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

<b>let </b><font color="Maroon" title="c1">M</font>, <font color="Maroon" title="c2">M1</font>, <font color="Maroon" title="c3">M2</font> be   <a href="matrix_1.html#NM1" title="MATRIX_1:NM.1">Matrix</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>

<b>assume </b><a NAME="E1:49"/><i><font color="Green" title="E43">A1</font></i>: 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">M1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">M2</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c4">M3</font> =  <a href="matrixr1.html#K1" title="MATRIXR1:func.1">MXR2MXF</a> <font color="Maroon" title="c2">M1</font>;<br/>
<b>set </b><font color="Maroon" title="c5">M4</font> =  <a href="matrixr1.html#K1" title="MATRIXR1:func.1">MXR2MXF</a> <font color="Maroon" title="c3">M2</font>;<br/>
<a NAME="E2:49"/><i><font color="Green" title="E44">A2</font></i>: 
(  <a href="matrixr1.html#K1" title="MATRIXR1:func.1">MXR2MXF</a> <font color="Maroon" title="c2">M1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">M1</font> &amp;  <a href="matrixr1.html#K1" title="MATRIXR1:func.1">MXR2MXF</a> <font color="Maroon" title="c3">M2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M2</font> )
 
<b>by </b><i><a class="ref" href="matrixr1.html#D1" title="MATRIXR1:def.1">MATRIXR1:def 1</a></i>;<br/>
<a NAME="E3:49"/>
 <a href="matrixr1.html#K2" title="MATRIXR1:func.2">MXF2MXR</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixr1.html#K1" title="MATRIXR1:func.1">MXR2MXF</a> <font color="Maroon" title="c2">M1</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p2">(<span class="default"><a href="matrixr1.html#K1" title="MATRIXR1:func.1">MXR2MXF</a> <font color="Maroon" title="c3">M2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">M1</font> <a href="matrixr1.html#K6" title="MATRIXR1:func.6">*</a> <font color="Maroon" title="c3">M2</font>
 
<b>by </b><i><a class="ref" href="matrixr1.html#D6" title="MATRIXR1:def.6">MATRIXR1:def 6</a></i>;<br/>
<a NAME="E4:49"/><b>then </b><i><font color="Green" title="E45">A3</font></i>: 
<span class="p1">(<span class="default"><a href="matrixr1.html#K1" title="MATRIXR1:func.1">MXR2MXF</a> <font color="Maroon" title="c2">M1</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K1" title="MATRIXR1:func.1">MXR2MXF</a> <font color="Maroon" title="c3">M2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">M1</font> <a href="matrixr1.html#K6" title="MATRIXR1:func.6">*</a> <font color="Maroon" title="c3">M2</font>
 
<b>by </b><i><a class="ref" href="matrixr1.html#D2" title="MATRIXR1:def.2">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" title="E46">A4</font></i>: 
<font color="Maroon" title="c1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">M1</font> <a href="matrixr1.html#K6" title="MATRIXR1:func.6">*</a> <font color="Maroon" title="c3">M2</font>
 ;<br/><b>hence </b><a NAME="E2:49_1"/><i><font color="Green" title="E47">A5</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">M1</font>
 <b>by </b><i><a class="txt" href="matrprob.html#E4:49"><i><font color="Green" title="E45">A3</font></i></a>, <a class="txt" href="matrprob.html#E2:49"><i><font color="Green" title="E44">A2</font></i></a>, <a class="txt" href="matrprob.html#E1:49"><i><font color="Green" title="E43">A1</font></i></a>, <a class="ref" href="matrix_3.html#D4" title="MATRIX_3:def.4">MATRIX_3:def 4</a></i>;<br/><b>thus </b><a NAME="E3:49_1"/><i><font color="Green" title="E48">A6</font></i>: 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">M2</font>
 <b>by </b><i><a class="txt" href="matrprob.html#E1:49_1"><i><font color="Green" title="E46">A4</font></i></a>, <a class="txt" href="matrprob.html#E4:49"><i><font color="Green" title="E45">A3</font></i></a>, <a class="txt" href="matrprob.html#E2:49"><i><font color="Green" title="E44">A2</font></i></a>, <a class="txt" href="matrprob.html#E1:49"><i><font color="Green" title="E43">A1</font></i></a>, <a class="ref" href="matrix_3.html#D4" title="MATRIX_3:def.4">MATRIX_3:def 4</a></i>;<br/><b>thus </b><a NAME="E4:49_1"/>
for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c1">M</font> holds <br/><font color="Maroon" title="c1">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c2">M1</font>,<font color="Olive" title="b1">i</font></span>)</span> <a href="matrprob.html#NK7" title="MATRPROB:NK.7">"*"</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c3">M2</font>,<font color="Olive" title="b2">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" title="c6">i</font>, <font color="Maroon" title="c7">j</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>



<b>assume </b><a NAME="E1:49_1_1"/><i><font color="Green" title="E49">A7</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c1">M</font>
 ;<br/>

<a NAME="E2:49_1_1"/><i><font color="Green" title="E50">A8</font></i>: 
( <font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">M1</font></span>)</span> &amp; <font color="Maroon" title="c7">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">M2</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="matrprob.html#E1:49_1_1"><i><font color="Green" title="E49">A7</font></i></a>, <a class="ref" href="matrprob.html#T12" target="_self" title="MATRPROB:th.12">Th12</a>, <a class="txt" href="matrprob.html#E2:49_1"><i><font color="Green" title="E47">A5</font></i></a>, <a class="txt" href="matrprob.html#E3:49_1"><i><font color="Green" title="E48">A6</font></i></a></i>;<br/>
<a NAME="E3:49_1_1"/><b>then </b>
<font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">M1</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E4:49_1_1"/><b>then </b><i><font color="Green" title="E51">A9</font></i>: 
 <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K1" title="MATRIXR1:func.1">MXR2MXF</a> <font color="Maroon" title="c2">M1</font></span>)</span>,<font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c2">M1</font>,<font color="Maroon" title="c6">i</font>
 
<b>by </b><i><a class="ref" href="matrprob.html#T16" target="_self" title="MATRPROB:th.16">Th16</a>, <a class="txt" href="matrprob.html#E2:49"><i><font color="Green" title="E44">A2</font></i></a></i>;<br/>
<a NAME="E5:49_1_1"/><i><font color="Green" title="E52">A10</font></i>: 
 <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K1" title="MATRIXR1:func.1">MXR2MXF</a> <font color="Maroon" title="c3">M2</font></span>)</span>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c3">M2</font>,<font color="Maroon" title="c7">j</font>
 
<b>by </b><i><a class="txt" href="matrprob.html#E2:49_1_1"><i><font color="Green" title="E50">A8</font></i></a>, <a class="ref" href="matrprob.html#T17" target="_self" title="MATRPROB:th.17">Th17</a>, <a class="txt" href="matrprob.html#E2:49"><i><font color="Green" title="E44">A2</font></i></a></i>;<br/>
<i><font color="Green" title="E53">A11</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c2">M1</font>,<font color="Maroon" title="c6">i</font></span>)</span> = 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">M1</font>
<b>by </b><i><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c3">M2</font>,<font color="Maroon" title="c7">j</font></span>)</span>
<b>by </b><i><a class="txt" href="matrprob.html#E1:49"><i><font color="Green" title="E43">A1</font></i></a>, <a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></i>
;<br/>
<a NAME="E7:49_1_1"/>
<font color="Maroon" title="c1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixr1.html#K2" title="MATRIXR1:func.2">MXF2MXR</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixr1.html#K1" title="MATRIXR1:func.1">MXR2MXF</a> <font color="Maroon" title="c2">M1</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p2">(<span class="default"><a href="matrixr1.html#K1" title="MATRIXR1:func.1">MXR2MXF</a> <font color="Maroon" title="c3">M2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="matrprob.html#E1:49_1"><i><font color="Green" title="E46">A4</font></i></a>, <a class="ref" href="matrixr1.html#D6" title="MATRIXR1:def.6">MATRIXR1:def 6</a></i>;<br/>
<a NAME="E8:49_1_1"/><b>then </b>
<font color="Maroon" title="c1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K1" title="MATRIXR1:func.1">MXR2MXF</a> <font color="Maroon" title="c2">M1</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K1" title="MATRIXR1:func.1">MXR2MXF</a> <font color="Maroon" title="c3">M2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="matrixr1.html#D2" title="MATRIXR1:def.2">MATRIXR1:def 2</a></i>;<br/>
<a NAME="E9:49_1_1"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixr1.html#K1" title="MATRIXR1:func.1">MXR2MXF</a> <font color="Maroon" title="c2">M1</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p2">(<span class="default"><a href="matrixr1.html#K1" title="MATRIXR1:func.1">MXR2MXF</a> <font color="Maroon" title="c3">M2</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c1">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixr1.html#K1" title="MATRIXR1:func.1">MXR2MXF</a> <font color="Maroon" title="c2">M1</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p2">(<span class="default"><a href="matrixr1.html#K1" title="MATRIXR1:func.1">MXR2MXF</a> <font color="Maroon" title="c3">M2</font></span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> )
 
<b>by </b><i><a class="txt" href="matrprob.html#E1:49_1_1"><i><font color="Green" title="E49">A7</font></i></a>, <a class="ref" href="matrixr1.html#T23" title="MATRIXR1:th.23">MATRIXR1:23</a></i>;<br/>
<b>hence </b><font color="Maroon" title="c1">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> = 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="matrixr1.html#K1" title="MATRIXR1:func.1">MXR2MXF</a> <font color="Maroon" title="c2">M1</font></span>)</span>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="fvsum_1.html#K13" title="FVSUM_1:func.13">"*"</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p2">(<span class="default"><a href="matrixr1.html#K1" title="MATRIXR1:func.1">MXR2MXF</a> <font color="Maroon" title="c3">M2</font></span>)</span>,<font color="Maroon" title="c7">j</font></span>)</span>
<b>by </b><i><a class="txt" href="matrprob.html#E2:49"><i><font color="Green" title="E44">A2</font></i></a>, <a class="txt" href="matrprob.html#E1:49"><i><font color="Green" title="E43">A1</font></i></a>, <a class="ref" href="matrix_3.html#D4" title="MATRIX_3:def.4">MATRIX_3:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c2">M1</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="matrprob.html#NK7" title="MATRPROB:NK.7">"*"</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c3">M2</font>,<font color="Maroon" title="c7">j</font></span>)</span>
<b>by </b><i><a class="txt" href="matrprob.html#E6:49_1_1"><i><font color="Green" title="E53">A11</font></i></a>, <a class="txt" href="matrprob.html#E4:49_1_1"><i><font color="Green" title="E51">A9</font></i></a>, <a class="txt" href="matrprob.html#E5:49_1_1"><i><font color="Green" title="E52">A10</font></i></a>, <a class="ref" href="matrprob.html#T38" target="_self" title="MATRPROB:th.38">Th38</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" title="E46">A12</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">M1</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">M2</font> &amp; ( for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c1">M</font> holds <br/><font color="Maroon" title="c1">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c2">M1</font>,<font color="Olive" title="b1">i</font></span>)</span> <a href="matrprob.html#NK7" title="MATRPROB:NK.7">"*"</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c3">M2</font>,<font color="Olive" title="b2">j</font></span>)</span> ) )
 ;<br/>

<b>set </b><font color="Maroon" title="c6">MM</font> =  <a href="matrixr1.html#K1" title="MATRIXR1:func.1">MXR2MXF</a> <font color="Maroon" title="c1">M</font>;<br/>
<a NAME="E7:49"/><i><font color="Green" title="E47">A13</font></i>: 
 <a href="matrixr1.html#K1" title="MATRIXR1:func.1">MXR2MXF</a> <font color="Maroon" title="c1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">M</font>
 
<b>by </b><i><a class="ref" href="matrixr1.html#D1" title="MATRIXR1:def.1">MATRIXR1:def 1</a></i>;<br/>
<a NAME="E8:49"/>
 <a href="matrixr1.html#K1" title="MATRIXR1:func.1">MXR2MXF</a> <font color="Maroon" title="c1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K1" title="MATRIXR1:func.1">MXR2MXF</a> <font color="Maroon" title="c2">M1</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K1" title="MATRIXR1:func.1">MXR2MXF</a> <font color="Maroon" title="c3">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" title="E48">A14</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K1" title="MATRIXR1:func.1">MXR2MXF</a> <font color="Maroon" title="c1">M</font></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">M</font>
<b>by </b><i><a class="ref" href="matrixr1.html#D1" title="MATRIXR1:def.1">MATRIXR1:def 1</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K1" title="MATRIXR1:func.1">MXR2MXF</a> <font color="Maroon" title="c2">M1</font></span>)</span>
<b>by </b><i><a class="txt" href="matrprob.html#E6:49"><i><font color="Green" title="E46">A12</font></i></a>, <a class="ref" href="matrixr1.html#D1" title="MATRIXR1:def.1">MATRIXR1:def 1</a></i>
;<br/>
<i><font color="Green" title="E49">A15</font></i>:  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K1" title="MATRIXR1:func.1">MXR2MXF</a> <font color="Maroon" title="c1">M</font></span>)</span> = 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c1">M</font>
<b>by </b><i><a class="ref" href="matrixr1.html#D1" title="MATRIXR1:def.1">MATRIXR1:def 1</a></i>
<br/>.= 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K1" title="MATRIXR1:func.1">MXR2MXF</a> <font color="Maroon" title="c3">M2</font></span>)</span>
<b>by </b><i><a class="txt" href="matrprob.html#E6:49"><i><font color="Green" title="E46">A12</font></i></a>, <a class="ref" href="matrixr1.html#D1" title="MATRIXR1:def.1">MATRIXR1:def 1</a></i>
;<br/>
<a NAME="E3:49_2"/>
for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K1" title="MATRIXR1:func.1">MXR2MXF</a> <font color="Maroon" title="c1">M</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="matrixr1.html#K1" title="MATRIXR1:func.1">MXR2MXF</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="matrixr1.html#K1" title="MATRIXR1:func.1">MXR2MXF</a> <font color="Maroon" title="c2">M1</font></span>)</span>,<font color="Olive" title="b1">i</font></span>)</span> <a href="fvsum_1.html#K13" title="FVSUM_1:func.13">"*"</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p2">(<span class="default"><a href="matrixr1.html#K1" title="MATRIXR1:func.1">MXR2MXF</a> <font color="Maroon" title="c3">M2</font></span>)</span>,<font color="Olive" title="b2">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" title="c7">i</font>, <font color="Maroon" title="c8">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:49_2_3"/><i><font color="Green" title="E50">A16</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K1" title="MATRIXR1:func.1">MXR2MXF</a> <font color="Maroon" title="c1">M</font></span>)</span>
 ;<br/>

<a NAME="E2:49_2_3"/><i><font color="Green" title="E51">A17</font></i>: 
( <font color="Maroon" title="c7">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <font color="Maroon" title="c8">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E3:49_2_3"/><b>then </b><i><font color="Green" title="E52">A18</font></i>: 
( <font color="Maroon" title="c7">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="matrixr1.html#K1" title="MATRIXR1:func.1">MXR2MXF</a> <font color="Maroon" title="c2">M1</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c8">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p2">(<span class="default"><a href="matrixr1.html#K1" title="MATRIXR1:func.1">MXR2MXF</a> <font color="Maroon" title="c3">M2</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="matrprob.html#E1:49_2_3"><i><font color="Green" title="E50">A16</font></i></a>, <a class="ref" href="matrprob.html#T12" target="_self" title="MATRPROB:th.12">Th12</a>, <a class="txt" href="matrprob.html#E1:49_2"><i><font color="Green" title="E48">A14</font></i></a>, <a class="txt" href="matrprob.html#E2:49_2"><i><font color="Green" title="E49">A15</font></i></a></i>;<br/>
<a NAME="E4:49_2_3"/><b>then </b>
<font color="Maroon" title="c7">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K1" title="MATRIXR1:func.1">MXR2MXF</a> <font color="Maroon" title="c2">M1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E5:49_2_3"/><b>then </b><i><font color="Green" title="E53">A19</font></i>: 
 <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K1" title="MATRIXR1:func.1">MXR2MXF</a> <font color="Maroon" title="c2">M1</font></span>)</span>,<font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c2">M1</font>,<font color="Maroon" title="c7">i</font>
 
<b>by </b><i><a class="ref" href="matrprob.html#T16" target="_self" title="MATRPROB:th.16">Th16</a>, <a class="txt" href="matrprob.html#E2:49"><i><font color="Green" title="E44">A2</font></i></a></i>;<br/>
<a NAME="E6:49_2_3"/><i><font color="Green" title="E54">A20</font></i>: 
 <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K1" title="MATRIXR1:func.1">MXR2MXF</a> <font color="Maroon" title="c3">M2</font></span>)</span>,<font color="Maroon" title="c8">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c3">M2</font>,<font color="Maroon" title="c8">j</font>
 
<b>by </b><i><a class="txt" href="matrprob.html#E3:49_2_3"><i><font color="Green" title="E52">A18</font></i></a>, <a class="ref" href="matrprob.html#T17" target="_self" title="MATRPROB:th.17">Th17</a>, <a class="txt" href="matrprob.html#E2:49"><i><font color="Green" title="E44">A2</font></i></a></i>;<br/>
<i><font color="Green" title="E55">A21</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c2">M1</font>,<font color="Maroon" title="c7">i</font></span>)</span> = 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">M1</font>
<b>by </b><i><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c3">M2</font>,<font color="Maroon" title="c8">j</font></span>)</span>
<b>by </b><i><a class="txt" href="matrprob.html#E1:49"><i><font color="Green" title="E43">A1</font></i></a>, <a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></i>
;<br/>
<a NAME="E8:49_2_3"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c1">M</font> &amp; <span class="p1">(<span class="default"><a href="matrixr1.html#K1" title="MATRIXR1:func.1">MXR2MXF</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</font> )
 
<b>by </b><i><a class="txt" href="matrprob.html#E7:49"><i><font color="Green" title="E47">A13</font></i></a>, <a class="txt" href="matrprob.html#E1:49_2_3"><i><font color="Green" title="E50">A16</font></i></a>, <a class="ref" href="matrixr1.html#T23" title="MATRIXR1:th.23">MATRIXR1:23</a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><a href="matrixr1.html#K1" title="MATRIXR1:func.1">MXR2MXF</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</font> = 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c2">M1</font>,<font color="Maroon" title="c7">i</font></span>)</span> <a href="matrprob.html#NK7" title="MATRPROB:NK.7">"*"</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c3">M2</font>,<font color="Maroon" title="c8">j</font></span>)</span>
<b>by </b><i><a class="txt" href="matrprob.html#E6:49"><i><font color="Green" title="E46">A12</font></i></a>, <a class="txt" href="matrprob.html#E2:49_2_3"><i><font color="Green" title="E51">A17</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="matrixr1.html#K1" title="MATRIXR1:func.1">MXR2MXF</a> <font color="Maroon" title="c2">M1</font></span>)</span>,<font color="Maroon" title="c7">i</font></span>)</span> <a href="fvsum_1.html#K13" title="FVSUM_1:func.13">"*"</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p2">(<span class="default"><a href="matrixr1.html#K1" title="MATRIXR1:func.1">MXR2MXF</a> <font color="Maroon" title="c3">M2</font></span>)</span>,<font color="Maroon" title="c8">j</font></span>)</span>
<b>by </b><i><a class="txt" href="matrprob.html#E5:49_2_3"><i><font color="Green" title="E53">A19</font></i></a>, <a class="txt" href="matrprob.html#E6:49_2_3"><i><font color="Green" title="E54">A20</font></i></a>, <a class="txt" href="matrprob.html#E7:49_2_3"><i><font color="Green" title="E55">A21</font></i></a>, <a class="ref" href="matrprob.html#T38" target="_self" title="MATRPROB:th.38">Th38</a></i>
;<br/><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:49_2"/>
 <a href="matrixr1.html#K1" title="MATRIXR1:func.1">MXR2MXF</a> <font color="Maroon" title="c1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K1" title="MATRIXR1:func.1">MXR2MXF</a> <font color="Maroon" title="c2">M1</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K1" title="MATRIXR1:func.1">MXR2MXF</a> <font color="Maroon" title="c3">M2</font></span>)</span>
 <b>by </b><i><a class="txt" href="matrprob.html#E1:49_2"><i><font color="Green" title="E48">A14</font></i></a>, <a class="txt" href="matrprob.html#E2:49_2"><i><font color="Green" title="E49">A15</font></i></a>, <a class="txt" href="matrprob.html#E2:49"><i><font color="Green" title="E44">A2</font></i></a>, <a class="txt" href="matrprob.html#E1:49"><i><font color="Green" title="E43">A1</font></i></a>, <a class="ref" href="matrix_3.html#D4" title="MATRIX_3:def.4">MATRIX_3:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:49"/><b>then </b>
<font color="Maroon" title="c1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixr1.html#K2" title="MATRIXR1:func.2">MXF2MXR</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixr1.html#K1" title="MATRIXR1:func.1">MXR2MXF</a> <font color="Maroon" title="c2">M1</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p2">(<span class="default"><a href="matrixr1.html#K1" title="MATRIXR1:func.1">MXR2MXF</a> <font color="Maroon" title="c3">M2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="matrprob.html#E7:49"><i><font color="Green" title="E47">A13</font></i></a>, <a class="ref" href="matrixr1.html#D2" title="MATRIXR1:def.2">MATRIXR1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E10:49"/>
<font color="Maroon" title="c1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">M1</font> <a href="matrixr1.html#K6" title="MATRIXR1:func.6">*</a> <font color="Maroon" title="c3">M2</font>
 <b>by </b><i><a class="ref" href="matrixr1.html#D6" title="MATRIXR1:def.6">MATRIXR1:def 6</a></i>;<br/>


</div>
