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<b>let </b><font color="Maroon">G1</font> be   <a href="matrix_1.html#NM4">Matrix</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>, <font color="Maroon">G2</font> be   <a href="matrix_1.html#NM4">Matrix</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:27_1_2"/><i><font color="Green">E64</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">v1</font> &amp;  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">v2</font> &amp; ( for <font color="Olive">n</font>, <font color="Olive">m</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">n</font>,<font color="Olive">m</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">G1</font> holds <br/><font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">n</font>,<font color="Olive">m</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">v1</font> <a href="goboard1.html#K1">.</a> <font color="Olive">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">v2</font> <a href="goboard1.html#K1">.</a> <font color="Olive">m</font></span>)</span></span><a href="euclid.html#K23">]|</a></span> ) )
 <b>and </b><br/><a NAME="E2:27_1_2"/><i><font color="Green">E65</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">v1</font> &amp;  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">v2</font> &amp; ( for <font color="Olive">n</font>, <font color="Olive">m</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">n</font>,<font color="Olive">m</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">G2</font> holds <br/><font color="Maroon">G2</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">n</font>,<font color="Olive">m</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">v1</font> <a href="goboard1.html#K1">.</a> <font color="Olive">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">v2</font> <a href="goboard1.html#K1">.</a> <font color="Olive">m</font></span>)</span></span><a href="euclid.html#K23">]|</a></span> ) )
 ;<br/>

<a NAME="E3:27_1_2"/><i><font color="Green">E66</font></i>: 
(  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">G1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">G1</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> &amp;  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">G2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">G2</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G2</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<b>by </b><i><a class="ref" href="matrix_1.html#D5">MATRIX_1:def 5</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">m</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:27_1_2_1"/><i><font color="Green">E67</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">n</font>,<font color="Maroon">m</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">G1</font>
 ;<br/><b>reconsider </b><font color="Maroon">m'</font> = <font color="Maroon">m</font>, <font color="Maroon">n'</font> = <font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of  <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="E3:27_1_2_1"/><i><font color="Green">E68</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G1</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">G1</font></span>)</span> &amp;  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G2</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">G2</font></span>)</span> )
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">r</font> = <font color="Maroon">v1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">n'</font>, <font color="Maroon">s</font> = <font color="Maroon">v2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">m'</font> as   <a href="real_1.html#NM1">Real</a> ;<br/><a NAME="E5:27_1_2_1"/>
( <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">n</font>,<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">s</font></span><a href="euclid.html#K23">]|</a></span> &amp; <font color="Maroon">G2</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">n</font>,<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">s</font></span><a href="euclid.html#K23">]|</a></span> )
 <b>by </b><i>, , , , </i>;<br/><b>hence </b><a NAME="E6:27_1_2_1"/>
<font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">n</font>,<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G2</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">n</font>,<font color="Maroon">m</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:27_1_2"/>
<font color="Maroon">G1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G2</font>
 <b>by </b><i>, , <a class="ref" href="matrix_1.html#T21">MATRIX_1:21</a></i>;<br/>


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