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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="matrix_1.html#NM1">Matrix</a> of <a href="numbers.html#K1">REAL</a> ;<br/>



<b>assume </b><a NAME="E1:36"/><i><font color="Green">E16</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 ;<br/>

<a NAME="E2:36"/><i><font color="Green">E17</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="matrixr1.html#K7" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K4">@</a> </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">c<sub>2</sub></font> <a href="matrix_1.html#K4">@</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="matrixr1.html#T27" target="_self">Th27</a></i>;<br/>
<a NAME="E3:36"/><i><font color="Green">E18</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K4">@</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="matrix_2.html#T12">MATRIX_2:12</a>, <a class="txt" href="matrixr1.html#E1:36"><i><font color="Green">E16</font></i></a></i>;<br/>
<a NAME="E4:36"/><i><font color="Green">E19</font></i>: 
 <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="matrixr1.html#K7" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="matrixr1.html#T27" target="_self">Th27</a></i>;<br/>
<a NAME="E5:36"/><i><font color="Green">E20</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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"><font color="Maroon">c<sub>1</sub></font> <a href="matrixr1.html#K7" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K4">@</a> </span>)</span></span>)</span> iff <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></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"><font color="Maroon">c<sub>1</sub></font> <a href="matrixr1.html#K7" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>



<a NAME="E1:36_1"/><i><font color="Green">E21</font></i>: 
 <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="matrixr1.html#K7" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K4">@</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K4">@</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="matrixr1.html#T28" target="_self">Th28</a></i>;<br/>
<a NAME="E2:36_1"/>
 <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="matrixr1.html#K7" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="matrixr1.html#T28" target="_self">Th28</a></i>;<br/>
<b>hence </b><a NAME="E3:36_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></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"><font color="Maroon">c<sub>1</sub></font> <a href="matrixr1.html#K7" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K4">@</a> </span>)</span></span>)</span> iff <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></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"><font color="Maroon">c<sub>1</sub></font> <a href="matrixr1.html#K7" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="matrixr1.html#E1:36_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="matrix_1.html#D7">MATRIX_1:def 7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:36"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></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">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></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"><font color="Maroon">c<sub>1</sub></font> <a href="matrixr1.html#K7" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="matrixr1.html#K7" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K4">@</a> </span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="matrixr1.html#K7" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>



<b>assume </b><a NAME="E1:36_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></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"><font color="Maroon">c<sub>1</sub></font> <a href="matrixr1.html#K7" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:36_2"/><b>then </b><i><font color="Green">E21</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></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">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="matrixr1.html#T28" target="_self">Th28</a></i>;<br/>
<a NAME="E3:36_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></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"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K4">@</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="matrixr1.html#E2:36_2"><i><font color="Green">E21</font></i></a>, <a class="ref" href="matrix_1.html#D7">MATRIX_1:def 7</a></i>;<br/>
<a NAME="E4:36_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="matrixr1.html#K7" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K4">@</a> </span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K4">@</a> </span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="matrixr1.html#T29" target="_self">Th29</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="matrixr1.html#K7" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K4">@</a> </span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> = 
<font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="matrixr1.html#E2:36_2"><i><font color="Green">E21</font></i></a>, <a class="ref" href="matrix_1.html#D7">MATRIX_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="matrixr1.html#K7" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="matrixr1.html#T29" target="_self">Th29</a>, <a class="txt" href="matrixr1.html#E2:36_2"><i><font color="Green">E21</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E6:36_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="matrixr1.html#K7" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K4">@</a> </span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="matrixr1.html#K7" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:36"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="matrixr1.html#K7" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K4">@</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="matrixr1.html#K7" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K4">@</a> </span>)</span>
 <b>by </b><i><a class="ref" href="matrix_1.html#D7">MATRIX_1:def 7</a>, <a class="txt" href="matrixr1.html#E2:36"><i><font color="Green">E17</font></i></a>, <a class="txt" href="matrixr1.html#E3:36"><i><font color="Green">E18</font></i></a>, <a class="txt" href="matrixr1.html#E4:36"><i><font color="Green">E19</font></i></a>, <a class="txt" href="matrixr1.html#E5:36"><i><font color="Green">E20</font></i></a></i>;<br/>


</div>
