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

<b>let </b><font color="Maroon">D1</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">D2</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">M1</font> be   <a href="matrix_1.html#NM1">Matrix</a> of <font color="Maroon">D1</font>;<br/><b>let </b><font color="Maroon">M2</font> be   <a href="matrix_1.html#NM1">Matrix</a> of <font color="Maroon">D2</font>;<br/>





<b>assume </b><a NAME="E1:19"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">M1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M2</font>
 ;<br/>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:19_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">M1</font></span>)</span>
 ;<br/><i><font color="Green">E49</font></i>:  <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">M1</font>,<font color="Maroon">j</font></span>)</span> = 
 <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">M1</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <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>
<b>by </b><i><a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>
<br/>.= 
 <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">M2</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</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="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/><a NAME="E3:19_1"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <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">M1</font>,<font color="Maroon">j</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M1</font>,<font color="Maroon">j</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</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> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:19_1_2"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">k</font> <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">M1</font>,<font color="Maroon">j</font></span>)</span>
 ;<br/>

<a NAME="E2:19_1_2"/>
<font color="Maroon">k</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#K9">Col</a> <font color="Maroon">M1</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:19_1_2"/><b>then </b>
<font color="Maroon">k</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>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>;<br/>
<a NAME="E4:19_1_2"/><b>then </b><i><font color="Green">E51</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">k</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> &amp; <font color="Maroon">k</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/>
<b>hence </b><span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M1</font>,<font color="Maroon">j</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> = 
<font color="Maroon">M1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">j</font>
<b>by </b><i><a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>
<br/>.= 
<font color="Maroon">M2</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">j</font>
<b>by </b><i>, , <a class="ref" href="matrixr1.html#T23">MATRIXR1:23</a></i>
<br/>.= 
<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> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i>, , <a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>
;<br/><br/>


</div><b>end;</b></div><b>hence </b><a NAME="E4:19_1"/>
 <a href="matrix_1.html#K9">Col</a> <font color="Maroon">M1</font>,<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="finseq_1.html#T17">FINSEQ_1:17</a></i>;<br/></div>
<b>end;</b></div>


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