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

<b>let </b><font color="Maroon">K</font> be   <a href="vectsp_1.html#NM2">Field</a>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="matrix_1.html#NM4">Matrix</a> of <font color="Maroon">K</font>, <font color="Maroon">B</font> be   <a href="matrix_1.html#NM4">Matrix</a> of <font color="Maroon">K</font>, <font color="Maroon">C</font> be   <a href="matrix_1.html#NM4">Matrix</a> of <font color="Maroon">K</font>;<br/>



<b>assume </b><a NAME="E1:59"/><i><font color="Green">E29</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">C</font> &amp;  <a href="matrix_1.html#K1">width</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">C</font> &amp;  <a href="matrix_1.html#K1">width</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">B</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">A</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">B</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 ;<br/>

<a NAME="E2:59"/><i><font color="Green">E30</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="matrix_3.html#K3">+</a> <font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">B</font> &amp;  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="matrix_3.html#K3">+</a> <font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">B</font> )
 
<b>by </b><i><a class="ref" href="matrix_3.html#D3">MATRIX_3:def 3</a></i>;<br/>
<a NAME="E3:59"/><b>then </b><i><font color="Green">E31</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="matrix_3.html#K3">+</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">A</font> &amp;  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="matrix_3.html#K3">+</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="matrix_3.html#K3">+</a> <font color="Maroon">C</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="matrix_3.html#D4">MATRIX_3:def 4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">M1</font> = <font color="Maroon">A</font> <a href="matrix_3.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="matrix_3.html#K3">+</a> <font color="Maroon">C</font></span>)</span> as   <a href="matrix_1.html#NM6">Matrix</a> of  <a href="finseq_1.html#K3">len</a> <font color="Maroon">A</font>, <a href="matrix_1.html#K1">width</a> <font color="Maroon">B</font>,<font color="Maroon">K</font> <b>by </b><i>, <a class="txt" href="matrix_4.html#E1:59"><i><font color="Green">E29</font></i></a>, <a class="ref" href="matrix_1.html#T20">MATRIX_1:20</a></i>;<br/>
<a NAME="E5:59"/><i><font color="Green">E32</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K4">*</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">A</font> &amp;  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K4">*</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">B</font> )
 
<b>by </b><i>, <a class="ref" href="matrix_3.html#D4">MATRIX_3:def 4</a></i>;<br/>
<a NAME="E6:59"/><i><font color="Green">E33</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K4">*</a> <font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">A</font> &amp;  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K4">*</a> <font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">C</font> )
 
<b>by </b><i>, <a class="ref" href="matrix_3.html#D4">MATRIX_3:def 4</a></i>;<br/>
<a NAME="E7:59"/><i><font color="Green">E34</font></i>: 
 <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">M1</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K4">*</a> <font color="Maroon">B</font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrix_4.html#E1:59"><i><font color="Green">E29</font></i></a>, <a class="txt" href="matrix_4.html#E2:59"><i><font color="Green">E30</font></i></a>, <a class="txt" href="matrix_4.html#E3:59"><i><font color="Green">E31</font></i></a>, </i>;<br/>
<a NAME="E8:59"/><i><font color="Green">E39</font></i>: 
 <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">M1</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K4">*</a> <font color="Maroon">C</font></span>)</span>
 
<b>by </b><i>, <a class="txt" href="matrix_4.html#E1:59"><i><font color="Green">E29</font></i></a>, <a class="txt" href="matrix_4.html#E2:59"><i><font color="Green">E30</font></i></a>, <a class="txt" href="matrix_4.html#E5:59"><i><font color="Green">E32</font></i></a>, </i>;<br/>
<a NAME="E9:59"/><i><font color="Green">E53</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K4">*</a> <font color="Maroon">B</font></span>)</span> <a href="matrix_3.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K4">*</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K4">*</a> <font color="Maroon">B</font></span>)</span> &amp;  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K4">*</a> <font color="Maroon">B</font></span>)</span> <a href="matrix_3.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K4">*</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K4">*</a> <font color="Maroon">B</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="matrix_3.html#D3">MATRIX_3:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">M2</font> = <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K4">*</a> <font color="Maroon">B</font></span>)</span> <a href="matrix_3.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K4">*</a> <font color="Maroon">C</font></span>)</span> as   <a href="matrix_1.html#NM6">Matrix</a> of  <a href="finseq_1.html#K3">len</a> <font color="Maroon">A</font>, <a href="matrix_1.html#K1">width</a> <font color="Maroon">B</font>,<font color="Maroon">K</font> <b>by </b><i>, <a class="txt" href="matrix_4.html#E3:59"><i><font color="Green">E31</font></i></a>, <a class="ref" href="matrix_1.html#T20">MATRIX_1:20</a></i>;<br/>
<a NAME="E11:59"/>
for <font color="Olive">i</font>, <font color="Olive">j</font> being  <a href="nat_1.html#NM1">Nat</a>  st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Olive">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">M1</font> holds <br/><font color="Maroon">M1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M2</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1"><b>proof </b></a><div class="add">

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



<b>assume </b><a NAME="E1:59_1"/><i><font color="Green">E54</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">M1</font>
 ;<br/>

<a NAME="E2:59_1"/>
 <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">A</font>,<font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">B</font>
 
<b>by </b><i>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/>
<a NAME="E3:59_1"/><b>then </b><i><font color="Green">E75</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">A</font>,<font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">B</font>,<font color="Maroon">j</font></span>)</span>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>;<br/>
<a NAME="E4:59_1"/>
 <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">A</font>,<font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">C</font>
 
<b>by </b><i>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/>
<a NAME="E5:59_1"/><b>then </b><i><font color="Green">E76</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">A</font>,<font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>;<br/>
<a NAME="E6:59_1"/><i><font color="Green">E77</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</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">B</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E7:59_1"/>
1 <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">B</font>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E8:59_1"/><b>then </b><i><font color="Green">E78</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="txt" href="matrix_4.html#E8:59"><i><font color="Green">E39</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E9:59_1"/>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">B</font></span>)</span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a>, <a class="txt" href="matrix_4.html#E1:59"><i><font color="Green">E29</font></i></a>, <a class="txt" href="matrix_4.html#E2:59"><i><font color="Green">E30</font></i></a>, </i>;<br/>
<a NAME="E10:59_1"/><b>then </b><i><font color="Green">E79</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<font color="Maroon">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="matrix_4.html#T8" target="_self">Th8</a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>reconsider </b><font color="Maroon">q1</font> =  <a href="matrix_1.html#K8">Line</a> <font color="Maroon">A</font>,<font color="Maroon">i</font>, <font color="Maroon">q2</font> =  <a href="matrix_1.html#K9">Col</a> <font color="Maroon">B</font>,<font color="Maroon">j</font>, <font color="Maroon">q3</font> =  <a href="matrix_1.html#K9">Col</a> <font color="Maroon">C</font>,<font color="Maroon">j</font> as    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">B</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">K</font> <b>by </b><i/>;<br/>
<i><font color="Green">E80</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">A</font>,<font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">B</font>,<font color="Maroon">j</font></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K12">mlt</a> <font color="Maroon">q1</font>,<font color="Maroon">q2</font></span>)</span>

<br/>.= 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">B</font>
<b>by </b><i><a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K12">mlt</a> <font color="Maroon">q1</font>,<font color="Maroon">q3</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">A</font>,<font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span></span>)</span>

;<br/>
<i><font color="Green">E81</font></i>: <font color="Maroon">M1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> = 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">A</font>,<font color="Maroon">i</font></span>)</span> <a href="fvsum_1.html#K13">"*"</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="matrix_3.html#K3">+</a> <font color="Maroon">C</font></span>)</span>,<font color="Maroon">j</font></span>)</span>
<b>by </b><i>, <a class="txt" href="matrix_4.html#E1:59"><i><font color="Green">E29</font></i></a>, , <a class="ref" href="matrix_3.html#D4">MATRIX_3:def 4</a></i>
<br/>.= 
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">A</font>,<font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p3">(<span class="default"><font color="Maroon">B</font> <a href="matrix_3.html#K3">+</a> <font color="Maroon">C</font></span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="fvsum_1.html#D10">FVSUM_1:def 10</a></i>
<br/>.= 
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">A</font>,<font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">B</font>,<font color="Maroon">j</font></span>)</span> <a href="fvsum_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="matrix_4.html#T9" target="_self">Th9</a>, <a class="ref" href="matrix_4.html#T1" target="_self">Th1</a></i>
<br/>.= 
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">A</font>,<font color="Maroon">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">B</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">A</font>,<font color="Maroon">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="matrix_4.html#T7" target="_self">Th7</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">A</font>,<font color="Maroon">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">B</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">A</font>,<font color="Maroon">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, </i>
;<br/>
<font color="Maroon">M2</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K4">*</a> <font color="Maroon">B</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K4">*</a> <font color="Maroon">C</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix_4.html#E6:59"><i><font color="Green">E33</font></i></a>, , <a class="ref" href="matrix_3.html#D3">MATRIX_3:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">A</font>,<font color="Maroon">i</font></span>)</span> <a href="fvsum_1.html#K13">"*"</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">B</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K4">*</a> <font color="Maroon">C</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span>
<b>by </b><i>, <a class="txt" href="matrix_4.html#E6:59"><i><font color="Green">E33</font></i></a>, , <a class="ref" href="matrix_3.html#D4">MATRIX_3:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">A</font>,<font color="Maroon">i</font></span>)</span> <a href="fvsum_1.html#K13">"*"</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">B</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">A</font>,<font color="Maroon">i</font></span>)</span> <a href="fvsum_1.html#K13">"*"</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="txt" href="matrix_4.html#E7:59"><i><font color="Green">E34</font></i></a>, , <a class="ref" href="matrix_3.html#D4">MATRIX_3:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">A</font>,<font color="Maroon">i</font></span>)</span> <a href="fvsum_1.html#K13">"*"</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">B</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">A</font>,<font color="Maroon">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="fvsum_1.html#D10">FVSUM_1:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">A</font>,<font color="Maroon">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">B</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">A</font>,<font color="Maroon">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="fvsum_1.html#D10">FVSUM_1:def 10</a></i>
;<br/>
<b>hence </b><a NAME="E15:59_1"/>
<font color="Maroon">M1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M2</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E12:59"/>
<font color="Maroon">A</font> <a href="matrix_3.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="matrix_3.html#K3">+</a> <font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K4">*</a> <font color="Maroon">B</font></span>)</span> <a href="matrix_3.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K4">*</a> <font color="Maroon">C</font></span>)</span>
 <b>by </b><i><a class="txt" href="matrix_4.html#E1:59"><i><font color="Green">E29</font></i></a>, <a class="txt" href="matrix_4.html#E2:59"><i><font color="Green">E30</font></i></a>, <a class="txt" href="matrix_4.html#E3:59"><i><font color="Green">E31</font></i></a>, <a class="ref" href="matrix_4.html#T2" target="_self">Th2</a>, <a class="ref" href="matrix_1.html#T21">MATRIX_1:21</a></i>;<br/>


</div>
