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<b>let </b><font color="Maroon">i</font> be   <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">j</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">M1</font> be   <a href="matrix_1.html#NM1">Matrix</a> of <a href="numbers.html#K2">COMPLEX</a> , <font color="Maroon">M2</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">E35</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">M1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">M2</font> &amp;  <a href="matrix_1.html#K1">width</a> <font color="Maroon">M1</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">M2</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</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> )
 ;<br/>

<a NAME="E2:15"/><i><font color="Green">E36</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">M2</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">M2</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:15"/><i><font color="Green">E37</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</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">M2</font>
 
<b>by </b><i>, </i>;<br/>
<i><font color="Green">E38</font></i>:  <a href="matrix_5.html#K1">COMPLEX2Field</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_5.html#K5">-</a> <font color="Maroon">M2</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">M1</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">M2</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">M1</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">M2</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">E40</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</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> <span class="p1">(<span class="default"><a href="matrix_5.html#K1">COMPLEX2Field</a> <font color="Maroon">M1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="matrix_5.html#D1">MATRIX_5:def 1</a>, </i>;<br/>
<a NAME="E6:15"/><i><font color="Green">E41</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</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> <span class="p1">(<span class="default"><a href="matrix_5.html#K1">COMPLEX2Field</a> <font color="Maroon">M2</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="matrix_5.html#D1">MATRIX_5:def 1</a></i>;<br/>
<b>set </b><font color="Maroon">m</font> =  <a href="matrix_5.html#K1">COMPLEX2Field</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_5.html#K5">-</a> <font color="Maroon">M2</font></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon">m'</font> =  <a href="matrix_5.html#K1">COMPLEX2Field</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_5.html#K5">-</a> <font color="Maroon">M2</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">m1</font> =  <a href="matrix_5.html#K1">COMPLEX2Field</a> <font color="Maroon">M1</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">m2</font> =  <a href="matrix_5.html#K1">COMPLEX2Field</a> <font color="Maroon">M2</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">E42</font></i>: <font color="Maroon">M1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> = 
<font color="Maroon">m1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</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">M1</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>
<b>by </b><i><a class="ref" href="complfld.html#D1">COMPLFLD:def 1</a></i>
;<br/>
<i><font color="Green">E43</font></i>: <font color="Maroon">M2</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> = 
<font color="Maroon">m2</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</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">M2</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</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">E53</font></i>: 
 <a href="binop_2.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</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">M2</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span>
 
<b>by </b><i><a class="ref" href="complfld.html#T4">COMPLFLD:4</a>, </i>;<br/>
<i><font color="Green">E54</font></i>: <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span> <a href="binop_2.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</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">M2</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</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">M1</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</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">M2</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="complfld.html#T3">COMPLFLD:3</a></i>
;<br/>
<span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_5.html#K5">-</a> <font color="Maroon">M2</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> = 
<font color="Maroon">m'</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</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">M1</font> <a href="matrix_5.html#K5">-</a> <font color="Maroon">M2</font></span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</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">M1</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">M2</font></span>)</span></span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>
<b>by </b><i><a class="ref" href="matrix_4.html#D1">MATRIX_4: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">M1</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</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">M2</font></span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span>
<b>by </b><i>, <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">M1</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</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">M2</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="matrix_3.html#D2">MATRIX_3:def 2</a>, </i>
;<br/>
<b>hence </b><a NAME="E15:15"/>
<span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_5.html#K5">-</a> <font color="Maroon">M2</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span> <a href="binop_2.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span>
 <b>by </b><i/>;<br/>


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