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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="matrix_1.html#NM1">Matrix</a> of <a href="numbers.html#K2">COMPLEX</a> ;<br/>

<b>assume </b><a NAME="E1:16"/><i><font color="Green">E14</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>

<a NAME="E2:16"/><i><font color="Green">E15</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="matrix_5.html#K5">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrixc1.html#K1" target="_self">*'</a> </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>1</sub></font> <a href="matrix_5.html#K5">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="matrixc1.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E3:16"/><i><font color="Green">E16</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="matrixc1.html#K1" target="_self">*'</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="matrixc1.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E4:16"/><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>2</sub></font> <a href="matrixc1.html#K1" target="_self">*'</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="matrixc1.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E5:16"/><i><font color="Green">E18</font></i>: 
 <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="matrixc1.html#K1" target="_self">*'</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="matrixc1.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E6:16"/><i><font color="Green">E19</font></i>: 
 <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrixc1.html#K1" target="_self">*'</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="matrixc1.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E7:16"/><i><font color="Green">E20</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="matrix_5.html#K5">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="matrixc1.html#T13" target="_self">Th13</a></i>;<br/>
<a NAME="E8:16"/><i><font color="Green">E21</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="matrixc1.html#K1" target="_self">*'</a> </span>)</span> <a href="matrix_5.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrixc1.html#K1" target="_self">*'</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>1</sub></font> <a href="matrixc1.html#K1" target="_self">*'</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="matrixc1.html#T13" target="_self">Th13</a></i>;<br/>
<a NAME="E9:16"/><i><font color="Green">E22</font></i>: 
 <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="matrix_5.html#K5">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrixc1.html#K1" target="_self">*'</a> </span>)</span> <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="matrix_5.html#K5">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="matrixc1.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E10:16"/><i><font color="Green">E23</font></i>: 
 <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="matrix_5.html#K5">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="matrixc1.html#T13" target="_self">Th13</a></i>;<br/>
<a NAME="E11:16"/><i><font color="Green">E24</font></i>: 
 <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="matrixc1.html#K1" target="_self">*'</a> </span>)</span> <a href="matrix_5.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrixc1.html#K1" target="_self">*'</a> </span>)</span></span>)</span> <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="matrixc1.html#K1" target="_self">*'</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="matrixc1.html#T13" target="_self">Th13</a></i>;<br/>
<a NAME="E12:16"/><i><font color="Green">E25</font></i>: 
 <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="matrixc1.html#K1" target="_self">*'</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="matrixc1.html#D1" target="_self">Def1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>now </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="E1:16_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"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="matrix_5.html#K5">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrixc1.html#K1" target="_self">*'</a> </span>)</span>
 ;<br/><a NAME="E2:16_1"/><b>then </b><i><font color="Green">E26</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="matrix_5.html#K5">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</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="matrix_5.html#K5">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="matrixc1.html#E2:16"><i><font color="Green">E15</font></i></a>, <a class="txt" href="matrixc1.html#E9:16"><i><font color="Green">E22</font></i></a>, <a class="ref" href="matrixc1.html#T1" target="_self">Th1</a></i>;<br/><a NAME="E3:16_1"/><b>then </b><i><font color="Green">E27</font></i>: 
<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="matrix_5.html#K5">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="matrixc1.html#T1" target="_self">Th1</a></i>;<br/><a NAME="E4:16_1"/><i><font color="Green">E28</font></i>: 
<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> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="matrixc1.html#T1" target="_self">Th1</a>, <a class="txt" href="matrixc1.html#E7:16"><i><font color="Green">E20</font></i></a>, <a class="txt" href="matrixc1.html#E10:16"><i><font color="Green">E23</font></i></a>, <a class="txt" href="matrixc1.html#E2:16_1"><i><font color="Green">E26</font></i></a></i>;<br/><a NAME="E5:16_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="matrixc1.html#K1" target="_self">*'</a> </span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</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="matrixc1.html#K1" target="_self">*'</a> </span>)</span> )
 <b>by </b><i><a class="ref" href="matrixc1.html#D1" target="_self">Def1</a>, <a class="txt" href="matrixc1.html#E7:16"><i><font color="Green">E20</font></i></a>, <a class="txt" href="matrixc1.html#E10:16"><i><font color="Green">E23</font></i></a>, <a class="txt" href="matrixc1.html#E2:16_1"><i><font color="Green">E26</font></i></a></i>;<br/><a NAME="E6:16_1"/><b>then </b><i><font color="Green">E29</font></i>: 
<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="matrixc1.html#K1" target="_self">*'</a> </span>)</span>
 <b>by </b><i><a class="ref" href="matrixc1.html#T1" target="_self">Th1</a></i>;<br/><a NAME="E7:16_1"/><i><font color="Green">E30</font></i>: 
<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> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="matrixc1.html#T1" target="_self">Th1</a>, <a class="txt" href="matrixc1.html#E7:16"><i><font color="Green">E20</font></i></a>, <a class="txt" href="matrixc1.html#E10:16"><i><font color="Green">E23</font></i></a>, <a class="txt" href="matrixc1.html#E2:16_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="matrixc1.html#E1:16"><i><font color="Green">E14</font></i></a></i>;<br/><a NAME="E8:16_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="matrix_5.html#K5">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrixc1.html#K1" target="_self">*'</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> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="matrix_5.html#K5">-</a> <font color="Maroon">c<sub>2</sub></font></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> <a href="complex1.html#K15">*'</a> 
 <b>by </b><i><a class="ref" href="matrixc1.html#D1" target="_self">Def1</a>, <a class="txt" href="matrixc1.html#E3:16_1"><i><font color="Green">E27</font></i></a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="matrix_5.html#K5">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrixc1.html#K1" target="_self">*'</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 class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <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> <a href="binop_2.html#K4">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <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></span>)</span> <a href="complex1.html#K15">*'</a> 
<b>by </b><i><a class="ref" href="matrixc1.html#T14" target="_self">Th14</a>, <a class="txt" href="matrixc1.html#E4:16_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="matrixc1.html#E1:16"><i><font color="Green">E14</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <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> <a href="complex1.html#K15">*'</a> </span>)</span> <a href="binop_2.html#K4">-</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#K3">*</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="complex1.html#K15">*'</a> </span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T120">COMPLEX1:120</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="matrixc1.html#K1" target="_self">*'</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> <a href="binop_2.html#K4">-</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#K3">*</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="complex1.html#K15">*'</a> </span>)</span>
<b>by </b><i><a class="ref" href="matrixc1.html#D1" target="_self">Def1</a>, <a class="txt" href="matrixc1.html#E4:16_1"><i><font color="Green">E28</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="matrixc1.html#K1" target="_self">*'</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> <a href="binop_2.html#K4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrixc1.html#K1" target="_self">*'</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="matrixc1.html#D1" target="_self">Def1</a>, <a class="txt" href="matrixc1.html#E7:16_1"><i><font color="Green">E30</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="matrixc1.html#K1" target="_self">*'</a> </span>)</span> <a href="matrix_5.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrixc1.html#K1" target="_self">*'</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>
<b>by </b><i><a class="ref" href="matrixc1.html#T14" target="_self">Th14</a>, <a class="txt" href="matrixc1.html#E6:16_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="matrixc1.html#E1:16"><i><font color="Green">E14</font></i></a>, <a class="txt" href="matrixc1.html#E3:16"><i><font color="Green">E16</font></i></a>, <a class="txt" href="matrixc1.html#E4:16"><i><font color="Green">E17</font></i></a>, <a class="txt" href="matrixc1.html#E5:16"><i><font color="Green">E18</font></i></a>, <a class="txt" href="matrixc1.html#E6:16"><i><font color="Green">E19</font></i></a></i>
;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E14:16"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="matrix_5.html#K5">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrixc1.html#K1" target="_self">*'</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="matrixc1.html#K1" target="_self">*'</a> </span>)</span> <a href="matrix_5.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrixc1.html#K1" target="_self">*'</a> </span>)</span>
 <b>by </b><i><a class="ref" href="matrix_1.html#T21">MATRIX_1:21</a>, <a class="txt" href="matrixc1.html#E2:16"><i><font color="Green">E15</font></i></a>, <a class="txt" href="matrixc1.html#E7:16"><i><font color="Green">E20</font></i></a>, <a class="txt" href="matrixc1.html#E8:16"><i><font color="Green">E21</font></i></a>, <a class="txt" href="matrixc1.html#E3:16"><i><font color="Green">E16</font></i></a>, <a class="txt" href="matrixc1.html#E9:16"><i><font color="Green">E22</font></i></a>, <a class="txt" href="matrixc1.html#E10:16"><i><font color="Green">E23</font></i></a>, <a class="txt" href="matrixc1.html#E11:16"><i><font color="Green">E24</font></i></a>, <a class="txt" href="matrixc1.html#E12:16"><i><font color="Green">E25</font></i></a></i>;<br/>


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