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

<b>let </b><font color="Maroon">c<sub>1</sub></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">E49</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<a NAME="E2:62"/><i><font color="Green">E50</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">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E1:62"><i><font color="Green">E49</font></i></a>, <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">c<sub>1</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a>, <a class="txt" href="matrixc1.html#E2:62"><i><font color="Green">E50</font></i></a></i>;<br/>
<a NAME="E4:62"/><b>then </b><i><font color="Green">E51</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</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">c<sub>1</sub></font></span>)</span> <a href="binop_2.html#K10">-</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>] means for <font color="Olive">b<sub>1</sub></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="binop_2.html#K23">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></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">b<sub>1</sub></font></span>)</span>;<br/>
<a NAME="E5:62"/>
for <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></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">b<sub>1</sub></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">b<sub>1</sub></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">c<sub>2</sub></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">E52</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">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:62_1"/><b>then </b><i><font color="Green">E53</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">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="matrixc1.html#D10" target="_self">Def10</a></i>;<br/>
<a NAME="E3:62_1"/><i><font color="Green">E54</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">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="matrixc1.html#D11" target="_self">Def11</a></i>;<br/>
<a NAME="E4:62_1"/><i><font color="Green">E55</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">c<sub>2</sub></font>,1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>2</sub></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">E56</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">E57</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><span class="p1">(<span class="default"><a href="matrixc1.html#K12" target="_self">ColSum</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>2</sub></font>,1</span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Olive">b<sub>1</sub></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">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:62_1_1"/><i><font color="Green">E58</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>

<a NAME="E2:62_1_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>3</sub></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">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E1:62_1_1"><i><font color="Green">E58</font></i></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">E60</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">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></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="txt" href="matrixc1.html#E1:62_1"><i><font color="Green">E52</font></i></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">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E5:62_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a>, <a class="txt" href="matrixc1.html#E1:62_1"><i><font color="Green">E52</font></i></a></i>;<br/>
<span class="p1">(<span class="default"><a href="matrixc1.html#K12" target="_self">ColSum</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">c<sub>3</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="matrixc1.html#D11" target="_self">Def11</a>, <a class="txt" href="matrixc1.html#E2:62_1_1"><i><font color="Green">E59</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_5.html#K9">.</a> 1
<b>by </b><i><a class="ref" href="matrixc1.html#T50" target="_self">Th50</a>, <a class="txt" href="matrixc1.html#E3:62_1_1"><i><font color="Green">E60</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">c<sub>3</sub></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">c<sub>2</sub></font>,1</span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a>, <a class="txt" href="matrixc1.html#E2:62_1_1"><i><font color="Green">E59</font></i></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">c<sub>2</sub></font></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>2</sub></font>,1</span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E7:62_1"/><i><font color="Green">E58</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">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T5">FINSEQ_1:5</a>, <a class="txt" href="matrixc1.html#E1:62_1"><i><font color="Green">E52</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">c<sub>2</sub></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">c<sub>2</sub></font>,1</span>)</span>
<b>by </b><i><a class="txt" href="matrixc1.html#E3:62_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="matrixc1.html#E4:62_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a>, <a class="txt" href="matrixc1.html#E6:62_1"><i><font color="Green">E57</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrixc1.html#K11" target="_self">LineSum</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_5.html#K9">.</a> 1
<b>by </b><i><a class="ref" href="matrixc1.html#D10" target="_self">Def10</a>, <a class="txt" href="matrixc1.html#E7:62_1"><i><font color="Green">E58</font></i></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">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="matrixc1.html#T50" target="_self">Th50</a>, <a class="txt" href="matrixc1.html#E2:62_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">c<sub>2</sub></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">c<sub>2</sub></font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E6:62"/><b>then </b><i><font color="Green">E52</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<a NAME="E7:62"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="binop_2.html#K23">+</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">c<sub>2</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:62_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>]
 ;<br/>

<a NAME="E2:62_2"/>
for <font color="Olive">b<sub>1</sub></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">c<sub>2</sub></font> <a href="binop_2.html#K23">+</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="Olive">b<sub>1</sub></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">b<sub>1</sub></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">b<sub>1</sub></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">c<sub>3</sub></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">E55</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</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">c<sub>3</sub></font>
 ;<br/>

<a NAME="E2:62_2_1"/><i><font color="Green">E56</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a>, <a class="txt" href="matrixc1.html#E1:62_2_1"><i><font color="Green">E55</font></i></a></i>;<br/>
<a NAME="E3:62_2_1"/><i><font color="Green">E57</font></i>: 
1 <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>2</sub></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">E58</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></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>, <a class="txt" href="matrixc1.html#E1:62_2_1"><i><font color="Green">E55</font></i></a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1;<br/>
<b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E6:62_2_1"/><i><font color="Green">E59</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>3</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K2">COMPLEX</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="matrix_1.html#T9">MATRIX_1:9</a></i>;<br/>
<a NAME="E7:62_2_1"/><i><font color="Green">E60</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T21">FINSEQ_5:21</a></i>;<br/>
<a NAME="E8:62_2_1"/><i><font color="Green">E61</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></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">c<sub>3</sub></font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K2">COMPLEX</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="matrixc1.html#E6:62_2_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="matrixc1.html#E7:62_2_1"><i><font color="Green">E60</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> 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>, <a class="txt" href="matrixc1.html#E8:62_2_1"><i><font color="Green">E61</font></i></a></i>;<br/>
<a NAME="E10:62_2_1"/><i><font color="Green">E62</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a>, <a class="txt" href="matrixc1.html#E2:62_2_1"><i><font color="Green">E56</font></i></a></i>;<br/>
<a NAME="E11:62_2_1"/><i><font color="Green">E63</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="txt" href="matrixc1.html#E3:62_2_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a>, <a class="txt" href="matrixc1.html#E2:62_2_1"><i><font color="Green">E56</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E13:62_2_1"/><i><font color="Green">E64</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>3</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="ref" href="matrix_1.html#D1">MATRIX_1:def 1</a></i>;<br/>
<a NAME="E14:62_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">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E4:62_2_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E15:62_2_1"/><b>then </b>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E16:62_2_1"/><b>then </b><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>3</sub></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">c<sub>3</sub></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">c<sub>8</sub></font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E18:62_2_1"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> 1 &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="matrixc1.html#E13:62_2_1"><i><font color="Green">E64</font></i></a></i>;<br/>
<a NAME="E19:62_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">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E11:62_2_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E20:62_2_1"/><b>then </b>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E21:62_2_1"/><b>then </b><i><font color="Green">E67</font></i>: 
<font color="Maroon">c<sub>6</sub></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">c<sub>6</sub></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">c<sub>9</sub></font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E23:62_2_1"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> 1 &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="matrixc1.html#E13:62_2_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="matrixc1.html#E7:62_2_1"><i><font color="Green">E60</font></i></a></i>;<br/>
<a NAME="E24:62_2_1"/><i><font color="Green">E69</font></i>: 
 <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D4">MATRIX_1:def 4</a>, <a class="txt" href="matrixc1.html#E11:62_2_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="matrixc1.html#E23:62_2_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="matrixc1.html#E21:62_2_1"><i><font color="Green">E67</font></i></a></i>;<br/>
<a NAME="E25:62_2_1"/><i><font color="Green">E70</font></i>: 
 <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>3</sub></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_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="matrixc1.html#E16:62_2_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="matrixc1.html#E18:62_2_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="matrixc1.html#E24:62_2_1"><i><font color="Green">E69</font></i></a></i>;<br/>
<a NAME="E26:62_2_1"/><i><font color="Green">E71</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>3</sub></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">c<sub>6</sub></font></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="binop_2.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span>,<font color="Olive">b<sub>1</sub></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">c<sub>3</sub></font></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Olive">b<sub>1</sub></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">c<sub>10</sub></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">E72</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<a NAME="E2:62_2_1_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>10</sub></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">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="txt" href="matrixc1.html#E1:62_2_1_1"><i><font color="Green">E72</font></i></a></i>;<br/>
<a NAME="E3:62_2_1_1"/><i><font color="Green">E74</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</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">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E1:62_2_1"><i><font color="Green">E55</font></i></a>, <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">E75</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a>, <a class="txt" href="matrixc1.html#E3:62_2_1_1"><i><font color="Green">E74</font></i></a></i>;<br/>
<a NAME="E5:62_2_1_1"/><i><font color="Green">E76</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</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">c<sub>3</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E3:62_2_1_1"><i><font color="Green">E74</font></i></a>, <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">E77</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</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">c<sub>3</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a>, <a class="txt" href="matrixc1.html#E5:62_2_1_1"><i><font color="Green">E76</font></i></a></i>;<br/>
<a NAME="E7:62_2_1_1"/><i><font color="Green">E78</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">c<sub>3</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></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">c<sub>11</sub></font> =  <a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>10</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>12</sub></font> =  <a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>10</sub></font>;<br/>
<i><font color="Green">E79</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">c<sub>6</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span> = 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a>, <a class="txt" href="matrixc1.html#E2:62_2_1"><i><font color="Green">E56</font></i></a></i>
;<br/>
<a NAME="E9:62_2_1_1"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1 <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">c<sub>3</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E2:62_2_1"><i><font color="Green">E56</font></i></a>, <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">E81</font></i>: 
 <a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
 
<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">E82</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">c<sub>3</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></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">c<sub>6</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E8:62_2_1_1"><i><font color="Green">E79</font></i></a>, <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a>, <a class="txt" href="matrixc1.html#E9:62_2_1_1"><i><font color="Green">E80</font></i></a></i>;<br/>
<a NAME="E2:62_2_1_1_2"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></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">c<sub>3</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></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">c<sub>3</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Olive">b<sub>1</sub></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">c<sub>13</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:62_2_1_1_2_1"/><i><font color="Green">E83</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>13</sub></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">c<sub>3</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> )
 ;<br/>

<a NAME="E2:62_2_1_1_2_1"/><i><font color="Green">E84</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1 )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a>, <a class="txt" href="matrixc1.html#E9:62_2_1_1"><i><font color="Green">E80</font></i></a>, <a class="txt" href="matrixc1.html#E1:62_2_1_1_2_1"><i><font color="Green">E83</font></i></a></i>;<br/>
<a NAME="E3:62_2_1_1_2_1"/><i><font color="Green">E85</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1 )
 
<b>by </b><i><a class="txt" href="matrixc1.html#E2:62_2_1_1_2_1"><i><font color="Green">E84</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">E86</font></i>: 
<font color="Maroon">c<sub>13</sub></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">c<sub>3</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="txt" href="matrixc1.html#E1:62_2_1_1_2_1"><i><font color="Green">E83</font></i></a></i>;<br/>
<a NAME="E5:62_2_1_1_2_1"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">c<sub>13</sub></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">c<sub>3</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a>, <a class="txt" href="matrixc1.html#E4:62_2_1_1_2_1"><i><font color="Green">E86</font></i></a></i>;<br/>
<a NAME="E6:62_2_1_1_2_1"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">c<sub>13</sub></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">c<sub>3</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D15">FINSEQ_1:def 15</a>, <a class="txt" href="matrixc1.html#E5:62_2_1_1_2_1"><i><font color="Green">E87</font></i></a></i>;<br/>
<a NAME="E7:62_2_1_1_2_1"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">c<sub>13</sub></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">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E1:62_2_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="matrixc1.html#E3:62_2_1_1_2_1"><i><font color="Green">E85</font></i></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">E90</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a>, <a class="txt" href="matrixc1.html#E7:62_2_1_1_2_1"><i><font color="Green">E89</font></i></a></i>;<br/>
<a NAME="E9:62_2_1_1_2_1"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E2:62_2_1_1_2_1"><i><font color="Green">E84</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">E92</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a>, <a class="txt" href="matrixc1.html#E10:62_2_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="matrixc1.html#E9:62_2_1_1_2_1"><i><font color="Green">E91</font></i></a></i>;<br/>
<a NAME="E11:62_2_1_1_2_1"/><i><font color="Green">E93</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="matrixc1.html#E2:62_2_1_1_2_1"><i><font color="Green">E84</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a>, <a class="txt" href="matrixc1.html#E1:62_2_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="matrixc1.html#E1:62_2_1_1"><i><font color="Green">E72</font></i></a></i>;<br/>
<a NAME="E12:62_2_1_1_2_1"/><i><font color="Green">E94</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>10</sub></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">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="matrixc1.html#T1" target="_self">Th1</a>, <a class="txt" href="matrixc1.html#E11:62_2_1_1_2_1"><i><font color="Green">E93</font></i></a></i>;<br/>
<a NAME="E13:62_2_1_1_2_1"/><i><font color="Green">E95</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a>, <a class="txt" href="matrixc1.html#E9:62_2_1_1"><i><font color="Green">E80</font></i></a>, <a class="txt" href="matrixc1.html#E1:62_2_1_1_2_1"><i><font color="Green">E83</font></i></a>, <a class="txt" href="matrixc1.html#E10:62_2_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="matrix_1.html#D4">MATRIX_1:def 4</a>, <a class="txt" href="matrixc1.html#E1:62_2_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="matrixc1.html#E16:62_2_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="matrixc1.html#E18:62_2_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="matrixc1.html#E24:62_2_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="matrixc1.html#E1:62_2_1_1"><i><font color="Green">E72</font></i></a></i>;<br/>
<a NAME="E14:62_2_1_1_2_1"/><i><font color="Green">E96</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>10</sub></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">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="matrixc1.html#T1" target="_self">Th1</a>, <a class="txt" href="matrixc1.html#E13:62_2_1_1_2_1"><i><font color="Green">E95</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>14</sub></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">E97</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="matrix_5.html#K9">.</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="ref" href="matrix_1.html#D6">MATRIX_1:def 6</a>, <a class="txt" href="matrixc1.html#E12:62_2_1_1_2_1"><i><font color="Green">E94</font></i></a></i>;<br/>
<a NAME="E17:62_2_1_1_2_1"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T121">FINSEQ_3:121</a>, <a class="txt" href="matrixc1.html#E2:62_2_1_1_2_1"><i><font color="Green">E84</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">c<sub>3</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">c<sub>13</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></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">c<sub>3</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="ref" href="funct_1.html#T70">FUNCT_1:70</a>, <a class="txt" href="matrixc1.html#E6:62_2_1_1_2_1"><i><font color="Green">E88</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a>, <a class="txt" href="matrixc1.html#E8:62_2_1_1_2_1"><i><font color="Green">E90</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="matrixc1.html#E14:62_2_1_1_2_1"><i><font color="Green">E96</font></i></a>, <a class="ref" href="matrix_1.html#D6">MATRIX_1:def 6</a>, <a class="txt" href="matrixc1.html#E16:62_2_1_1_2_1"><i><font color="Green">E97</font></i></a>, <a class="txt" href="matrixc1.html#E17:62_2_1_1_2_1"><i><font color="Green">E98</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="txt" href="matrixc1.html#E10:62_2_1_1_2_1"><i><font color="Green">E92</font></i></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">c<sub>3</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">c<sub>13</sub></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">c<sub>6</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a>, <a class="txt" href="matrixc1.html#E1:62_2_1_1_2"><i><font color="Green">E82</font></i></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">c<sub>6</sub></font></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="binop_2.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span>,<font color="Maroon">c<sub>10</sub></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">c<sub>6</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span>,<font color="Maroon">c<sub>10</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="matrixc1.html#D11" target="_self">Def11</a>, <a class="txt" href="matrixc1.html#E2:62_2_1_1"><i><font color="Green">E73</font></i></a>, <a class="txt" href="matrixc1.html#E25:62_2_1"><i><font color="Green">E70</font></i></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">c<sub>6</sub></font>,<font color="Maroon">c<sub>10</sub></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">c<sub>3</sub></font>,<font color="Maroon">c<sub>10</sub></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">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a>, <a class="txt" href="matrixc1.html#E4:62_2_1_1"><i><font color="Green">E75</font></i></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">c<sub>6</sub></font>,<font color="Maroon">c<sub>10</sub></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">c<sub>3</sub></font>,<font color="Maroon">c<sub>10</sub></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">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a>, <a class="txt" href="matrixc1.html#E6:62_2_1_1"><i><font color="Green">E77</font></i></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">c<sub>3</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="matrixc1.html#T51" target="_self">Th51</a>, <a class="txt" href="matrixc1.html#E1:62_2_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="matrixc1.html#E7:62_2_1_1"><i><font color="Green">E78</font></i></a>, <a class="txt" href="matrixc1.html#E10:62_2_1_1"><i><font color="Green">E81</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrixc1.html#K12" target="_self">ColSum</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="ref" href="matrixc1.html#D11" target="_self">Def11</a>, <a class="txt" href="matrixc1.html#E2:62_2_1_1"><i><font color="Green">E73</font></i></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">c<sub>6</sub></font></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="binop_2.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span>,<font color="Maroon">c<sub>10</sub></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">c<sub>3</sub></font></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E27:62_2_1"/><i><font color="Green">E72</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</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">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E1:62_2_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="finseq_1.html#T5">FINSEQ_1:5</a></i>;<br/>
<a NAME="E28:62_2_1"/><i><font color="Green">E73</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</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">c<sub>3</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E27:62_2_1"><i><font color="Green">E72</font></i></a>, <a class="ref" href="matrixc1.html#D10" target="_self">Def10</a></i>;<br/>
<a NAME="E29:62_2_1"/><i><font color="Green">E74</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</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">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E28:62_2_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E30:62_2_1"/><i><font color="Green">E75</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">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="matrixc1.html#D11" target="_self">Def11</a></i>;<br/>
<i><font color="Green">E76</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">c<sub>6</sub></font></span>)</span> = 
 <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="ref" href="matrixc1.html#D11" target="_self">Def11</a></i>
<br/>.= 
 <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>3</sub></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_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="matrixc1.html#E16:62_2_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="matrixc1.html#E18:62_2_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="matrixc1.html#E24:62_2_1"><i><font color="Green">E69</font></i></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">c<sub>3</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</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="E32:62_2_1"/><i><font color="Green">E77</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">c<sub>3</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/>
<i><font color="Green">E78</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">c<sub>6</sub></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">c<sub>3</sub></font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> = 
 <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="complsp2.html#T6">COMPLSP2:6</a>, <a class="txt" href="matrixc1.html#E30:62_2_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="matrixc1.html#E32:62_2_1"><i><font color="Green">E77</font></i></a>, <a class="txt" href="matrixc1.html#E25:62_2_1"><i><font color="Green">E70</font></i></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">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="matrixc1.html#D11" target="_self">Def11</a></i>
;<br/>
<a NAME="E34:62_2_1"/><i><font color="Green">E79</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></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">c<sub>3</sub></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">c<sub>6</sub></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">c<sub>3</sub></font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Olive">b<sub>1</sub></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">c<sub>3</sub></font></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Olive">b<sub>1</sub></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">c<sub>10</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:62_2_1_4"/><i><font color="Green">E80</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>10</sub></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">c<sub>3</sub></font></span>)</span> )
 ;<br/>

<a NAME="E2:62_2_1_4"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">c<sub>10</sub></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">c<sub>3</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E1:62_2_1_4"><i><font color="Green">E80</font></i></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">E82</font></i>: 
<font color="Maroon">c<sub>10</sub></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">c<sub>6</sub></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">c<sub>3</sub></font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E1:62_2_1_4"><i><font color="Green">E80</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="txt" href="matrixc1.html#E33:62_2_1"><i><font color="Green">E78</font></i></a></i>;<br/>
<a NAME="E4:62_2_1_4"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">c<sub>10</sub></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">c<sub>6</sub></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">c<sub>3</sub></font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E3:62_2_1_4"><i><font color="Green">E82</font></i></a>, <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">E84</font></i>: 
<font color="Maroon">c<sub>10</sub></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">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E25:62_2_1"><i><font color="Green">E70</font></i></a>, <a class="ref" href="matrixc1.html#D11" target="_self">Def11</a>, <a class="txt" href="matrixc1.html#E2:62_2_1_4"><i><font color="Green">E81</font></i></a></i>;<br/>
<a NAME="E6:62_2_1_4"/><i><font color="Green">E85</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="txt" href="matrixc1.html#E5:62_2_1_4"><i><font color="Green">E84</font></i></a>, <a class="txt" href="matrixc1.html#E25:62_2_1"><i><font color="Green">E70</font></i></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">c<sub>6</sub></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">c<sub>3</sub></font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">c<sub>10</sub></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">c<sub>6</sub></font></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">c<sub>10</sub></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">c<sub>3</sub></font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="matrixc1.html#E4:62_2_1_4"><i><font color="Green">E83</font></i></a>, <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">c<sub>6</sub></font></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="binop_2.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span>,<font color="Maroon">c<sub>10</sub></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#E5:62_2_1_4"><i><font color="Green">E84</font></i></a>, <a class="txt" href="matrixc1.html#E25:62_2_1"><i><font color="Green">E70</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrixc1.html#K12" target="_self">ColSum</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="matrixc1.html#E26:62_2_1"><i><font color="Green">E71</font></i></a>, <a class="txt" href="matrixc1.html#E6:62_2_1_4"><i><font color="Green">E85</font></i></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">c<sub>6</sub></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">c<sub>3</sub></font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">c<sub>10</sub></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">c<sub>3</sub></font></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon">c<sub>10</sub></font> =  <a href="matrixc1.html#K11" target="_self">LineSum</a> <font color="Maroon">c<sub>6</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>11</sub></font> =  <a href="matrixc1.html#K11" target="_self">LineSum</a> <font color="Maroon">c<sub>3</sub></font>;<br/>
<a NAME="E35:62_2_1"/><i><font color="Green">E80</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">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="matrixc1.html#D10" target="_self">Def10</a></i>;<br/>
<a NAME="E36:62_2_1"/><i><font color="Green">E81</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">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="txt" href="matrixc1.html#E1:62_2_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="matrixc1.html#D10" target="_self">Def10</a></i>;<br/>
<a NAME="E37:62_2_1"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1 <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">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a>, <a class="txt" href="matrixc1.html#E36:62_2_1"><i><font color="Green">E81</font></i></a></i>;<br/>
<i><font color="Green">E83</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">c<sub>6</sub></font></span>)</span> = 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="ref" href="matrixc1.html#D10" target="_self">Def10</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a>, <a class="txt" href="matrixc1.html#E2:62_2_1"><i><font color="Green">E56</font></i></a></i>
;<br/>
<a NAME="E39:62_2_1"/>
 <a href="matrixc1.html#K11" target="_self">LineSum</a> <font color="Maroon">c<sub>6</sub></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">c<sub>3</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
 
<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">E84</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">c<sub>3</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></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">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E38:62_2_1"><i><font color="Green">E83</font></i></a>, <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a>, <a class="txt" href="matrixc1.html#E37:62_2_1"><i><font color="Green">E82</font></i></a></i>;<br/>
<a NAME="E2:62_2_1_6"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></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">c<sub>3</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></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">c<sub>3</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Olive">b<sub>1</sub></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">c<sub>6</sub></font></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Olive">b<sub>1</sub></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">c<sub>12</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:62_2_1_6_1"/><i><font color="Green">E85</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>12</sub></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">c<sub>3</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> )
 ;<br/>

<a NAME="E2:62_2_1_6_1"/><i><font color="Green">E86</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1 )
 
<b>by </b><i><a class="txt" href="matrixc1.html#E1:62_2_1_6"><i><font color="Green">E84</font></i></a>, <a class="txt" href="matrixc1.html#E1:62_2_1_6_1"><i><font color="Green">E85</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a>, <a class="txt" href="matrixc1.html#E38:62_2_1"><i><font color="Green">E83</font></i></a></i>;<br/>
<a NAME="E3:62_2_1_6_1"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">c<sub>12</sub></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">c<sub>3</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="txt" href="matrixc1.html#E1:62_2_1_6_1"><i><font color="Green">E85</font></i></a></i>;<br/>
<a NAME="E4:62_2_1_6_1"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">c<sub>12</sub></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">c<sub>3</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a>, <a class="txt" href="matrixc1.html#E3:62_2_1_6_1"><i><font color="Green">E87</font></i></a></i>;<br/>
<a NAME="E5:62_2_1_6_1"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">c<sub>12</sub></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">c<sub>3</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D15">FINSEQ_1:def 15</a>, <a class="txt" href="matrixc1.html#E4:62_2_1_6_1"><i><font color="Green">E88</font></i></a></i>;<br/>
<a NAME="E6:62_2_1_6_1"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">c<sub>12</sub></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">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E1:62_2_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="matrixc1.html#E2:62_2_1_6_1"><i><font color="Green">E86</font></i></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">E91</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E1:62_2_1_6"><i><font color="Green">E84</font></i></a>, <a class="txt" href="matrixc1.html#E1:62_2_1_6_1"><i><font color="Green">E85</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="txt" href="matrixc1.html#E38:62_2_1"><i><font color="Green">E83</font></i></a></i>;<br/>
<a NAME="E8:62_2_1_6_1"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">c<sub>12</sub></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">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a>, <a class="txt" href="matrixc1.html#E2:62_2_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="matrixc1.html#E7:62_2_1_6_1"><i><font color="Green">E91</font></i></a></i>;<br/>
<a NAME="E9:62_2_1_6_1"/><i><font color="Green">E93</font></i>: 
 <a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>12</sub></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">E94</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">c<sub>3</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span> = 
 <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>3</sub></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">c<sub>6</sub></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_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="matrixc1.html#E16:62_2_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="matrixc1.html#E18:62_2_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="matrixc1.html#E24:62_2_1"><i><font color="Green">E69</font></i></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">c<sub>6</sub></font>,<font color="Maroon">c<sub>12</sub></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">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></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">c<sub>3</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Olive">b<sub>1</sub></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">c<sub>13</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:62_2_1_6_1_1_2"/><i><font color="Green">E95</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>13</sub></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">c<sub>3</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span> )
 ;<br/>

<a NAME="E2:62_2_1_6_1_1_2"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">c<sub>13</sub></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">c<sub>3</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="txt" href="matrixc1.html#E1:62_2_1_6_1_1_2"><i><font color="Green">E95</font></i></a></i>;<br/>
<a NAME="E3:62_2_1_6_1_1_2"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">c<sub>13</sub></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">c<sub>3</sub></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#E2:62_2_1_6_1_1_2"><i><font color="Green">E96</font></i></a></i>;<br/>
<a NAME="E4:62_2_1_6_1_1_2"/><i><font color="Green">E98</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="matrixc1.html#E3:62_2_1_6_1_1_2"><i><font color="Green">E97</font></i></a>, <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">E99</font></i>: 
<font color="Maroon">c<sub>13</sub></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">c<sub>3</sub></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#E2:62_2_1_6_1_1_2"><i><font color="Green">E96</font></i></a></i>;<br/>
<a NAME="E6:62_2_1_6_1_1_2"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">c<sub>13</sub></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">c<sub>6</sub></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#E2:62_2_1_6_1_1_2"><i><font color="Green">E96</font></i></a>, <a class="txt" href="matrixc1.html#E25:62_2_1"><i><font color="Green">E70</font></i></a></i>;<br/>
<a NAME="E7:62_2_1_6_1_1_2"/><i><font color="Green">E101</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="matrixc1.html#E3:62_2_1_6_1_1_2"><i><font color="Green">E97</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="txt" href="matrixc1.html#E1:62_2_1_6"><i><font color="Green">E84</font></i></a>, <a class="txt" href="matrixc1.html#E1:62_2_1_6_1"><i><font color="Green">E85</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a>, <a class="txt" href="matrixc1.html#E38:62_2_1"><i><font color="Green">E83</font></i></a>, <a class="txt" href="matrixc1.html#E1:62_2_1"><i><font color="Green">E55</font></i></a></i>;<br/>
<a NAME="E8:62_2_1_6_1_1_2"/><i><font color="Green">E102</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></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">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="matrixc1.html#T1" target="_self">Th1</a>, <a class="txt" href="matrixc1.html#E4:62_2_1_6_1_1_2"><i><font color="Green">E98</font></i></a>, <a class="txt" href="matrixc1.html#E2:62_2_1_6_1"><i><font color="Green">E86</font></i></a>, <a class="txt" href="matrixc1.html#E1:62_2_1"><i><font color="Green">E55</font></i></a></i>;<br/>
<a NAME="E9:62_2_1_6_1_1_2"/><i><font color="Green">E103</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="txt" href="matrixc1.html#E1:62_2_1_6"><i><font color="Green">E84</font></i></a>, <a class="txt" href="matrixc1.html#E1:62_2_1_6_1"><i><font color="Green">E85</font></i></a>, <a class="ref" href="matrixc1.html#D10" target="_self">Def10</a>, <a class="ref" href="matrix_1.html#D4">MATRIX_1:def 4</a>, <a class="txt" href="matrixc1.html#E16:62_2_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="matrixc1.html#E18:62_2_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="matrixc1.html#E24:62_2_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="matrixc1.html#E7:62_2_1_6_1_1_2"><i><font color="Green">E101</font></i></a></i>;<br/>
<a NAME="E10:62_2_1_6_1_1_2"/><i><font color="Green">E104</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></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">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="matrixc1.html#T1" target="_self">Th1</a>, <a class="txt" href="matrixc1.html#E9:62_2_1_6_1_1_2"><i><font color="Green">E103</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>14</sub></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">E105</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="matrix_5.html#K9">.</a> <font color="Maroon">c<sub>13</sub></font> )
 <b>by </b><i><a class="ref" href="matrix_1.html#D6">MATRIX_1:def 6</a>, <a class="txt" href="matrixc1.html#E8:62_2_1_6_1_1_2"><i><font color="Green">E102</font></i></a></i>;<br/>
<a NAME="E13:62_2_1_6_1_1_2"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T121">FINSEQ_3:121</a>, <a class="txt" href="matrixc1.html#E1:62_2_1_6"><i><font color="Green">E84</font></i></a>, <a class="txt" href="matrixc1.html#E1:62_2_1_6_1"><i><font color="Green">E85</font></i></a>, <a class="txt" href="matrixc1.html#E38:62_2_1"><i><font color="Green">E83</font></i></a></i>;<br/>
<span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">c<sub>13</sub></font> = 
<font color="Maroon">c<sub>3</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a>, <a class="txt" href="matrixc1.html#E5:62_2_1_6_1_1_2"><i><font color="Green">E99</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="txt" href="matrixc1.html#E10:62_2_1_6_1_1_2"><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#E13:62_2_1_6_1_1_2"><i><font color="Green">E106</font></i></a>, <a class="txt" href="matrixc1.html#E12:62_2_1_6_1_1_2"><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">c<sub>6</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a>, <a class="txt" href="matrixc1.html#E6:62_2_1_6_1_1_2"><i><font color="Green">E100</font></i></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">c<sub>3</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">c<sub>13</sub></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">c<sub>3</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a>, <a class="txt" href="matrixc1.html#E1:62_2_1_6_1_1"><i><font color="Green">E94</font></i></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">c<sub>3</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">c<sub>12</sub></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">c<sub>3</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></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">c<sub>3</sub></font></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="ref" href="funct_1.html#T70">FUNCT_1:70</a>, <a class="txt" href="matrixc1.html#E5:62_2_1_6_1"><i><font color="Green">E89</font></i></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">c<sub>6</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="matrixc1.html#E9:62_2_1_6_1"><i><font color="Green">E93</font></i></a>, <a class="ref" href="matrixc1.html#D10" target="_self">Def10</a>, <a class="txt" href="matrixc1.html#E6:62_2_1_6_1"><i><font color="Green">E90</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrixc1.html#K11" target="_self">LineSum</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="txt" href="matrixc1.html#E8:62_2_1_6_1"><i><font color="Green">E92</font></i></a>, <a class="ref" href="matrixc1.html#D10" target="_self">Def10</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">c<sub>3</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">c<sub>12</sub></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">c<sub>6</sub></font></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">c<sub>12</sub></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">c<sub>6</sub></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">c<sub>3</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a>, <a class="txt" href="matrixc1.html#E1:62_2_1_6"><i><font color="Green">E84</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">c<sub>3</sub></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">c<sub>6</sub></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">c<sub>3</sub></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">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="matrixc1.html#T51" target="_self">Th51</a>, <a class="txt" href="matrixc1.html#E35:62_2_1"><i><font color="Green">E80</font></i></a>, <a class="txt" href="matrixc1.html#E1:62_2_1"><i><font color="Green">E55</font></i></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">c<sub>6</sub></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">c<sub>3</sub></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">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a>, <a class="txt" href="matrixc1.html#E29:62_2_1"><i><font color="Green">E74</font></i></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">c<sub>6</sub></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">c<sub>3</sub></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">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="matrixc1.html#E1:62_2"><i><font color="Green">E54</font></i></a>, <a class="txt" href="matrixc1.html#E10:62_2_1"><i><font color="Green">E62</font></i></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">c<sub>6</sub></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">c<sub>3</sub></font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="matrixc1.html#D10" target="_self">Def10</a>, <a class="txt" href="matrixc1.html#E27:62_2_1"><i><font color="Green">E72</font></i></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">c<sub>6</sub></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">c<sub>3</sub></font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="matrixc1.html#T37" target="_self">Th37</a>, <a class="txt" href="matrixc1.html#E31:62_2_1"><i><font color="Green">E76</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">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="matrixc1.html#E33:62_2_1"><i><font color="Green">E78</font></i></a>, <a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a>, <a class="txt" href="matrixc1.html#E34:62_2_1"><i><font color="Green">E79</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E41:62_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">c<sub>3</sub></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">c<sub>3</sub></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">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E8:62"/><i><font color="Green">E54</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="matrixc1.html#E6:62"><i><font color="Green">E52</font></i></a>, <a class="txt" href="matrixc1.html#E7:62"><i><font color="Green">E53</font></i></a>);<br/></i>
<a NAME="E9:62"/><i><font color="Green">E55</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">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</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">c<sub>1</sub></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">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="matrixc1.html#E8:62"><i><font color="Green">E54</font></i></a></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">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="matrixc1.html#E4:62"><i><font color="Green">E51</font></i></a>, <a class="txt" href="matrixc1.html#E9:62"><i><font color="Green">E55</font></i></a></i>;<br/>


</div>
