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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</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="E1:15"/><i><font color="Green">E14</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>4</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<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> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<a NAME="E2:15"/><i><font color="Green">E15</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="ref" href="matrixc1.html#T1" target="_self">Th1</a>, <a class="txt" href="matrixc1.html#E1:15"><i><font color="Green">E14</font></i></a></i>;<br/>
<a NAME="E3:15"/><i><font color="Green">E16</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<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> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="matrixc1.html#T1" target="_self">Th1</a>, <a class="txt" href="matrixc1.html#E2:15"><i><font color="Green">E15</font></i></a></i>;<br/>
<i><font color="Green">E17</font></i>:  <a href="matrix_5.html#K1">COMPLEX2Field</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrix_5.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> = 
 <a href="matrix_5.html#K1">COMPLEX2Field</a> <span class="p1">(<span class="default"><a href="matrix_5.html#K2">Field2COMPLEX</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_5.html#K1">COMPLEX2Field</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_4.html#K1">-</a> <span class="p3">(<span class="default"><a href="matrix_5.html#K1">COMPLEX2Field</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="matrix_5.html#D5">MATRIX_5:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix_5.html#K1">COMPLEX2Field</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_4.html#K1">-</a> <span class="p1">(<span class="default"><a href="matrix_5.html#K1">COMPLEX2Field</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_5.html#T6">MATRIX_5:6</a></i>
;<br/>
<a NAME="E5:15"/><i><font color="Green">E18</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<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"><a href="matrix_5.html#K1">COMPLEX2Field</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="matrix_5.html#D1">MATRIX_5:def 1</a>, <a class="txt" href="matrixc1.html#E1:15"><i><font color="Green">E14</font></i></a></i>;<br/>
<a NAME="E6:15"/><i><font color="Green">E19</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<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"><a href="matrix_5.html#K1">COMPLEX2Field</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E3:15"><i><font color="Green">E16</font></i></a>, <a class="ref" href="matrix_5.html#D1">MATRIX_5:def 1</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="matrix_5.html#K1">COMPLEX2Field</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrix_5.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> =  <a href="matrix_5.html#K1">COMPLEX2Field</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrix_5.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> as   <a href="matrix_1.html#NM1">Matrix</a> of <a href="numbers.html#K2">COMPLEX</a>  <b>by </b><i><a class="ref" href="complfld.html#D1">COMPLFLD:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> =  <a href="matrix_5.html#K1">COMPLEX2Field</a> <font color="Maroon">c<sub>3</sub></font> as   <a href="matrix_1.html#NM1">Matrix</a> of <a href="numbers.html#K2">COMPLEX</a>  <b>by </b><i><a class="ref" href="complfld.html#D1">COMPLFLD:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> =  <a href="matrix_5.html#K1">COMPLEX2Field</a> <font color="Maroon">c<sub>4</sub></font> as   <a href="matrix_1.html#NM1">Matrix</a> of <a href="numbers.html#K2">COMPLEX</a>  <b>by </b><i><a class="ref" href="complfld.html#D1">COMPLFLD:def 1</a></i>;<br/>
<i><font color="Green">E20</font></i>: <font color="Maroon">c<sub>3</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> = 
<font color="Maroon">c<sub>7</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="matrix_5.html#D1">MATRIX_5:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix_5.html#K1">COMPLEX2Field</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="complfld.html#D1">COMPLFLD:def 1</a></i>
;<br/>
<i><font color="Green">E21</font></i>: <font color="Maroon">c<sub>4</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> = 
<font color="Maroon">c<sub>8</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="matrix_5.html#D1">MATRIX_5:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix_5.html#K1">COMPLEX2Field</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="complfld.html#D1">COMPLFLD:def 1</a></i>
;<br/>
<a NAME="E12:15"/><i><font color="Green">E22</font></i>: 
 <a href="binop_2.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_5.html#K1">COMPLEX2Field</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="complfld.html#T4">COMPLFLD:4</a>, <a class="txt" href="matrixc1.html#E11:15"><i><font color="Green">E21</font></i></a></i>;<br/>
<i><font color="Green">E23</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_2.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_2.html#K3">+</a> <span class="p1">(<span class="default"><a href="binop_2.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_5.html#K1">COMPLEX2Field</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_5.html#K1">COMPLEX2Field</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="matrixc1.html#E10:15"><i><font color="Green">E20</font></i></a>, <a class="txt" href="matrixc1.html#E12:15"><i><font color="Green">E22</font></i></a>, <a class="ref" href="complfld.html#T3">COMPLFLD:3</a></i>
;<br/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrix_5.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> = 
<font color="Maroon">c<sub>6</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="matrix_5.html#D1">MATRIX_5:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix_5.html#K1">COMPLEX2Field</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrix_5.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="complfld.html#D1">COMPLFLD:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_5.html#K1">COMPLEX2Field</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_3.html#K3">+</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K2">-</a> <span class="p3">(<span class="default"><a href="matrix_5.html#K1">COMPLEX2Field</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="matrix_4.html#D1">MATRIX_4:def 1</a>, <a class="txt" href="matrixc1.html#E4:15"><i><font color="Green">E17</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_5.html#K1">COMPLEX2Field</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_3.html#K2">-</a> <span class="p3">(<span class="default"><a href="matrix_5.html#K1">COMPLEX2Field</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="matrixc1.html#E5:15"><i><font color="Green">E18</font></i></a>, <a class="ref" href="matrix_3.html#D3">MATRIX_3:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_5.html#K1">COMPLEX2Field</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_5.html#K1">COMPLEX2Field</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="matrix_3.html#D2">MATRIX_3:def 2</a>, <a class="txt" href="matrixc1.html#E6:15"><i><font color="Green">E19</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E15:15"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrix_5.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_2.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="matrixc1.html#E13:15"><i><font color="Green">E23</font></i></a></i>;<br/>


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