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

<b>let </b><font color="Maroon">M</font> be   <a href="matrix_1.html#NM1">Matrix</a> of <a href="numbers.html#K1">REAL</a> , <font color="Maroon">M1</font> be   <a href="matrix_1.html#NM1">Matrix</a> of <a href="numbers.html#K1">REAL</a> , <font color="Maroon">M2</font> be   <a href="matrix_1.html#NM1">Matrix</a> of <a href="numbers.html#K1">REAL</a> ;<br/>

<b>assume </b><a NAME="E1:49"/><i><font color="Green">E43</font></i>: 
 <a href="matrix_1.html#K1">width</a> <font color="Maroon">M1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">M2</font>
 ;<br/>

<b>set </b><font color="Maroon">M3</font> =  <a href="matrixr1.html#K1">MXR2MXF</a> <font color="Maroon">M1</font>;<br/>
<b>set </b><font color="Maroon">M4</font> =  <a href="matrixr1.html#K1">MXR2MXF</a> <font color="Maroon">M2</font>;<br/>
<a NAME="E2:49"/><i><font color="Green">E48</font></i>: 
(  <a href="matrixr1.html#K1">MXR2MXF</a> <font color="Maroon">M1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M1</font> &amp;  <a href="matrixr1.html#K1">MXR2MXF</a> <font color="Maroon">M2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M2</font> )
 
<b>by </b><i><a class="ref" href="matrixr1.html#D1">MATRIXR1:def 1</a></i>;<br/>
<a NAME="E3:49"/>
 <a href="matrixr1.html#K2">MXF2MXR</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixr1.html#K1">MXR2MXF</a> <font color="Maroon">M1</font></span>)</span> <a href="matrix_3.html#K4">*</a> <span class="p2">(<span class="default"><a href="matrixr1.html#K1">MXR2MXF</a> <font color="Maroon">M2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">M1</font> <a href="matrixr1.html#K6">*</a> <font color="Maroon">M2</font>
 
<b>by </b><i><a class="ref" href="matrixr1.html#D6">MATRIXR1:def 6</a></i>;<br/>
<a NAME="E4:49"/><b>then </b><i><font color="Green">E49</font></i>: 
<span class="p1">(<span class="default"><a href="matrixr1.html#K1">MXR2MXF</a> <font color="Maroon">M1</font></span>)</span> <a href="matrix_3.html#K4">*</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K1">MXR2MXF</a> <font color="Maroon">M2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">M1</font> <a href="matrixr1.html#K6">*</a> <font color="Maroon">M2</font>
 
<b>by </b><i><a class="ref" href="matrixr1.html#D2">MATRIXR1:def 2</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:49_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">M</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M1</font> <a href="matrixr1.html#K6">*</a> <font color="Maroon">M2</font>
 ;<br/><b>hence </b><a NAME="E2:49_1"/><i><font color="Green">E51</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">M</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">M1</font>
 <b>by </b><i>, , <a class="txt" href="matrprob.html#E1:49"><i><font color="Green">E43</font></i></a>, <a class="ref" href="matrix_3.html#D4">MATRIX_3:def 4</a></i>;<br/><b>thus </b><a NAME="E3:49_1"/><i><font color="Green">E58</font></i>: 
 <a href="matrix_1.html#K1">width</a> <font color="Maroon">M</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">M2</font>
 <b>by </b><i><a class="ref" href="matrprob.html#T1" target="_self">Th1</a>, , , <a class="txt" href="matrprob.html#E1:49"><i><font color="Green">E43</font></i></a>, <a class="ref" href="matrix_3.html#D4">MATRIX_3:def 4</a></i>;<br/><b>thus </b><a NAME="E4:49_1"/>
for <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">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">M</font> holds <br/><font color="Maroon">M</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M1</font>,<font color="Olive">i</font></span>)</span> <a href="matrprob.html#NK7" target="_self">"*"</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M2</font>,<font color="Olive">j</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1"><b>proof </b></a><div class="add">

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



<b>assume </b><a NAME="E1:49_1_1"/><i><font color="Green">E59</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">M</font>
 ;<br/>

<a NAME="E2:49_1_1"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">i</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">M1</font></span>)</span> &amp; <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">M2</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="matrprob.html#E4:49"><i><font color="Green">E49</font></i></a>, , , <a class="txt" href="matrprob.html#E2:49"><i><font color="Green">E48</font></i></a></i>;<br/>
<a NAME="E3:49_1_1"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">M1</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E4:49_1_1"/><b>then </b><i><font color="Green">E61</font></i>: 
 <a href="matrix_1.html#K8">Line</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K1">MXR2MXF</a> <font color="Maroon">M1</font></span>)</span>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K8">Line</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:49_1_1"/><i><font color="Green">E62</font></i>: 
 <a href="matrix_1.html#K9">Col</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K1">MXR2MXF</a> <font color="Maroon">M2</font></span>)</span>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K9">Col</a> <font color="Maroon">M2</font>,<font color="Maroon">j</font>
 
<b>by </b><i><a class="txt" href="matrprob.html#E1:49_1"><i><font color="Green">E50</font></i></a>, , </i>;<br/>
<i><font color="Green">E63</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">M1</font>,<font color="Maroon">i</font></span>)</span> = 
 <a href="matrix_1.html#K1">width</a> <font color="Maroon">M1</font>
<b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>
<br/>.= 
 <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">M2</font>,<font color="Maroon">j</font></span>)</span>
<b>by </b><i><a class="txt" href="matrprob.html#E1:49"><i><font color="Green">E43</font></i></a>, <a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>
;<br/>
<a NAME="E7:49_1_1"/>
<font color="Maroon">M</font> <a href="hidden.html#R1">=</a>  <a href="matrixr1.html#K2">MXF2MXR</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixr1.html#K1">MXR2MXF</a> <font color="Maroon">M1</font></span>)</span> <a href="matrix_3.html#K4">*</a> <span class="p2">(<span class="default"><a href="matrixr1.html#K1">MXR2MXF</a> <font color="Maroon">M2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="matrprob.html#T1" target="_self">Th1</a>, <a class="ref" href="matrixr1.html#D6">MATRIXR1:def 6</a></i>;<br/>
<a NAME="E8:49_1_1"/><b>then </b>
<font color="Maroon">M</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K1">MXR2MXF</a> <font color="Maroon">M1</font></span>)</span> <a href="matrix_3.html#K4">*</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K1">MXR2MXF</a> <font color="Maroon">M2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="matrixr1.html#D2">MATRIXR1:def 2</a></i>;<br/>
<a NAME="E9:49_1_1"/><b>then </b>
( <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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixr1.html#K1">MXR2MXF</a> <font color="Maroon">M1</font></span>)</span> <a href="matrix_3.html#K4">*</a> <span class="p2">(<span class="default"><a href="matrixr1.html#K1">MXR2MXF</a> <font color="Maroon">M2</font></span>)</span></span>)</span> &amp; <font color="Maroon">M</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixr1.html#K1">MXR2MXF</a> <font color="Maroon">M1</font></span>)</span> <a href="matrix_3.html#K4">*</a> <span class="p2">(<span class="default"><a href="matrixr1.html#K1">MXR2MXF</a> <font color="Maroon">M2</font></span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> )
 
<b>by </b><i><a class="txt" href="matrprob.html#E4:49"><i><font color="Green">E49</font></i></a>, <a class="ref" href="matrixr1.html#T23">MATRIXR1:23</a></i>;<br/>
<b>hence </b><font color="Maroon">M</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> <span class="p2">(<span class="default"><a href="matrixr1.html#K1">MXR2MXF</a> <font color="Maroon">M1</font></span>)</span>,<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"><a href="matrixr1.html#K1">MXR2MXF</a> <font color="Maroon">M2</font></span>)</span>,<font color="Maroon">j</font></span>)</span>
<b>by </b><i>, <a class="txt" href="matrprob.html#E1:49"><i><font color="Green">E43</font></i></a>, <a class="ref" href="matrix_3.html#D4">MATRIX_3:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font></span>)</span> <a href="matrprob.html#NK7" target="_self">"*"</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M2</font>,<font color="Maroon">j</font></span>)</span>
<b>by </b><i>, <a class="txt" href="matrprob.html#E2:49_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="matrprob.html#T2" target="_self">Th2</a>, </i>
;<br/><br/>


</div><b>end;</b></div></div>
<b>end;</b></div>

<b>assume </b><a NAME="E6:49"/><i><font color="Green">E77</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">M</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">M1</font> &amp;  <a href="matrix_1.html#K1">width</a> <font color="Maroon">M</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">M2</font> &amp; ( for <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">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">M</font> holds <br/><font color="Maroon">M</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M1</font>,<font color="Olive">i</font></span>)</span> <a href="matrprob.html#NK7" target="_self">"*"</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M2</font>,<font color="Olive">j</font></span>)</span> ) )
 ;<br/>

<b>set </b><font color="Maroon">MM</font> =  <a href="matrixr1.html#K1">MXR2MXF</a> <font color="Maroon">M</font>;<br/>
<a NAME="E7:49"/><i><font color="Green">E117</font></i>: 
 <a href="matrixr1.html#K1">MXR2MXF</a> <font color="Maroon">M</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="matrixr1.html#D1">MATRIXR1:def 1</a></i>;<br/>
<a NAME="E8:49"/>
 <a href="matrixr1.html#K1">MXR2MXF</a> <font color="Maroon">M</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K1">MXR2MXF</a> <font color="Maroon">M1</font></span>)</span> <a href="matrix_3.html#K4">*</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K1">MXR2MXF</a> <font color="Maroon">M2</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_2"><b>proof </b></a><div class="add">

<i><font color="Green">E118</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K1">MXR2MXF</a> <font color="Maroon">M</font></span>)</span> = 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">M</font>
<b>by </b><i><a class="ref" href="matrixr1.html#D1">MATRIXR1:def 1</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K1">MXR2MXF</a> <font color="Maroon">M1</font></span>)</span>
<b>by </b><i><a class="ref" href="matrprob.html#T1" target="_self">Th1</a>, <a class="ref" href="matrixr1.html#D1">MATRIXR1:def 1</a></i>
;<br/>
<i><font color="Green">E119</font></i>:  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K1">MXR2MXF</a> <font color="Maroon">M</font></span>)</span> = 
 <a href="matrix_1.html#K1">width</a> <font color="Maroon">M</font>
<b>by </b><i><a class="ref" href="matrixr1.html#D1">MATRIXR1:def 1</a></i>
<br/>.= 
 <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K1">MXR2MXF</a> <font color="Maroon">M2</font></span>)</span>
<b>by </b><i><a class="ref" href="matrprob.html#T1" target="_self">Th1</a>, <a class="ref" href="matrixr1.html#D1">MATRIXR1:def 1</a></i>
;<br/>
<a NAME="E3:49_2"/>
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> <span class="p1">(<span class="default"><a href="matrixr1.html#K1">MXR2MXF</a> <font color="Maroon">M</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="matrixr1.html#K1">MXR2MXF</a> <font color="Maroon">M</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><a href="matrixr1.html#K1">MXR2MXF</a> <font color="Maroon">M1</font></span>)</span>,<font color="Olive">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"><a href="matrixr1.html#K1">MXR2MXF</a> <font color="Maroon">M2</font></span>)</span>,<font color="Olive">j</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_2_3"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:49_2_3"/><i><font color="Green">E120</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> <span class="p1">(<span class="default"><a href="matrixr1.html#K1">MXR2MXF</a> <font color="Maroon">M</font></span>)</span>
 ;<br/>

<a NAME="E2:49_2_3"/><i><font color="Green">E121</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  )
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E3:49_2_3"/><b>then </b><i><font color="Green">E122</font></i>: 
( <font color="Maroon">i</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="matrixr1.html#K1">MXR2MXF</a> <font color="Maroon">M1</font></span>)</span></span>)</span> &amp; <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> <span class="p2">(<span class="default"><a href="matrixr1.html#K1">MXR2MXF</a> <font color="Maroon">M2</font></span>)</span></span>)</span> )
 
<b>by </b><i>, , <a class="txt" href="matrprob.html#E2:49"><i><font color="Green">E48</font></i></a>, <a class="txt" href="matrprob.html#E4:49"><i><font color="Green">E49</font></i></a></i>;<br/>
<a NAME="E4:49_2_3"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K1">MXR2MXF</a> <font color="Maroon">M1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E5:49_2_3"/><b>then </b><i><font color="Green">E123</font></i>: 
 <a href="matrix_1.html#K8">Line</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K1">MXR2MXF</a> <font color="Maroon">M1</font></span>)</span>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K8">Line</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:49_2_3"/><i><font color="Green">E124</font></i>: 
 <a href="matrix_1.html#K9">Col</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K1">MXR2MXF</a> <font color="Maroon">M2</font></span>)</span>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K9">Col</a> <font color="Maroon">M2</font>,<font color="Maroon">j</font>
 
<b>by </b><i><a class="ref" href="matrprob.html#T2" target="_self">Th2</a>, , </i>;<br/>
<i><font color="Green">E125</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">M1</font>,<font color="Maroon">i</font></span>)</span> = 
 <a href="matrix_1.html#K1">width</a> <font color="Maroon">M1</font>
<b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>
<br/>.= 
 <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">M2</font>,<font color="Maroon">j</font></span>)</span>
<b>by </b><i><a class="txt" href="matrprob.html#E1:49"><i><font color="Green">E43</font></i></a>, <a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>
;<br/>
<a NAME="E8:49_2_3"/>
( <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">M</font> &amp; <span class="p1">(<span class="default"><a href="matrixr1.html#K1">MXR2MXF</a> <font color="Maroon">M</font></span>)</span> <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">M</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> )
 
<b>by </b><i>, , <a class="ref" href="matrixr1.html#T23">MATRIXR1:23</a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><a href="matrixr1.html#K1">MXR2MXF</a> <font color="Maroon">M</font></span>)</span> <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">M1</font>,<font color="Maroon">i</font></span>)</span> <a href="matrprob.html#NK7" target="_self">"*"</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M2</font>,<font color="Maroon">j</font></span>)</span>
<b>by </b><i><a class="ref" href="matrprob.html#T1" target="_self">Th1</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><a href="matrixr1.html#K1">MXR2MXF</a> <font color="Maroon">M1</font></span>)</span>,<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"><a href="matrixr1.html#K1">MXR2MXF</a> <font color="Maroon">M2</font></span>)</span>,<font color="Maroon">j</font></span>)</span>
<b>by </b><i>, , , </i>
;<br/><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:49_2"/>
 <a href="matrixr1.html#K1">MXR2MXF</a> <font color="Maroon">M</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K1">MXR2MXF</a> <font color="Maroon">M1</font></span>)</span> <a href="matrix_3.html#K4">*</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K1">MXR2MXF</a> <font color="Maroon">M2</font></span>)</span>
 <b>by </b><i><a class="txt" href="matrprob.html#E2:49"><i><font color="Green">E48</font></i></a>, <a class="txt" href="matrprob.html#E4:49"><i><font color="Green">E49</font></i></a>, , <a class="txt" href="matrprob.html#E1:49"><i><font color="Green">E43</font></i></a>, <a class="ref" href="matrix_3.html#D4">MATRIX_3:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:49"/><b>then </b>
<font color="Maroon">M</font> <a href="hidden.html#R1">=</a>  <a href="matrixr1.html#K2">MXF2MXR</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixr1.html#K1">MXR2MXF</a> <font color="Maroon">M1</font></span>)</span> <a href="matrix_3.html#K4">*</a> <span class="p2">(<span class="default"><a href="matrixr1.html#K1">MXR2MXF</a> <font color="Maroon">M2</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="matrixr1.html#D2">MATRIXR1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E10:49"/>
<font color="Maroon">M</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M1</font> <a href="matrixr1.html#K6">*</a> <font color="Maroon">M2</font>
 <b>by </b><i><a class="ref" href="matrixr1.html#D6">MATRIXR1:def 6</a></i>;<br/>


</div>
