a class="txt" href="matrixc1.html#E9:61_2_1_1"><i><font color="Green" title="E72">A38</font></i></a>, <a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E3:61_2_1_1_2_1"/><b>then </b><i><font color="Green" title="E77">A43</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">n</font> &amp; <font color="Maroon" title="c13">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">k1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 )
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E4:61_2_1_1_2_1"/>
<font color="Maroon" title="c13">n</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"><span class="p3">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c3">N</font>,<font color="Maroon" title="c10">j</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">k1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E1:61_2_1_1_2_1"><i><font color="Green" title="E75">A41</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E5:61_2_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c13">n</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"><span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c3">N</font>,<font color="Maroon" title="c10">j</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">k1</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/>
<a NAME="E6:61_2_1_1_2_1"/><b>then </b><i><font color="Green" title="E78">A44</font></i>: 
<font color="Maroon" title="c13">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c3">N</font>,<font color="Maroon" title="c10">j</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c4">k1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D15" title="FINSEQ_1:def.15">FINSEQ_1:def 15</a></i>;<br/>
<a NAME="E7:61_2_1_1_2_1"/>
<font color="Maroon" title="c13">n</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="c3">N</font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E1:61_2_1"><i><font color="Green" title="E49">A15</font></i></a>, <a class="txt" href="matrixc1.html#E3:61_2_1_1_2_1"><i><font color="Green" title="E77">A43</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E8:61_2_1_1_2_1"/><b>then </b><i><font color="Green" title="E79">A45</font></i>: 
<font color="Maroon" title="c13">n</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="c3">N</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E9:61_2_1_1_2_1"/>
<font color="Maroon" title="c13">n</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> <font color="Maroon" title="c4">k1</font>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E2:61_2_1_1_2_1"><i><font color="Green" title="E76">A42</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E10:61_2_1_1_2_1"/><b>then </b><i><font color="Green" title="E80">A46</font></i>: 
<font color="Maroon" title="c13">n</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="c6">N2</font>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E11:61_2_1"><i><font color="Green" title="E55">A21</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="E11:61_2_1_1_2_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">n</font> &amp; <font color="Maroon" title="c13">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">N</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">j</font> &amp; <font color="Maroon" title="c10">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">N</font> )
 
<b>by </b><i><a class="txt" href="matrixc1.html#E1:61_2_1"><i><font color="Green" title="E49">A15</font></i></a>, <a class="txt" href="matrixc1.html#E1:61_2_1_1"><i><font color="Green" title="E65">A31</font></i></a>, <a class="txt" href="matrixc1.html#E2:61_2_1_1_2_1"><i><font color="Green" title="E76">A42</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E12:61_2_1_1_2_1"/><b>then </b><i><font color="Green" title="E81">A47</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">n</font>,<font color="Maroon" title="c10">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="c3">N</font>
 
<b>by </b><i><a class="ref" href="matrixc1.html#T1" target="_self" title="MATRIXC1:th.1">Th1</a></i>;<br/>
<a NAME="E13:61_2_1_1_2_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">n</font> &amp; <font color="Maroon" title="c13">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">N2</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">j</font> &amp; <font color="Maroon" title="c10">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c6">N2</font> )
 
<b>by </b><i><a class="txt" href="matrixc1.html#E1:61_2_1"><i><font color="Green" title="E49">A15</font></i></a>, <a class="txt" href="matrixc1.html#E11:61_2_1"><i><font color="Green" title="E55">A21</font></i></a>, <a class="txt" href="matrixc1.html#E17:61_2_1"><i><font color="Green" title="E58">A24</font></i></a>, <a class="txt" href="matrixc1.html#E19:61_2_1"><i><font color="Green" title="E59">A25</font></i></a>, <a class="txt" href="matrixc1.html#E25:61_2_1"><i><font color="Green" title="E62">A28</font></i></a>, <a class="txt" href="matrixc1.html#E1:61_2_1_1"><i><font color="Green" title="E65">A31</font></i></a>, <a class="txt" href="matrixc1.html#E9:61_2_1_1"><i><font color="Green" title="E72">A38</font></i></a>, <a class="txt" href="matrixc1.html#E1:61_2_1_1_2_1"><i><font color="Green" title="E75">A41</font></i></a>, <a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a>, <a class="ref" href="matrix_1.html#D4" title="MATRIX_1:def.4">MATRIX_1:def 4</a></i>;<br/>
<a NAME="E14:61_2_1_1_2_1"/><b>then </b><i><font color="Green" title="E82">A48</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">n</font>,<font color="Maroon" title="c10">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="c6">N2</font>
 
<b>by </b><i><a class="ref" href="matrixc1.html#T1" target="_self" title="MATRIXC1:th.1">Th1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c14">pn</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> <b> such that </b><br/><a NAME="E16:61_2_1_1_2_1"/><i><font color="Green" title="E83">A49</font></i>: 
( <font color="Maroon" title="c14">pn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">N</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">n</font> &amp; <font color="Maroon" title="c3">N</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">n</font>,<font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">pn</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c10">j</font> )
 <b>by </b><i><a class="txt" href="matrixc1.html#E12:61_2_1_1_2_1"><i><font color="Green" title="E81">A47</font></i></a>, <a class="ref" href="matrix_1.html#D6" title="MATRIX_1:def.6">MATRIX_1:def 6</a></i>;<br/>
<a NAME="E17:61_2_1_1_2_1"/><i><font color="Green" title="E84">A50</font></i>: 
<font color="Maroon" title="c6">N2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">N</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">n</font>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E2:61_2_1_1_2_1"><i><font color="Green" title="E76">A42</font></i></a>, <a class="ref" href="finseq_3.html#T121" title="FINSEQ_3:th.121">FINSEQ_3:121</a></i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c3">N</font>,<font color="Maroon" title="c10">j</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">k1</font></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c13">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c3">N</font>,<font color="Maroon" title="c10">j</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c4">k1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">n</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D15" title="FINSEQ_1:def.15">FINSEQ_1:def 15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c3">N</font>,<font color="Maroon" title="c10">j</font></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c13">n</font>
<b>by </b><i><a class="txt" href="matrixc1.html#E6:61_2_1_1_2_1"><i><font color="Green" title="E78">A44</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>
<br/>.= 
<font color="Maroon" title="c3">N</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">n</font>,<font color="Maroon" title="c10">j</font>
<b>by </b><i><a class="txt" href="matrixc1.html#E8:61_2_1_1_2_1"><i><font color="Green" title="E79">A45</font></i></a>, <a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></i>
<br/>.= 
<font color="Maroon" title="c6">N2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">n</font>,<font color="Maroon" title="c10">j</font>
<b>by </b><i><a class="txt" href="matrixc1.html#E14:61_2_1_1_2_1"><i><font color="Green" title="E82">A48</font></i></a>, <a class="txt" href="matrixc1.html#E16:61_2_1_1_2_1"><i><font color="Green" title="E83">A49</font></i></a>, <a class="txt" href="matrixc1.html#E17:61_2_1_1_2_1"><i><font color="Green" title="E84">A50</font></i></a>, <a class="ref" href="matrix_1.html#D6" title="MATRIX_1:def.6">MATRIX_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c6">N2</font>,<font color="Maroon" title="c10">j</font></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c13">n</font>
<b>by </b><i><a class="txt" href="matrixc1.html#E10:61_2_1_1_2_1"><i><font color="Green" title="E80">A46</font></i></a>, <a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></i>
;<br/>
<b>hence </b><a NAME="E19:61_2_1_1_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c3">N</font>,<font color="Maroon" title="c10">j</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">k1</font></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c13">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c6">N2</font>,<font color="Maroon" title="c10">j</font></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c13">n</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:61_2_1_1_2"/>
 <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c6">N2</font>,<font color="Maroon" title="c10">j</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c3">N</font>,<font color="Maroon" title="c10">j</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">k1</font>
 <b>by </b><i><a class="txt" href="matrixc1.html#E1:61_2_1_1_2"><i><font color="Green" title="E74">A40</font></i></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></i>;<br/>


</div><b>end;</b></div>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixc1.html#K11" title="MATRIXC1:func.11">ColSum</a> <font color="Maroon" title="c6">N2</font></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c10">j</font></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">N</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c10">j</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K17" title="RVSUM_1:func.17">Sum</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c6">N2</font>,<font color="Maroon" title="c10">j</font></span>)</span></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">N</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c10">j</font></span>)</span>
<b>by </b><i><a class="txt" href="matrixc1.html#E26:61_2_1"><i><font color="Green" title="E63">A29</font></i></a>, <a class="txt" href="matrixc1.html#E2:61_2_1_1"><i><font color="Green" title="E66">A32</font></i></a>, <a class="ref" href="matrixc1.html#D11" target="_self" title="MATRIXC1:def.11">Def11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K17" title="RVSUM_1:func.17">Sum</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c6">N2</font>,<font color="Maroon" title="c10">j</font></span>)</span></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c3">N</font>,<font color="Maroon" title="c10">j</font></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="matrixc1.html#E4:61_2_1_1"><i><font color="Green" title="E68">A34</font></i></a>, <a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K17" title="RVSUM_1:func.17">Sum</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c6">N2</font>,<font color="Maroon" title="c10">j</font></span>)</span></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c3">N</font>,<font color="Maroon" title="c10">j</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="matrixc1.html#E6:61_2_1_1"><i><font color="Green" title="E69">A35</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
 <a href="rvsum_1.html#K17" title="RVSUM_1:func.17">Sum</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c3">N</font>,<font color="Maroon" title="c10">j</font></span>)</span>
<b>by </b><i><a class="txt" href="matrixc1.html#E1:61_2_1"><i><font color="Green" title="E49">A15</font></i></a>, <a class="txt" href="matrixc1.html#E7:61_2_1_1"><i><font color="Green" title="E70">A36</font></i></a>, <a class="txt" href="matrixc1.html#E10:61_2_1_1"><i><font color="Green" title="E73">A39</font></i></a>, <a class="ref" href="matrixc1.html#T51" target="_self" title="MATRIXC1:th.51">Th51</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrixc1.html#K11" title="MATRIXC1:func.11">ColSum</a> <font color="Maroon" title="c3">N</font></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c10">j</font>
<b>by </b><i><a class="txt" href="matrixc1.html#E2:61_2_1_1"><i><font color="Green" title="E66">A32</font></i></a>, <a class="ref" href="matrixc1.html#D11" target="_self" title="MATRIXC1:def.11">Def11</a></i>
;<br/>
<b>hence </b><a NAME="E12:61_2_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixc1.html#K11" title="MATRIXC1:func.11">ColSum</a> <font color="Maroon" title="c6">N2</font></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c10">j</font></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">N</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c10">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K11" title="MATRIXC1:func.11">ColSum</a> <font color="Maroon" title="c3">N</font></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c10">j</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E28:61_2_1"/><i><font color="Green" title="E65">A51</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <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="c3">N</font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E1:61_2_1"><i><font color="Green" title="E49">A15</font></i></a>, <a class="ref" href="finseq_1.html#T5" title="FINSEQ_1:th.5">FINSEQ_1:5</a></i>;<br/>
<a NAME="E29:61_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <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="matrixc1.html#K10" title="MATRIXC1:func.10">LineSum</a> <font color="Maroon" title="c3">N</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="matrixc1.html#D10" target="_self" title="MATRIXC1:def.10">Def10</a></i>;<br/>
<a NAME="E30:61_2_1"/><b>then </b><i><font color="Green" title="E66">A52</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <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="matrixc1.html#K10" title="MATRIXC1:func.10">LineSum</a> <font color="Maroon" title="c3">N</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/>
<a NAME="E31:61_2_1"/><i><font color="Green" title="E67">A53</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K11" title="MATRIXC1:func.11">ColSum</a> <font color="Maroon" title="c6">N2</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="c6">N2</font>
 
<b>by </b><i><a class="ref" href="matrixc1.html#D11" target="_self" title="MATRIXC1:def.11">Def11</a></i>;<br/>
<i><font color="Green" title="E68">A54</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K11" title="MATRIXC1:func.11">ColSum</a> <font color="Maroon" title="c6">N2</font></span>)</span> = 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c6">N2</font>
<b>by </b><i><a class="ref" href="matrixc1.html#D11" target="_self" title="MATRIXC1:def.11">Def11</a></i>
<br/>.= 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">N</font>
<b>by </b><i><a class="txt" href="matrixc1.html#E1:61_2_1"><i><font color="Green" title="E49">A15</font></i></a>, <a class="txt" href="matrixc1.html#E17:61_2_1"><i><font color="Green" title="E58">A24</font></i></a>, <a class="txt" href="matrixc1.html#E19:61_2_1"><i><font color="Green" title="E59">A25</font></i></a>, <a class="txt" href="matrixc1.html#E25:61_2_1"><i><font color="Green" title="E62">A28</font></i></a>, <a class="ref" href="matrix_1.html#D4" title="MATRIX_1:def.4">MATRIX_1:def 4</a></i>
<br/>.= 
 <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="c3">N</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</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="E33:61_2_1"/>
 <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="c3">N</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></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="c3">N</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/>
<b>then </b><i><font color="Green" title="E69">A55</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixc1.html#K11" title="MATRIXC1:func.11">ColSum</a> <font color="Maroon" title="c6">N2</font></span>)</span> <a href="complsp1.html#K3" title="COMPLSP1:func.3">+</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">N</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span> = 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">N</font>
<b>by </b><i><a class="txt" href="matrixc1.html#E26:61_2_1"><i><font color="Green" title="E63">A29</font></i></a>, <a class="txt" href="matrixc1.html#E31:61_2_1"><i><font color="Green" title="E67">A53</font></i></a>, <a class="ref" href="complsp2.html#T6" title="COMPLSP2:th.6">COMPLSP2:6</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K11" title="MATRIXC1:func.11">ColSum</a> <font color="Maroon" title="c3">N</font></span>)</span>
<b>by </b><i><a class="ref" href="matrixc1.html#D11" target="_self" title="MATRIXC1:def.11">Def11</a></i>
;<br/>
<a NAME="E35:61_2_1"/><i><font color="Green" title="E70">A56</font></i>: 
for <font color="Olive" title="b1">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K11" title="MATRIXC1:func.11">ColSum</a> <font color="Maroon" title="c3">N</font></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixc1.html#K11" title="MATRIXC1:func.11">ColSum</a> <font color="Maroon" title="c6">N2</font></span>)</span> <a href="complsp1.html#K3" title="COMPLSP1:func.3">+</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">N</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K11" title="MATRIXC1:func.11">ColSum</a> <font color="Maroon" title="c3">N</font></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Olive" title="b1">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:61_2_1_4"/><i><font color="Green" title="E71">A57</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">j</font> &amp; <font color="Maroon" title="c10">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K11" title="MATRIXC1:func.11">ColSum</a> <font color="Maroon" title="c3">N</font></span>)</span> )
 ;<br/>

<a NAME="E2:61_2_1_4"/><b>then </b><i><font color="Green" title="E72">A58</font></i>: 
<font color="Maroon" title="c10">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="matrixc1.html#K11" title="MATRIXC1:func.11">ColSum</a> <font color="Maroon" title="c3">N</font></span>)</span></span>)</span>
 
<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="E3:61_2_1_4"/>
<font color="Maroon" title="c10">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"><span class="p3">(<span class="default"><a href="matrixc1.html#K11" title="MATRIXC1:func.11">ColSum</a> <font color="Maroon" title="c6">N2</font></span>)</span> <a href="complsp1.html#K3" title="COMPLSP1:func.3">+</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">N</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E34:61_2_1"><i><font color="Green" title="E69">A55</font></i></a>, <a class="txt" href="matrixc1.html#E1:61_2_1_4"><i><font color="Green" title="E71">A57</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E4:61_2_1_4"/><b>then </b><i><font color="Green" title="E73">A59</font></i>: 
<font color="Maroon" title="c10">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"><span class="p2">(<span class="default"><a href="matrixc1.html#K11" title="MATRIXC1:func.11">ColSum</a> <font color="Maroon" title="c6">N2</font></span>)</span> <a href="complsp1.html#K3" title="COMPLSP1:func.3">+</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">N</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></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/>
<a NAME="E5:61_2_1_4"/><i><font color="Green" title="E74">A60</font></i>: 
<font color="Maroon" title="c10">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="c6">N2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E26:61_2_1"><i><font color="Green" title="E63">A29</font></i></a>, <a class="txt" href="matrixc1.html#E2:61_2_1_4"><i><font color="Green" title="E72">A58</font></i></a>, <a class="ref" href="matrixc1.html#D11" target="_self" title="MATRIXC1:def.11">Def11</a></i>;<br/>
<a NAME="E6:61_2_1_4"/><b>then </b><i><font color="Green" title="E75">A61</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">j</font> &amp; <font color="Maroon" title="c10">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">N</font> )
 
<b>by </b><i><a class="txt" href="matrixc1.html#E26:61_2_1"><i><font color="Green" title="E63">A29</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixc1.html#K11" title="MATRIXC1:func.11">ColSum</a> <font color="Maroon" title="c6">N2</font></span>)</span> <a href="complsp1.html#K3" title="COMPLSP1:func.3">+</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">N</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c10">j</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixc1.html#K11" title="MATRIXC1:func.11">ColSum</a> <font color="Maroon" title="c6">N2</font></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c10">j</font></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">N</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c10">j</font></span>)</span>
<b>by </b><i><a class="txt" href="matrixc1.html#E4:61_2_1_4"><i><font color="Green" title="E73">A59</font></i></a>, <a class="ref" href="complsp2.html#T1" title="COMPLSP2:th.1">COMPLSP2:1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixc1.html#K11" title="MATRIXC1:func.11">ColSum</a> <font color="Maroon" title="c6">N2</font></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c10">j</font></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">N</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c10">j</font></span>)</span>
<b>by </b><i><a class="txt" href="matrixc1.html#E26:61_2_1"><i><font color="Green" title="E63">A29</font></i></a>, <a class="txt" href="matrixc1.html#E5:61_2_1_4"><i><font color="Green" title="E74">A60</font></i></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrixc1.html#K11" title="MATRIXC1:func.11">ColSum</a> <font color="Maroon" title="c3">N</font></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c10">j</font>
<b>by </b><i><a class="txt" href="matrixc1.html#E27:61_2_1"><i><font color="Green" title="E64">A30</font></i></a>, <a class="txt" href="matrixc1.html#E6:61_2_1_4"><i><font color="Green" title="E75">A61</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E8:61_2_1_4"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixc1.html#K11" title="MATRIXC1:func.11">ColSum</a> <font color="Maroon" title="c6">N2</font></span>)</span> <a href="complsp1.html#K3" title="COMPLSP1:func.3">+</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">N</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K11" title="MATRIXC1:func.11">ColSum</a> <font color="Maroon" title="c3">N</font></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c10">j</font>
 ;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon" title="c10">g1</font> =  <a href="matrixc1.html#K10" title="MATRIXC1:func.10">LineSum</a> <font color="Maroon" title="c6">N2</font>;<br/>
<b>set </b><font color="Maroon" title="c11">f3</font> =  <a href="matrixc1.html#K10" title="MATRIXC1:func.10">LineSum</a> <font color="Maroon" title="c3">N</font>;<br/>
<a NAME="E36:61_2_1"/><i><font color="Green" title="E71">A62</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K10" title="MATRIXC1:func.10">LineSum</a> <font color="Maroon" title="c3">N</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="c3">N</font>
 
<b>by </b><i><a class="ref" href="matrixc1.html#D10" target="_self" title="MATRIXC1:def.10">Def10</a></i>;<br/>
<a NAME="E37:61_2_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K10" title="MATRIXC1:func.10">LineSum</a> <font color="Maroon" title="c3">N</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">k1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<b>by </b><i><a class="txt" href="matrixc1.html#E1:61_2_1"><i><font color="Green" title="E49">A15</font></i></a>, <a class="ref" href="matrixc1.html#D10" target="_self" title="MATRIXC1:def.10">Def10</a></i>;<br/>
<a NAME="E38:61_2_1"/><b>then </b><i><font color="Green" title="E72">A63</font></i>: 
<font color="Maroon" title="c4">k1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K10" title="MATRIXC1:func.10">LineSum</a> <font color="Maroon" title="c3">N</font></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<i><font color="Green" title="E73">A64</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K10" title="MATRIXC1:func.10">LineSum</a> <font color="Maroon" title="c6">N2</font></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">N2</font>
<b>by </b><i><a class="ref" href="matrixc1.html#D10" target="_self" title="MATRIXC1:def.10">Def10</a></i>
<br/>.= 
<font color="Maroon" title="c4">k1</font>
<b>by </b><i><a class="txt" href="matrixc1.html#E2:61_2_1"><i><font color="Green" title="E50">A16</font></i></a>, <a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a></i>
;<br/>
<a NAME="E40:61_2_1"/>
 <a href="matrixc1.html#K10" title="MATRIXC1:func.10">LineSum</a> <font color="Maroon" title="c6">N2</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K10" title="MATRIXC1:func.10">LineSum</a> <font color="Maroon" title="c3">N</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">k1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2_1_6"><b>proof </b></a><div class="add">

<a NAME="E1:61_2_1_6"/><i><font color="Green" title="E74">A65</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixc1.html#K10" title="MATRIXC1:func.10">LineSum</a> <font color="Maroon" title="c3">N</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">k1</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> <span class="p1">(<span class="default"><a href="matrixc1.html#K10" title="MATRIXC1:func.10">LineSum</a> <font color="Maroon" title="c6">N2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E38:61_2_1"><i><font color="Green" title="E72">A63</font></i></a>, <a class="txt" href="matrixc1.html#E39:61_2_1"><i><font color="Green" title="E73">A64</font></i></a>, <a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E2:61_2_1_6"/>
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixc1.html#K10" title="MATRIXC1:func.10">LineSum</a> <font color="Maroon" title="c3">N</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">k1</font></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixc1.html#K10" title="MATRIXC1:func.10">LineSum</a> <font color="Maroon" title="c3">N</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">k1</font></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K10" title="MATRIXC1:func.10">LineSum</a> <font color="Maroon" title="c6">N2</font></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Olive" title="b1">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2_1_6_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:61_2_1_6_1"/><i><font color="Green" title="E75">A66</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">n</font> &amp; <font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixc1.html#K10" title="MATRIXC1:func.10">LineSum</a> <font color="Maroon" title="c3">N</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">k1</font></span>)</span> )
 ;<br/>

<a NAME="E2:61_2_1_6_1"/><b>then </b><i><font color="Green" title="E76">A67</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">n</font> &amp; <font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">k1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 )
 
<b>by </b><i><a class="txt" href="matrixc1.html#E39:61_2_1"><i><font color="Green" title="E73">A64</font></i></a>, <a class="txt" href="matrixc1.html#E1:61_2_1_6"><i><font color="Green" title="E74">A65</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E3:61_2_1_6_1"/>
<font color="Maroon" title="c12">n</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"><span class="p3">(<span class="default"><a href="matrixc1.html#K10" title="MATRIXC1:func.10">LineSum</a> <font color="Maroon" title="c3">N</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">k1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E1:61_2_1_6_1"><i><font color="Green" title="E75">A66</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E4:61_2_1_6_1"/><b>then </b>
<font color="Maroon" title="c12">n</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"><span class="p2">(<span class="default"><a href="matrixc1.html#K10" title="MATRIXC1:func.10">LineSum</a> <font color="Maroon" title="c3">N</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">k1</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/>
<a NAME="E5:61_2_1_6_1"/><b>then </b><i><font color="Green" title="E77">A68</font></i>: 
<font color="Maroon" title="c12">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixc1.html#K10" title="MATRIXC1:func.10">LineSum</a> <font color="Maroon" title="c3">N</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c4">k1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D15" title="FINSEQ_1:def.15">FINSEQ_1:def 15</a></i>;<br/>
<a NAME="E6:61_2_1_6_1"/><i><font color="Green" title="E78">A69</font></i>: 
<font color="Maroon" title="c12">n</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="c3">N</font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E1:61_2_1"><i><font color="Green" title="E49">A15</font></i></a>, <a class="txt" href="matrixc1.html#E2:61_2_1_6_1"><i><font color="Green" title="E76">A67</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E7:61_2_1_6_1"/>
<font color="Maroon" title="c12">n</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> <font color="Maroon" title="c4">k1</font>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E39:61_2_1"><i><font color="Green" title="E73">A64</font></i></a>, <a class="txt" href="matrixc1.html#E1:61_2_1_6"><i><font color="Green" title="E74">A65</font></i></a>, <a class="txt" href="matrixc1.html#E1:61_2_1_6_1"><i><font color="Green" title="E75">A66</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E8:61_2_1_6_1"/><b>then </b><i><font color="Green" title="E79">A70</font></i>: 
<font color="Maroon" title="c12">n</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="c6">N2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E2:61_2_1"><i><font color="Green" title="E50">A16</font></i></a>, <a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E9:61_2_1_6_1"/><i><font color="Green" title="E80">A71</font></i>: 
 <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">N</font>,<font color="Maroon" title="c12">n</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c6">N2</font>,<font color="Maroon" title="c12">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2_1_6_1_1"><b>proof </b></a><div class="add">

<i><font color="Green" title="E81">A72</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="c3">N</font>,<font color="Maroon" title="c12">n</font></span>)</span> = 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">N</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 href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c6">N2</font>
<b>by </b><i><a class="txt" href="matrixc1.html#E1:61_2_1"><i><font color="Green" title="E49">A15</font></i></a>, <a class="txt" href="matrixc1.html#E17:61_2_1"><i><font color="Green" title="E58">A24</font></i></a>, <a class="txt" href="matrixc1.html#E19:61_2_1"><i><font color="Green" title="E59">A25</font></i></a>, <a class="txt" href="matrixc1.html#E25:61_2_1"><i><font color="Green" title="E62">A28</font></i></a>, <a class="ref" href="matrix_1.html#D4" title="MATRIX_1:def.4">MATRIX_1:def 4</a></i>
<br/>.= 
 <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="c6">N2</font>,<font color="Maroon" title="c12">n</font></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="E2:61_2_1_6_1_1"/>
for <font color="Olive" title="b1">n1</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n1</font> &amp; <font color="Olive" title="b1">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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="c3">N</font>,<font color="Maroon" title="c12">n</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">N</font>,<font color="Maroon" title="c12">n</font></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Olive" title="b1">n1</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="c6">N2</font>,<font color="Maroon" title="c12">n</font></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Olive" title="b1">n1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2_1_6_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">n1</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:61_2_1_6_1_1_2"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">n1</font> &amp; <font color="Maroon" title="c13">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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="c3">N</font>,<font color="Maroon" title="c12">n</font></span>)</span> )
 ;<br/>

<a NAME="E2:61_2_1_6_1_1_2"/><b>then </b><i><font color="Green" title="E82">A73</font></i>: 
<font color="Maroon" title="c13">n1</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="c3">N</font>,<font color="Maroon" title="c12">n</font></span>)</span></span>)</span>
 
<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="E3:61_2_1_6_1_1_2"/><b>then </b><i><font color="Green" title="E83">A74</font></i>: 
<font color="Maroon" title="c13">n1</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="c3">N</font></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="E4:61_2_1_6_1_1_2"/><b>then </b><i><font color="Green" title="E84">A75</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">n1</font> &amp; <font color="Maroon" title="c13">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">N</font> )
 
<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="E5:61_2_1_6_1_1_2"/><i><font color="Green" title="E85">A76</font></i>: 
<font color="Maroon" title="c13">n1</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="c3">N</font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E2:61_2_1_6_1_1_2"><i><font color="Green" title="E82">A73</font></i></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></i>;<br/>
<a NAME="E6:61_2_1_6_1_1_2"/><i><font color="Green" title="E86">A77</font></i>: 
<font color="Maroon" title="c13">n1</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="c6">N2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E26:61_2_1"><i><font color="Green" title="E63">A29</font></i></a>, <a class="txt" href="matrixc1.html#E2:61_2_1_6_1_1_2"><i><font color="Green" title="E82">A73</font></i></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></i>;<br/>
<a NAME="E7:61_2_1_6_1_1_2"/><i><font color="Green" title="E87">A78</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">n</font> &amp; <font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">N</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">n1</font> &amp; <font color="Maroon" title="c13">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">N</font> )
 
<b>by </b><i><a class="txt" href="matrixc1.html#E1:61_2_1"><i><font color="Green" title="E49">A15</font></i></a>, <a class="txt" href="matrixc1.html#E39:61_2_1"><i><font color="Green" title="E73">A64</font></i></a>, <a class="txt" href="matrixc1.html#E1:61_2_1_6"><i><font color="Green" title="E74">A65</font></i></a>, <a class="txt" href="matrixc1.html#E1:61_2_1_6_1"><i><font color="Green" title="E75">A66</font></i></a>, <a class="txt" href="matrixc1.html#E3:61_2_1_6_1_1_2"><i><font color="Green" title="E83">A74</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E8:61_2_1_6_1_1_2"/><i><font color="Green" title="E88">A79</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">n</font>,<font color="Maroon" title="c13">n1</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="c3">N</font>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E1:61_2_1"><i><font color="Green" title="E49">A15</font></i></a>, <a class="txt" href="matrixc1.html#E2:61_2_1_6_1"><i><font color="Green" title="E76">A67</font></i></a>, <a class="txt" href="matrixc1.html#E4:61_2_1_6_1_1_2"><i><font color="Green" title="E84">A75</font></i></a>, <a class="ref" href="matrixc1.html#T1" target="_self" title="MATRIXC1:th.1">Th1</a></i>;<br/>
<a NAME="E9:61_2_1_6_1_1_2"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">n</font> &amp; <font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">N2</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">n1</font> &amp; <font color="Maroon" title="c13">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c6">N2</font> )
 
<b>by </b><i><a class="txt" href="matrixc1.html#E17:61_2_1"><i><font color="Green" title="E58">A24</font></i></a>, <a class="txt" href="matrixc1.html#E19:61_2_1"><i><font color="Green" title="E59">A25</font></i></a>, <a class="txt" href="matrixc1.html#E25:61_2_1"><i><font color="Green" title="E62">A28</font></i></a>, <a class="txt" href="matrixc1.html#E1:61_2_1_6"><i><font color="Green" title="E74">A65</font></i></a>, <a class="txt" href="matrixc1.html#E1:61_2_1_6_1"><i><font color="Green" title="E75">A66</font></i></a>, <a class="txt" href="matrixc1.html#E7:61_2_1_6_1_1_2"><i><font color="Green" title="E87">A78</font></i></a>, <a class="ref" href="matrixc1.html#D10" target="_self" title="MATRIXC1:def.10">Def10</a>, <a class="ref" href="matrix_1.html#D4" title="MATRIX_1:def.4">MATRIX_1:def 4</a></i>;<br/>
<a NAME="E10:61_2_1_6_1_1_2"/><b>then </b><i><font color="Green" title="E89">A80</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">n</font>,<font color="Maroon" title="c13">n1</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="c6">N2</font>
 
<b>by </b><i><a class="ref" href="matrixc1.html#T1" target="_self" title="MATRIXC1:th.1">Th1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c14">pn</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> <b> such that </b><br/><a NAME="E12:61_2_1_6_1_1_2"/><i><font color="Green" title="E90">A81</font></i>: 
( <font color="Maroon" title="c14">pn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">N</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">n</font> &amp; <font color="Maroon" title="c3">N</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">n</font>,<font color="Maroon" title="c13">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">pn</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c13">n1</font> )
 <b>by </b><i><a class="txt" href="matrixc1.html#E8:61_2_1_6_1_1_2"><i><font color="Green" title="E88">A79</font></i></a>, <a class="ref" href="matrix_1.html#D6" title="MATRIX_1:def.6">MATRIX_1:def 6</a></i>;<br/>
<a NAME="E13:61_2_1_6_1_1_2"/><i><font color="Green" title="E91">A82</font></i>: 
<font color="Maroon" title="c6">N2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">N</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">n</font>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E39:61_2_1"><i><font color="Green" title="E73">A64</font></i></a>, <a class="txt" href="matrixc1.html#E1:61_2_1_6"><i><font color="Green" title="E74">A65</font></i></a>, <a class="txt" href="matrixc1.html#E1:61_2_1_6_1"><i><font color="Green" title="E75">A66</font></i></a>, <a class="ref" href="finseq_3.html#T121" title="FINSEQ_3:th.121">FINSEQ_3:121</a></i>;<br/>
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">N</font>,<font color="Maroon" title="c12">n</font></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c13">n1</font> = 
<font color="Maroon" title="c3">N</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">n</font>,<font color="Maroon" title="c13">n1</font>
<b>by </b><i><a class="txt" href="matrixc1.html#E5:61_2_1_6_1_1_2"><i><font color="Green" title="E85">A76</font></i></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></i>
<br/>.= 
<font color="Maroon" title="c6">N2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">n</font>,<font color="Maroon" title="c13">n1</font>
<b>by </b><i><a class="txt" href="matrixc1.html#E10:61_2_1_6_1_1_2"><i><font color="Green" title="E89">A80</font></i></a>, <a class="txt" href="matrixc1.html#E12:61_2_1_6_1_1_2"><i><font color="Green" title="E90">A81</font></i></a>, <a class="txt" href="matrixc1.html#E13:61_2_1_6_1_1_2"><i><font color="Green" title="E91">A82</font></i></a>, <a class="ref" href="matrix_1.html#D6" title="MATRIX_1:def.6">MATRIX_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c6">N2</font>,<font color="Maroon" title="c12">n</font></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c13">n1</font>
<b>by </b><i><a class="txt" href="matrixc1.html#E6:61_2_1_6_1_1_2"><i><font color="Green" title="E86">A77</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="E15:61_2_1_6_1_1_2"/>
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">N</font>,<font color="Maroon" title="c12">n</font></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c13">n1</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="c6">N2</font>,<font color="Maroon" title="c12">n</font></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c13">n1</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:61_2_1_6_1_1"/>
 <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">N</font>,<font color="Maroon" title="c12">n</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c6">N2</font>,<font color="Maroon" title="c12">n</font>
 <b>by </b><i><a class="txt" href="matrixc1.html#E1:61_2_1_6_1_1"><i><font color="Green" title="E81">A72</font></i></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></i>;<br/>


</div><b>end;</b></div>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixc1.html#K10" title="MATRIXC1:func.10">LineSum</a> <font color="Maroon" title="c3">N</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">k1</font></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c12">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixc1.html#K10" title="MATRIXC1:func.10">LineSum</a> <font color="Maroon" title="c3">N</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c4">k1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">n</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D15" title="FINSEQ_1:def.15">FINSEQ_1:def 15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrixc1.html#K10" title="MATRIXC1:func.10">LineSum</a> <font color="Maroon" title="c3">N</font></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c12">n</font>
<b>by </b><i><a class="txt" href="matrixc1.html#E5:61_2_1_6_1"><i><font color="Green" title="E77">A68</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>
<br/>.= 
 <a href="rvsum_1.html#K17" title="RVSUM_1:func.17">Sum</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c6">N2</font>,<font color="Maroon" title="c12">n</font></span>)</span>
<b>by </b><i><a class="txt" href="matrixc1.html#E6:61_2_1_6_1"><i><font color="Green" title="E78">A69</font></i></a>, <a class="txt" href="matrixc1.html#E9:61_2_1_6_1"><i><font color="Green" title="E80">A71</font></i></a>, <a class="ref" href="matrixc1.html#D10" target="_self" title="MATRIXC1:def.10">Def10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrixc1.html#K10" title="MATRIXC1:func.10">LineSum</a> <font color="Maroon" title="c6">N2</font></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c12">n</font>
<b>by </b><i><a class="txt" href="matrixc1.html#E8:61_2_1_6_1"><i><font color="Green" title="E79">A70</font></i></a>, <a class="ref" href="matrixc1.html#D10" target="_self" title="MATRIXC1:def.10">Def10</a></i>
;<br/>
<b>hence </b><a NAME="E11:61_2_1_6_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixc1.html#K10" title="MATRIXC1:func.10">LineSum</a> <font color="Maroon" title="c3">N</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">k1</font></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c12">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K10" title="MATRIXC1:func.10">LineSum</a> <font color="Maroon" title="c6">N2</font></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c12">n</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:61_2_1_6"/>
 <a href="matrixc1.html#K10" title="MATRIXC1:func.10">LineSum</a> <font color="Maroon" title="c6">N2</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K10" title="MATRIXC1:func.10">LineSum</a> <font color="Maroon" title="c3">N</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">k1</font>
 <b>by </b><i><a class="txt" href="matrixc1.html#E1:61_2_1_6"><i><font color="Green" title="E74">A65</font></i></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></i>;<br/>


</div><b>end;</b></div>
<b>then </b> <a href="rvsum_1.html#K17" title="RVSUM_1:func.17">Sum</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K10" title="MATRIXC1:func.10">LineSum</a> <font color="Maroon" title="c3">N</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K17" title="RVSUM_1:func.17">Sum</a> <span class="p2">(<span class="default"><a href="matrixc1.html#K10" title="MATRIXC1:func.10">LineSum</a> <font color="Maroon" title="c6">N2</font></span>)</span></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixc1.html#K10" title="MATRIXC1:func.10">LineSum</a> <font color="Maroon" title="c3">N</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="matrixc1.html#E1:61_2_1"><i><font color="Green" title="E49">A15</font></i></a>, <a class="txt" href="matrixc1.html#E36:61_2_1"><i><font color="Green" title="E71">A62</font></i></a>, <a class="ref" href="matrixc1.html#T51" target="_self" title="MATRIXC1:th.51">Th51</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K17" title="RVSUM_1:func.17">Sum</a> <span class="p2">(<span class="default"><a href="matrixc1.html#K10" title="MATRIXC1:func.10">LineSum</a> <font color="Maroon" title="c6">N2</font></span>)</span></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixc1.html#K10" title="MATRIXC1:func.10">LineSum</a> <font color="Maroon" title="c3">N</font></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="matrixc1.html#E30:61_2_1"><i><font color="Green" title="E66">A52</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K17" title="RVSUM_1:func.17">Sum</a> <span class="p2">(<span class="default"><a href="matrixc1.html#K11" title="MATRIXC1:func.11">ColSum</a> <font color="Maroon" title="c6">N2</font></span>)</span></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixc1.html#K10" title="MATRIXC1:func.10">LineSum</a> <font color="Maroon" title="c3">N</font></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="matrixc1.html#E1:61_2"><i><font color="Green" title="E48">A14</font></i></a>, <a class="txt" href="matrixc1.html#E11:61_2_1"><i><font color="Green" title="E55">A21</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K17" title="RVSUM_1:func.17">Sum</a> <span class="p2">(<span class="default"><a href="matrixc1.html#K11" title="MATRIXC1:func.11">ColSum</a> <font color="Maroon" title="c6">N2</font></span>)</span></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K17" title="RVSUM_1:func.17">Sum</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">N</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="matrixc1.html#E28:61_2_1"><i><font color="Green" title="E65">A51</font></i></a>, <a class="ref" href="matrixc1.html#D10" target="_self" title="MATRIXC1:def.10">Def10</a></i>
<br/>.= 
 <a href="rvsum_1.html#K17" title="RVSUM_1:func.17">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixc1.html#K11" title="MATRIXC1:func.11">ColSum</a> <font color="Maroon" title="c6">N2</font></span>)</span> <a href="complsp1.html#K3" title="COMPLSP1:func.3">+</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">N</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="matrixc1.html#E32:61_2_1"><i><font color="Green" title="E68">A54</font></i></a>, <a class="ref" href="matrixc1.html#T37" target="_self" title="MATRIXC1:th.37">Th37</a></i>
<br/>.= 
 <a href="rvsum_1.html#K17" title="RVSUM_1:func.17">Sum</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K11" title="MATRIXC1:func.11">ColSum</a> <font color="Maroon" title="c3">N</font></span>)</span>
<b>by </b><i><a class="txt" href="matrixc1.html#E34:61_2_1"><i><font color="Green" title="E69">A55</font></i></a>, <a class="txt" href="matrixc1.html#E35:61_2_1"><i><font color="Green" title="E70">A56</font></i></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></i>
;<br/>
<b>hence </b><a NAME="E42:61_2_1"/>
 <a href="rvsum_1.html#K17" title="RVSUM_1:func.17">Sum</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K10" title="MATRIXC1:func.10">LineSum</a> <font color="Maroon" title="c3">N</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K17" title="RVSUM_1:func.17">Sum</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K11" title="MATRIXC1:func.11">ColSum</a> <font color="Maroon" title="c3">N</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:61_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E8:61"/>
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="matrixc1.html#E6:61"><i><font color="Green" title="E46">A12</font></i></a>, <a class="txt" href="matrixc1.html#E7:61"><i><font color="Green" title="E47">A13</font></i></a>);<br/></i>
<a NAME="E9:61"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <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> implies  <a href="rvsum_1.html#K17" title="RVSUM_1:func.17">Sum</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K10" title="MATRIXC1:func.10">LineSum</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K17" title="RVSUM_1:func.17">Sum</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K11" title="MATRIXC1:func.11">ColSum</a> <font color="Maroon" title="c1">M</font></span>)</span> )
 
;<br/>
<b>hence </b><a NAME="E10:61"/>
 <a href="rvsum_1.html#K17" title="RVSUM_1:func.17">Sum</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K10" title="MATRIXC1:func.10">LineSum</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K17" title="RVSUM_1:func.17">Sum</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K11" title="MATRIXC1:func.11">ColSum</a> <font color="Maroon" title="c1">M</font></span>)</span>
 <b>by </b><i><a class="txt" href="matrixc1.html#E4:61"><i><font color="Green" title="E45">A1</font></i></a></i>;<br/>


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