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

<b>let </b><font color="Maroon">M</font> be   <a href="matrix_1.html#NM1">Matrix</a> of <a href="numbers.html#K2">COMPLEX</a> ;<br/>

<b>assume </b><a NAME="E1:62"/><i><font color="Green">E35</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">M</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<a NAME="E2:62"/><i><font color="Green">E36</font></i>: 
0 <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">M</font>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T26">NAT_1:26</a></i>;<br/>
<a NAME="E3:62"/>
<span class="p1">(<span class="default">0 <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K10">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">M</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a>, </i>;<br/>
<a NAME="E4:62"/><b>then </b><i><font color="Green">E37</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">M</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">M</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b>for <font color="Olive">N</font> being  <a href="matrix_1.html#NM1">Matrix</a> of <a href="numbers.html#K2">COMPLEX</a>   st <font color="Maroon">a<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">N</font> holds <br/> <a href="matrixc1.html#K6" target="_self">Sum</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K11" target="_self">LineSum</a> <font color="Olive">N</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrixc1.html#K6" target="_self">Sum</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K12" target="_self">ColSum</a> <font color="Olive">N</font></span>)</span>;<br/>
<a NAME="E5:62"/>
for <font color="Olive">N</font> being  <a href="matrix_1.html#NM1">Matrix</a> of <a href="numbers.html#K2">COMPLEX</a>   st 0 <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">N</font> holds <br/> <a href="matrixc1.html#K6" target="_self">Sum</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K11" target="_self">LineSum</a> <font color="Olive">N</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrixc1.html#K6" target="_self">Sum</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K12" target="_self">ColSum</a> <font color="Olive">N</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">N</font> be   <a href="matrix_1.html#NM1">Matrix</a> of <a href="numbers.html#K2">COMPLEX</a> ;<br/>

<b>assume </b><a NAME="E1:62_1"/><i><font color="Green">E38</font></i>: 
0 <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">N</font>
 ;<br/>

<a NAME="E2:62_1"/><b>then </b><i><font color="Green">E40</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K11" target="_self">LineSum</a> <font color="Maroon">N</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="matrixc1.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E3:62_1"/><i><font color="Green">E41</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K12" target="_self">ColSum</a> <font color="Maroon">N</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">N</font>
 
<b>by </b><i><a class="ref" href="matrixc1.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E4:62_1"/><i><font color="Green">E42</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">N</font>,1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">N</font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/>
<a NAME="E5:62_1"/><i><font color="Green">E43</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T5">FINSEQ_1:5</a></i>;<br/>
<a NAME="E6:62_1"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">N</font> holds <br/><span class="p1">(<span class="default"><a href="matrixc1.html#K12" target="_self">ColSum</a> <font color="Maroon">N</font></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">N</font>,1</span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Olive">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:62_1_1"/><i><font color="Green">E54</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">N</font> )
 ;<br/>

<a NAME="E2:62_1_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">N</font></span>)</span>
 
<b>by </b><i><a class="ref" href="matrixc1.html#T7" target="_self">Th7</a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E3:62_1_1"/><i><font color="Green">E56</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">N</font>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a>, <a class="ref" href="matrixc1.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E4:62_1_1"/><i><font color="Green">E61</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">N</font>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E3:62_1_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a>, <a class="ref" href="matrixc1.html#T5" target="_self">Th5</a></i>;<br/>
<span class="p1">(<span class="default"><a href="matrixc1.html#K12" target="_self">ColSum</a> <font color="Maroon">N</font></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">j</font> = 
 <a href="matrixc1.html#K6" target="_self">Sum</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">N</font>,<font color="Maroon">j</font></span>)</span>
<b>by </b><i><a class="ref" href="matrixc1.html#T2" target="_self">Th2</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">N</font>,<font color="Maroon">j</font></span>)</span> <a href="matrix_5.html#K9">.</a> 1
<b>by </b><i><a class="ref" href="matrixc1.html#T3" target="_self">Th3</a>, </i>
<br/>.= 
<font color="Maroon">N</font> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">j</font>
<b>by </b><i><a class="txt" href="matrixc1.html#E4:62_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">N</font>,1</span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">j</font>
<b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a>, </i>
;<br/>
<b>hence </b><a NAME="E6:62_1_1"/>
<span class="p1">(<span class="default"><a href="matrixc1.html#K12" target="_self">ColSum</a> <font color="Maroon">N</font></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">N</font>,1</span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">j</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E7:62_1"/><i><font color="Green">E62</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">N</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T5">FINSEQ_1:5</a>, <a class="ref" href="matrixc1.html#T5" target="_self">Th5</a></i>;<br/>
 <a href="matrixc1.html#K6" target="_self">Sum</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K12" target="_self">ColSum</a> <font color="Maroon">N</font></span>)</span> = 
 <a href="matrixc1.html#K6" target="_self">Sum</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">N</font>,1</span>)</span>
<b>by </b><i>, , <a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a>, <a class="ref" href="matrixc1.html#T6" target="_self">Th6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrixc1.html#K11" target="_self">LineSum</a> <font color="Maroon">N</font></span>)</span> <a href="matrix_5.html#K9">.</a> 1
<b>by </b><i><a class="ref" href="matrixc1.html#T1" target="_self">Th1</a>, <a class="ref" href="matrixc1.html#T7" target="_self">Th7</a></i>
<br/>.= 
 <a href="matrixc1.html#K6" target="_self">Sum</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K11" target="_self">LineSum</a> <font color="Maroon">N</font></span>)</span>
<b>by </b><i><a class="ref" href="matrixc1.html#T3" target="_self">Th3</a>, <a class="txt" href="matrixc1.html#E6:62_1"><i><font color="Green">E53</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E9:62_1"/>
 <a href="matrixc1.html#K6" target="_self">Sum</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K11" target="_self">LineSum</a> <font color="Maroon">N</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrixc1.html#K6" target="_self">Sum</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K12" target="_self">ColSum</a> <font color="Maroon">N</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E6:62"/><b>then </b><i><font color="Green">E63</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<a NAME="E7:62"/><i><font color="Green">E64</font></i>: 
for <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="xcmplx_0.html#K2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:62_2"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/>

<a NAME="E2:62_2"/>
for <font color="Olive">N</font> being  <a href="matrix_1.html#NM1">Matrix</a> of <a href="numbers.html#K2">COMPLEX</a>   st <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">N</font> holds <br/> <a href="matrixc1.html#K6" target="_self">Sum</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K11" target="_self">LineSum</a> <font color="Olive">N</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrixc1.html#K6" target="_self">Sum</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K12" target="_self">ColSum</a> <font color="Olive">N</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">N</font> be   <a href="matrix_1.html#NM1">Matrix</a> of <a href="numbers.html#K2">COMPLEX</a> ;<br/>

<b>assume </b><a NAME="E1:62_2_1"/><i><font color="Green">E103</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">N</font>
 ;<br/>

<a NAME="E2:62_2_1"/><i><font color="Green">E104</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">N</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a>, </i>;<br/>
<a NAME="E3:62_2_1"/><i><font color="Green">E105</font></i>: 
1 <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E4:62_2_1"/><i><font color="Green">E106</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">N</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a>, </i>;<br/>
<b>reconsider </b><font color="Maroon">k1</font> = <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1 as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<b>consider </b><font color="Maroon">n</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E7:62_2_1"/><i><font color="Green">E107</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">N</font> holds <br/> ex <font color="Olive">p</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K2">COMPLEX</a>  st <br/>( <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="ref" href="matrix_1.html#T9">MATRIX_1:9</a></i>;<br/>
<a NAME="E8:62_2_1"/><i><font color="Green">E108</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k1</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">N</font>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T21">FINSEQ_5:21</a></i>;<br/>
<a NAME="E9:62_2_1"/><i><font color="Green">E109</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k1</font></span>)</span> holds <br/> ex <font color="Olive">p</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K2">COMPLEX</a>  st <br/>( <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 
<b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">N2</font> = <font color="Maroon">N</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k1</font> as   <a href="matrix_1.html#NM1">Matrix</a> of <a href="numbers.html#K2">COMPLEX</a>  <b>by </b><i><a class="ref" href="matrix_1.html#T9">MATRIX_1:9</a>, </i>;<br/>
<a NAME="E11:62_2_1"/><i><font color="Green">E110</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">N2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a>, </i>;<br/>
<a NAME="E12:62_2_1"/><i><font color="Green">E111</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">N2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="matrixc1.html#T6" target="_self">Th6</a>, <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a>, </i>;<br/>
<b>consider </b><font color="Maroon">n3</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E14:62_2_1"/><i><font color="Green">E113</font></i>: 
for <font color="Olive">x2</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x2</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">N</font> holds <br/> ex <font color="Olive">s</font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <br/>( <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Olive">x2</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n3</font> )
 <b>by </b><i><a class="ref" href="matrix_1.html#D1">MATRIX_1:def 1</a></i>;<br/>
<a NAME="E15:62_2_1"/>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">N</font></span>)</span>
 
<b>by </b><i><a class="ref" href="matrixc1.html#T7" target="_self">Th7</a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E16:62_2_1"/><b>then </b>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">N</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E17:62_2_1"/><b>then </b><i><font color="Green">E114</font></i>: 
<font color="Maroon">N</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">N</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<b>then consider </b><font color="Maroon">s2</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E19:62_2_1"/><i><font color="Green">E115</font></i>: 
( <font color="Maroon">s2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N</font> <a href="funct_1.html#K1">.</a> 1 &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">s2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n3</font> )
 <b>by </b><i><a class="txt" href="matrixc1.html#E7:62"><i><font color="Green">E64</font></i></a></i>;<br/>
<a NAME="E20:62_2_1"/>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">N2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E6:62"><i><font color="Green">E63</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E21:62_2_1"/><b>then </b>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">N2</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E22:62_2_1"/><b>then </b><i><font color="Green">E116</font></i>: 
<font color="Maroon">N2</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">N2</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<b>then consider </b><font color="Maroon">s3</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E24:62_2_1"/><i><font color="Green">E117</font></i>: 
( <font color="Maroon">s3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N2</font> <a href="funct_1.html#K1">.</a> 1 &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">s3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n3</font> )
 <b>by </b><i><a class="txt" href="matrixc1.html#E7:62"><i><font color="Green">E64</font></i></a>, </i>;<br/>
<a NAME="E25:62_2_1"/><i><font color="Green">E118</font></i>: 
 <a href="matrix_1.html#K1">width</a> <font color="Maroon">N2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n3</font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D4">MATRIX_1:def 4</a>, <a class="txt" href="matrixc1.html#E6:62"><i><font color="Green">E63</font></i></a>, <a class="ref" href="matrixc1.html#T11" target="_self">Th11</a>, <a class="ref" href="matrixc1.html#T10" target="_self">Th10</a></i>;<br/>
<a NAME="E26:62_2_1"/><i><font color="Green">E119</font></i>: 
 <a href="matrix_1.html#K1">width</a> <font color="Maroon">N2</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">N</font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D4">MATRIX_1:def 4</a>, , <a class="txt" href="matrixc1.html#E1:62_2"><i><font color="Green">E65</font></i></a>, <a class="ref" href="matrixc1.html#T9" target="_self">Th9</a>, <a class="ref" href="matrixc1.html#T13" target="_self">Th13</a></i>;<br/>
<a NAME="E27:62_2_1"/><i><font color="Green">E120</font></i>: 
for <font color="Olive">j</font> being  <a href="nat_1.html#NM1">Nat</a>  st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">N</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixc1.html#K12" target="_self">ColSum</a> <font color="Maroon">N2</font></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Olive">j</font></span>)</span> <a href="binop_2.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>,<font color="Olive">j</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K12" target="_self">ColSum</a> <font color="Maroon">N</font></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Olive">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">j</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:62_2_1_1"/><i><font color="Green">E121</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">N</font> )
 ;<br/>

<a NAME="E2:62_2_1_1"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">N</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, </i>;<br/>
<a NAME="E3:62_2_1_1"/><i><font color="Green">E156</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">N</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T6">FINSEQ_1:6</a></i>;<br/>
<a NAME="E4:62_2_1_1"/><i><font color="Green">E157</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">N</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a>, </i>;<br/>
<a NAME="E5:62_2_1_1"/><i><font color="Green">E158</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">N</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>;<br/>
<a NAME="E6:62_2_1_1"/><i><font color="Green">E159</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">N</font>,<font color="Maroon">j</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a>, </i>;<br/>
<a NAME="E7:62_2_1_1"/><i><font color="Green">E160</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">N</font>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">N</font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>;<br/>
<b>set </b><font color="Maroon">f</font> =  <a href="matrix_1.html#K9">Col</a> <font color="Maroon">N</font>,<font color="Maroon">j</font>;<br/>
<b>set </b><font color="Maroon">g</font> =  <a href="matrix_1.html#K9">Col</a> <font color="Maroon">N2</font>,<font color="Maroon">j</font>;<br/>
<i><font color="Green">E161</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">N2</font>,<font color="Maroon">j</font></span>)</span> = 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">N2</font>
<b>by </b><i><a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>
<br/>.= 
<font color="Maroon">k1</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a>, </i>
;<br/>
<a NAME="E9:62_2_1_1"/><i><font color="Green">E162</font></i>: 
<font color="Maroon">k1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">N</font>,<font color="Maroon">j</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>;<br/>
<a NAME="E10:62_2_1_1"/><i><font color="Green">E163</font></i>: 
 <a href="matrix_1.html#K9">Col</a> <font color="Maroon">N2</font>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">N</font>,<font color="Maroon">j</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_2_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:62_2_1_1_2"/><i><font color="Green">E164</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">N</font>,<font color="Maroon">j</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">N2</font>,<font color="Maroon">j</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a>, <a class="ref" href="matrixc1.html#D7" target="_self">Def7</a></i>;<br/>
<a NAME="E2:62_2_1_1_2"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> &amp; <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">N</font>,<font color="Maroon">j</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k1</font></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">N</font>,<font color="Maroon">j</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k1</font></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">N2</font>,<font color="Maroon">j</font></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Olive">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_2_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:62_2_1_1_2_1"/><i><font color="Green">E165</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">N</font>,<font color="Maroon">j</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k1</font></span>)</span> )
 ;<br/>

<a NAME="E2:62_2_1_1_2_1"/><i><font color="Green">E166</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k1</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a>, <a class="ref" href="matrixc1.html#D7" target="_self">Def7</a>, </i>;<br/>
<a NAME="E3:62_2_1_1_2_1"/><i><font color="Green">E167</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k1</font> <a href="binop_2.html#K23">+</a> 1 )
 
<b>by </b><i><a class="txt" href="matrixc1.html#E23"><i><font color="Green">Lemma84</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E4:62_2_1_1_2_1"/><i><font color="Green">E168</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">N</font>,<font color="Maroon">j</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, </i>;<br/>
<a NAME="E5:62_2_1_1_2_1"/><i><font color="Green">E169</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">N</font>,<font color="Maroon">j</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a>, <a class="ref" href="matrixc1.html#T18" target="_self">Th18</a></i>;<br/>
<a NAME="E6:62_2_1_1_2_1"/><i><font color="Green">E198</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">N</font>,<font color="Maroon">j</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D15">FINSEQ_1:def 15</a>, <a class="ref" href="matrixc1.html#T19" target="_self">Th19</a></i>;<br/>
<a NAME="E7:62_2_1_1_2_1"/><i><font color="Green">E199</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">N</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="matrixc1.html#T16" target="_self">Th16</a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E8:62_2_1_1_2_1"/><i><font color="Green">E200</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">N</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a>, </i>;<br/>
<a NAME="E9:62_2_1_1_2_1"/><i><font color="Green">E201</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k1</font>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E23"><i><font color="Green">Lemma84</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E10:62_2_1_1_2_1"/><i><font color="Green">E202</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">N2</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a>, , </i>;<br/>
<a NAME="E11:62_2_1_1_2_1"/><i><font color="Green">E203</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">N</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">N</font> )
 
<b>by </b><i><a class="txt" href="matrixc1.html#E23"><i><font color="Green">Lemma84</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a>, , </i>;<br/>
<a NAME="E12:62_2_1_1_2_1"/><i><font color="Green">E204</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">n</font>,<font color="Maroon">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">N</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E13:62_2_1_1_2_1"/><i><font color="Green">E205</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">N2</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">N2</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a>, <a class="ref" href="matrixc1.html#D7" target="_self">Def7</a>, , , <a class="ref" href="matrix_1.html#D4">MATRIX_1:def 4</a>, , <a class="txt" href="matrixc1.html#E1:62_2"><i><font color="Green">E65</font></i></a>, <a class="ref" href="matrixc1.html#T9" target="_self">Th9</a>, <a class="ref" href="matrixc1.html#T13" target="_self">Th13</a>, </i>;<br/>
<a NAME="E14:62_2_1_1_2_1"/><i><font color="Green">E206</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">n</font>,<font color="Maroon">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">N2</font>
 
<b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">pn</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K2">COMPLEX</a> <b> such that </b><br/><a NAME="E16:62_2_1_1_2_1"/><i><font color="Green">E208</font></i>: 
( <font color="Maroon">pn</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> &amp; <font color="Maroon">N</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">n</font>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">pn</font> <a href="matrix_5.html#K9">.</a> <font color="Maroon">j</font> )
 <b>by </b><i><a class="ref" href="matrix_1.html#D6">MATRIX_1:def 6</a>, <a class="ref" href="matrixc1.html#T22" target="_self">Th22</a></i>;<br/>
<a NAME="E17:62_2_1_1_2_1"/><i><font color="Green">E209</font></i>: 
<font color="Maroon">N2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T121">FINSEQ_3:121</a>, <a class="txt" href="matrixc1.html#E23"><i><font color="Green">Lemma84</font></i></a></i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">N</font>,<font color="Maroon">j</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k1</font></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">N</font>,<font color="Maroon">j</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k1</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D15">FINSEQ_1:def 15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">N</font>,<font color="Maroon">j</font></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="funct_1.html#T70">FUNCT_1:70</a>, </i>
<br/>.= 
<font color="Maroon">N</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">n</font>,<font color="Maroon">j</font>
<b>by </b><i><a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a>, <a class="ref" href="matrixc1.html#T20" target="_self">Th20</a></i>
<br/>.= 
<font color="Maroon">N2</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">n</font>,<font color="Maroon">j</font>
<b>by </b><i>, <a class="ref" href="matrix_1.html#D6">MATRIX_1:def 6</a>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">N2</font>,<font color="Maroon">j</font></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="matrixc1.html#T21" target="_self">Th21</a>, <a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>
;<br/>
<b>hence </b><a NAME="E19:62_2_1_1_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">N</font>,<font color="Maroon">j</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k1</font></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">N2</font>,<font color="Maroon">j</font></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">n</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:62_2_1_1_2"/>
 <a href="matrix_1.html#K9">Col</a> <font color="Maroon">N2</font>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">N</font>,<font color="Maroon">j</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k1</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T18">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#K12" target="_self">ColSum</a> <font color="Maroon">N2</font></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">j</font></span>)</span> <a href="binop_2.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>,<font color="Maroon">j</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="matrixc1.html#K6" target="_self">Sum</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">N2</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="binop_2.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>,<font color="Maroon">j</font></span>)</span>
<b>by </b><i><a class="ref" href="matrixc1.html#T2" target="_self">Th2</a>, , <a class="ref" href="matrixc1.html#T14" target="_self">Th14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrixc1.html#K6" target="_self">Sum</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">N2</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="binop_2.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">N</font>,<font color="Maroon">j</font></span>)</span> <a href="matrix_5.html#K9">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrixc1.html#K6" target="_self">Sum</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">N2</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="binop_2.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">N</font>,<font color="Maroon">j</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a>, </i>
<br/>.= 
 <a href="matrixc1.html#K6" target="_self">Sum</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">N</font>,<font color="Maroon">j</font></span>)</span>
<b>by </b><i>, , , </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrixc1.html#K12" target="_self">ColSum</a> <font color="Maroon">N</font></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">j</font>
<b>by </b><i><a class="ref" href="matrixc1.html#T2" target="_self">Th2</a>, </i>
;<br/>
<b>hence </b><a NAME="E12:62_2_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixc1.html#K12" target="_self">ColSum</a> <font color="Maroon">N2</font></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">j</font></span>)</span> <a href="binop_2.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K12" target="_self">ColSum</a> <font color="Maroon">N</font></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">j</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E28:62_2_1"/><i><font color="Green">E210</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">N</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T5">FINSEQ_1:5</a></i>;<br/>
<a NAME="E29:62_2_1"/><i><font color="Green">E211</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="matrixc1.html#K11" target="_self">LineSum</a> <font color="Maroon">N</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="matrixc1.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E30:62_2_1"/><i><font color="Green">E212</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K11" target="_self">LineSum</a> <font color="Maroon">N</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E31:62_2_1"/><i><font color="Green">E213</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K12" target="_self">ColSum</a> <font color="Maroon">N2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">N2</font>
 
<b>by </b><i><a class="ref" href="matrixc1.html#T2" target="_self">Th2</a></i>;<br/>
<i><font color="Green">E214</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K12" target="_self">ColSum</a> <font color="Maroon">N2</font></span>)</span> = 
 <a href="matrix_1.html#K1">width</a> <font color="Maroon">N2</font>
<b>by </b><i><a class="ref" href="matrixc1.html#T2" target="_self">Th2</a></i>
<br/>.= 
 <a href="matrix_1.html#K1">width</a> <font color="Maroon">N</font>
<b>by </b><i><a class="ref" href="matrix_1.html#D4">MATRIX_1:def 4</a>, , <a class="txt" href="matrixc1.html#E1:62_2"><i><font color="Green">E65</font></i></a>, <a class="ref" href="matrixc1.html#T9" target="_self">Th9</a>, <a class="ref" href="matrixc1.html#T13" target="_self">Th13</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">N</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>
;<br/>
<a NAME="E33:62_2_1"/><i><font color="Green">E215</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">N</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">N</font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/>
<i><font color="Green">E216</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixc1.html#K12" target="_self">ColSum</a> <font color="Maroon">N2</font></span>)</span> <a href="complsp1.html#K3">+</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">N</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span> = 
 <a href="matrix_1.html#K1">width</a> <font color="Maroon">N</font>
<b>by </b><i><a class="ref" href="complsp2.html#T6">COMPLSP2:6</a>, , , <a class="ref" href="matrixc1.html#T14" target="_self">Th14</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K12" target="_self">ColSum</a> <font color="Maroon">N</font></span>)</span>
<b>by </b><i><a class="ref" href="matrixc1.html#T2" target="_self">Th2</a></i>
;<br/>
<a NAME="E35:62_2_1"/><i><font color="Green">E217</font></i>: 
for <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K12" target="_self">ColSum</a> <font color="Maroon">N</font></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixc1.html#K12" target="_self">ColSum</a> <font color="Maroon">N2</font></span>)</span> <a href="complsp1.html#K3">+</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">N</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K12" target="_self">ColSum</a> <font color="Maroon">N</font></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Olive">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_2_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:62_2_1_4"/><i><font color="Green">E218</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K12" target="_self">ColSum</a> <font color="Maroon">N</font></span>)</span> )
 ;<br/>

<a NAME="E2:62_2_1_4"/><i><font color="Green">E219</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="matrixc1.html#K12" target="_self">ColSum</a> <font color="Maroon">N</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="matrixc1.html#D7" target="_self">Def7</a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E3:62_2_1_4"/><i><font color="Green">E220</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrixc1.html#K12" target="_self">ColSum</a> <font color="Maroon">N2</font></span>)</span> <a href="complsp1.html#K3">+</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">N</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="matrixc1.html#D7" target="_self">Def7</a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, </i>;<br/>
<a NAME="E4:62_2_1_4"/><i><font color="Green">E221</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixc1.html#K12" target="_self">ColSum</a> <font color="Maroon">N2</font></span>)</span> <a href="complsp1.html#K3">+</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">N</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E5:62_2_1_4"/><i><font color="Green">E222</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">N2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="matrixc1.html#T14" target="_self">Th14</a>, <a class="ref" href="matrixc1.html#T2" target="_self">Th2</a>, </i>;<br/>
<a NAME="E6:62_2_1_4"/><i><font color="Green">E223</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">N</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="txt" href="matrixc1.html#E23"><i><font color="Green">Lemma84</font></i></a>, <a class="ref" href="matrixc1.html#T14" target="_self">Th14</a></i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixc1.html#K12" target="_self">ColSum</a> <font color="Maroon">N2</font></span>)</span> <a href="complsp1.html#K3">+</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">N</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">j</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixc1.html#K12" target="_self">ColSum</a> <font color="Maroon">N2</font></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">j</font></span>)</span> <a href="binop_2.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">N</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">j</font></span>)</span>
<b>by </b><i>, <a class="ref" href="complsp2.html#T1">COMPLSP2:1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixc1.html#K12" target="_self">ColSum</a> <font color="Maroon">N2</font></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">j</font></span>)</span> <a href="binop_2.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>,<font color="Maroon">j</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a>, <a class="txt" href="matrixc1.html#E23"><i><font color="Green">Lemma84</font></i></a>, <a class="ref" href="matrixc1.html#T14" target="_self">Th14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrixc1.html#K12" target="_self">ColSum</a> <font color="Maroon">N</font></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">j</font>
<b>by </b><i>, <a class="ref" href="matrixc1.html#T16" target="_self">Th16</a></i>
;<br/>
<b>hence </b><a NAME="E8:62_2_1_4"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixc1.html#K12" target="_self">ColSum</a> <font color="Maroon">N2</font></span>)</span> <a href="complsp1.html#K3">+</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">N</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K12" target="_self">ColSum</a> <font color="Maroon">N</font></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">j</font>
 ;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon">g1</font> =  <a href="matrixc1.html#K11" target="_self">LineSum</a> <font color="Maroon">N2</font>;<br/>
<b>set </b><font color="Maroon">f3</font> =  <a href="matrixc1.html#K11" target="_self">LineSum</a> <font color="Maroon">N</font>;<br/>
<a NAME="E36:62_2_1"/><i><font color="Green">E225</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K11" target="_self">LineSum</a> <font color="Maroon">N</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">N</font>
 
<b>by </b><i><a class="ref" href="matrixc1.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E37:62_2_1"/><i><font color="Green">E226</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K11" target="_self">LineSum</a> <font color="Maroon">N</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k1</font> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i>, <a class="ref" href="matrixc1.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E38:62_2_1"/><i><font color="Green">E227</font></i>: 
<font color="Maroon">k1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K11" target="_self">LineSum</a> <font color="Maroon">N</font></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a>, </i>;<br/>
<i><font color="Green">E228</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K11" target="_self">LineSum</a> <font color="Maroon">N2</font></span>)</span> = 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">N2</font>
<b>by </b><i><a class="ref" href="matrixc1.html#T1" target="_self">Th1</a></i>
<br/>.= 
<font color="Maroon">k1</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a>, </i>
;<br/>
<a NAME="E40:62_2_1"/>
 <a href="matrixc1.html#K11" target="_self">LineSum</a> <font color="Maroon">N2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K11" target="_self">LineSum</a> <font color="Maroon">N</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_2_1_6"><b>proof </b></a><div class="add">

<a NAME="E1:62_2_1_6"/><i><font color="Green">E229</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixc1.html#K11" target="_self">LineSum</a> <font color="Maroon">N</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K11" target="_self">LineSum</a> <font color="Maroon">N2</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a>, </i>;<br/>
<a NAME="E2:62_2_1_6"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> &amp; <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixc1.html#K11" target="_self">LineSum</a> <font color="Maroon">N</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k1</font></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixc1.html#K11" target="_self">LineSum</a> <font color="Maroon">N</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k1</font></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K11" target="_self">LineSum</a> <font color="Maroon">N2</font></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Olive">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_2_1_6_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:62_2_1_6_1"/><i><font color="Green">E230</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixc1.html#K11" target="_self">LineSum</a> <font color="Maroon">N</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k1</font></span>)</span> )
 ;<br/>

<a NAME="E2:62_2_1_6_1"/><i><font color="Green">E231</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k1</font> <a href="binop_2.html#K23">+</a> 1 )
 
<b>by </b><i><a class="txt" href="matrixc1.html#E23"><i><font color="Green">Lemma84</font></i></a>, <a class="ref" href="matrixc1.html#T16" target="_self">Th16</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a>, </i>;<br/>
<a NAME="E3:62_2_1_6_1"/><i><font color="Green">E232</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrixc1.html#K11" target="_self">LineSum</a> <font color="Maroon">N</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="ref" href="matrixc1.html#T16" target="_self">Th16</a></i>;<br/>
<a NAME="E4:62_2_1_6_1"/><i><font color="Green">E233</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixc1.html#K11" target="_self">LineSum</a> <font color="Maroon">N</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a>, <a class="ref" href="matrixc1.html#T19" target="_self">Th19</a></i>;<br/>
<a NAME="E5:62_2_1_6_1"/><i><font color="Green">E234</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixc1.html#K11" target="_self">LineSum</a> <font color="Maroon">N</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D15">FINSEQ_1:def 15</a>, </i>;<br/>
<a NAME="E6:62_2_1_6_1"/><i><font color="Green">E235</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">N</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="matrixc1.html#T18" target="_self">Th18</a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E7:62_2_1_6_1"/><i><font color="Green">E236</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k1</font>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E23"><i><font color="Green">Lemma84</font></i></a>, <a class="ref" href="matrixc1.html#T16" target="_self">Th16</a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, </i>;<br/>
<a NAME="E8:62_2_1_6_1"/><i><font color="Green">E237</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">N2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a>, , </i>;<br/>
<a NAME="E9:62_2_1_6_1"/><i><font color="Green">E238</font></i>: 
 <a href="matrix_1.html#K8">Line</a> <font color="Maroon">N</font>,<font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K8">Line</a> <font color="Maroon">N2</font>,<font color="Maroon">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_2_1_6_1_1"><b>proof </b></a><div class="add">

<i><font color="Green">E239</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">N</font>,<font color="Maroon">n</font></span>)</span> = 
 <a href="matrix_1.html#K1">width</a> <font color="Maroon">N</font>
<b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>
<br/>.= 
 <a href="matrix_1.html#K1">width</a> <font color="Maroon">N2</font>
<b>by </b><i><a class="ref" href="matrix_1.html#D4">MATRIX_1:def 4</a>, , <a class="txt" href="matrixc1.html#E1:62_2"><i><font color="Green">E65</font></i></a>, <a class="ref" href="matrixc1.html#T9" target="_self">Th9</a>, <a class="ref" href="matrixc1.html#T13" target="_self">Th13</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">N2</font>,<font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>
;<br/>
<a NAME="E2:62_2_1_6_1_1"/>
for <font color="Olive">n1</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n1</font> &amp; <font color="Olive">n1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">N</font>,<font color="Maroon">n</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">N</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Olive">n1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">N2</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Olive">n1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_2_1_6_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:62_2_1_6_1_1_2"/><i><font color="Green">E241</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n1</font> &amp; <font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">N</font>,<font color="Maroon">n</font></span>)</span> )
 ;<br/>

<a NAME="E2:62_2_1_6_1_1_2"/><i><font color="Green">E242</font></i>: 
<font color="Maroon">n1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">N</font>,<font color="Maroon">n</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, </i>;<br/>
<a NAME="E3:62_2_1_6_1_1_2"/><i><font color="Green">E243</font></i>: 
<font color="Maroon">n1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">N</font></span>)</span>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a>, </i>;<br/>
<a NAME="E4:62_2_1_6_1_1_2"/><i><font color="Green">E244</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n1</font> &amp; <font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">N</font> )
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E5:62_2_1_6_1_1_2"/><i><font color="Green">E245</font></i>: 
<font color="Maroon">n1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">N</font></span>)</span>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a>, </i>;<br/>
<a NAME="E6:62_2_1_6_1_1_2"/><i><font color="Green">E246</font></i>: 
<font color="Maroon">n1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">N2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a>, , <a class="ref" href="matrixc1.html#T14" target="_self">Th14</a></i>;<br/>
<a NAME="E7:62_2_1_6_1_1_2"/><i><font color="Green">E247</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">N</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n1</font> &amp; <font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">N</font> )
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="txt" href="matrixc1.html#E23"><i><font color="Green">Lemma84</font></i></a>, <a class="ref" href="matrixc1.html#T16" target="_self">Th16</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a>, , </i>;<br/>
<a NAME="E8:62_2_1_6_1_1_2"/><i><font color="Green">E248</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">n</font>,<font color="Maroon">n1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">N</font>
 
<b>by </b><i>, , <a class="ref" href="matrixc1.html#T18" target="_self">Th18</a>, </i>;<br/>
<a NAME="E9:62_2_1_6_1_1_2"/><i><font color="Green">E249</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">N2</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n1</font> &amp; <font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">N2</font> )
 
<b>by </b><i><a class="txt" href="matrixc1.html#E23"><i><font color="Green">Lemma84</font></i></a>, <a class="ref" href="matrixc1.html#T16" target="_self">Th16</a>, <a class="ref" href="matrixc1.html#T1" target="_self">Th1</a>, <a class="ref" href="matrix_1.html#D4">MATRIX_1:def 4</a>, <a class="txt" href="matrixc1.html#E1:62_2"><i><font color="Green">E65</font></i></a>, <a class="ref" href="matrixc1.html#T9" target="_self">Th9</a>, <a class="ref" href="matrixc1.html#T13" target="_self">Th13</a>, </i>;<br/>
<a NAME="E10:62_2_1_6_1_1_2"/><i><font color="Green">E250</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">n</font>,<font color="Maroon">n1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">N2</font>
 
<b>by </b><i>, <a class="txt" href="matrixc1.html#E1:62_2_1"><i><font color="Green">E103</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">pn</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K2">COMPLEX</a> <b> such that </b><br/><a NAME="E12:62_2_1_6_1_1_2"/><i><font color="Green">E251</font></i>: 
( <font color="Maroon">pn</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> &amp; <font color="Maroon">N</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">n</font>,<font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">pn</font> <a href="matrix_5.html#K9">.</a> <font color="Maroon">n1</font> )
 <b>by </b><i><a class="ref" href="matrix_1.html#D6">MATRIX_1:def 6</a>, </i>;<br/>
<a NAME="E13:62_2_1_6_1_1_2"/><i><font color="Green">E252</font></i>: 
<font color="Maroon">N2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T121">FINSEQ_3:121</a>, <a class="txt" href="matrixc1.html#E23"><i><font color="Green">Lemma84</font></i></a>, <a class="ref" href="matrixc1.html#T16" target="_self">Th16</a>, </i>;<br/>
<span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">N</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">n1</font> = 
<font color="Maroon">N</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">n</font>,<font color="Maroon">n1</font>
<b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a>, <a class="ref" href="matrixc1.html#T23" target="_self">Th23</a></i>
<br/>.= 
<font color="Maroon">N2</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">n</font>,<font color="Maroon">n1</font>
<b>by </b><i><a class="txt" href="matrixc1.html#E2:62_2_1"><i><font color="Green">E104</font></i></a>, <a class="ref" href="matrix_1.html#D6">MATRIX_1:def 6</a>, <a class="txt" href="matrixc1.html#E4:62_2_1"><i><font color="Green">E106</font></i></a>, <a class="txt" href="matrixc1.html#E3:62_2_1"><i><font color="Green">E105</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">N2</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">n1</font>
<b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a>, </i>
;<br/>
<b>hence </b><a NAME="E15:62_2_1_6_1_1_2"/>
<span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">N</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">N2</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">n1</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:62_2_1_6_1_1"/>
 <a href="matrix_1.html#K8">Line</a> <font color="Maroon">N</font>,<font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K8">Line</a> <font color="Maroon">N2</font>,<font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a>, <a class="ref" href="matrixc1.html#T22" target="_self">Th22</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" target="_self">LineSum</a> <font color="Maroon">N</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k1</font></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixc1.html#K11" target="_self">LineSum</a> <font color="Maroon">N</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k1</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D15">FINSEQ_1:def 15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrixc1.html#K11" target="_self">LineSum</a> <font color="Maroon">N</font></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="funct_1.html#T70">FUNCT_1:70</a>, </i>
<br/>.= 
 <a href="matrixc1.html#K6" target="_self">Sum</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">N2</font>,<font color="Maroon">n</font></span>)</span>
<b>by </b><i>, <a class="ref" href="matrixc1.html#T1" target="_self">Th1</a>, <a class="ref" href="matrixc1.html#T20" target="_self">Th20</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrixc1.html#K11" target="_self">LineSum</a> <font color="Maroon">N2</font></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="matrixc1.html#T21" target="_self">Th21</a>, <a class="ref" href="matrixc1.html#T1" target="_self">Th1</a></i>
;<br/>
<b>hence </b><a NAME="E11:62_2_1_6_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixc1.html#K11" target="_self">LineSum</a> <font color="Maroon">N</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k1</font></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K11" target="_self">LineSum</a> <font color="Maroon">N2</font></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">n</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:62_2_1_6"/>
 <a href="matrixc1.html#K11" target="_self">LineSum</a> <font color="Maroon">N2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K11" target="_self">LineSum</a> <font color="Maroon">N</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k1</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a>, <a class="txt" href="matrixc1.html#E23"><i><font color="Green">Lemma84</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then </b> <a href="matrixc1.html#K6" target="_self">Sum</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K11" target="_self">LineSum</a> <font color="Maroon">N</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="matrixc1.html#K6" target="_self">Sum</a> <span class="p2">(<span class="default"><a href="matrixc1.html#K11" target="_self">LineSum</a> <font color="Maroon">N2</font></span>)</span></span>)</span> <a href="binop_2.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixc1.html#K11" target="_self">LineSum</a> <font color="Maroon">N</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="matrixc1.html#D7" target="_self">Def7</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrixc1.html#K6" target="_self">Sum</a> <span class="p2">(<span class="default"><a href="matrixc1.html#K11" target="_self">LineSum</a> <font color="Maroon">N2</font></span>)</span></span>)</span> <a href="binop_2.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixc1.html#K11" target="_self">LineSum</a> <font color="Maroon">N</font></span>)</span> <a href="matrix_5.html#K9">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrixc1.html#K6" target="_self">Sum</a> <span class="p2">(<span class="default"><a href="matrixc1.html#K12" target="_self">ColSum</a> <font color="Maroon">N2</font></span>)</span></span>)</span> <a href="binop_2.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixc1.html#K11" target="_self">LineSum</a> <font color="Maroon">N</font></span>)</span> <a href="matrix_5.html#K9">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrixc1.html#K6" target="_self">Sum</a> <span class="p2">(<span class="default"><a href="matrixc1.html#K12" target="_self">ColSum</a> <font color="Maroon">N2</font></span>)</span></span>)</span> <a href="binop_2.html#K3">+</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K6" target="_self">Sum</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">N</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="matrixc1.html#T1" target="_self">Th1</a>, </i>
<br/>.= 
 <a href="matrixc1.html#K6" target="_self">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixc1.html#K12" target="_self">ColSum</a> <font color="Maroon">N2</font></span>)</span> <a href="complsp1.html#K3">+</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">N</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="matrixc1.html#E1:62"><i><font color="Green">E35</font></i></a>, </i>
<br/>.= 
 <a href="matrixc1.html#K6" target="_self">Sum</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K12" target="_self">ColSum</a> <font color="Maroon">N</font></span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a>, </i>
;<br/>
<b>hence </b><a NAME="E42:62_2_1"/>
 <a href="matrixc1.html#K6" target="_self">Sum</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K11" target="_self">LineSum</a> <font color="Maroon">N</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrixc1.html#K6" target="_self">Sum</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K12" target="_self">ColSum</a> <font color="Maroon">N</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:62_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E8:62"/><i><font color="Green">E253</font></i>: 
for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2">NAT_1:sch 2</a>(<a class="ref" href="matrixc1.html#T5" target="_self">Th5</a>, );<br/></i>
<a NAME="E9:62"/><i><font color="Green">E254</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">M</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">M</font> implies  <a href="matrixc1.html#K6" target="_self">Sum</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K11" target="_self">LineSum</a> <font color="Maroon">M</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrixc1.html#K6" target="_self">Sum</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K12" target="_self">ColSum</a> <font color="Maroon">M</font></span>)</span> )
 
<b>by </b><i/>;<br/>
<b>thus </b><a NAME="E10:62"/>
 <a href="matrixc1.html#K6" target="_self">Sum</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K11" target="_self">LineSum</a> <font color="Maroon">M</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrixc1.html#K6" target="_self">Sum</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K12" target="_self">ColSum</a> <font color="Maroon">M</font></span>)</span>
 <b>by </b><i>, </i>;<br/>


</div>
