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<div class="add">

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

<b>assume </b><a NAME="E1:71"/><i><font color="Green">E35</font></i>: 
 <a href="matrix_1.html#K1">width</a> <font color="Maroon">M</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<a NAME="E2:71"/><i><font color="Green">E36</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">M</font> <a href="matrix_1.html#K4">@</a> </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">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E3:71"/><i><font color="Green">E37</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">M</font> <a href="matrix_1.html#K4">@</a> </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">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span>
 
<b>by </b><i/>;<br/>
<i><font color="Green">E38</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">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span> <a href="matrixc1.html#K1" target="_self">*'</a> </span>)</span> = 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="matrix_1.html#K1">width</a> <font color="Maroon">M</font>
<b>by </b><i><a class="ref" href="matrix_1.html#D7">MATRIX_1:def 7</a></i>
;<br/>
<i><font color="Green">E40</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">M</font> <a href="matrixc1.html#K1" target="_self">*'</a> </span>)</span> <a href="matrix_1.html#K4">@</a> </span>)</span> = 
 <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="matrixc1.html#K1" target="_self">*'</a> </span>)</span>
<b>by </b><i><a class="ref" href="matrix_1.html#D7">MATRIX_1:def 7</a></i>
<br/>.= 
 <a href="matrix_1.html#K1">width</a> <font color="Maroon">M</font>
<b>by </b><i/>
;<br/>
<i><font color="Green">E41</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">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span> <a href="matrixc1.html#K1" target="_self">*'</a> </span>)</span> = 
 <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">M</font>
<b>by </b><i><a class="ref" href="matrix_2.html#T12">MATRIX_2:12</a>, </i>
;<br/>
<a NAME="E7:71"/><i><font color="Green">E42</font></i>: 
 <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><font color="Maroon">M</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">M</font>
 
<b>by </b><i/>;<br/>
<i><font color="Green">E43</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">M</font> <a href="matrixc1.html#K1" target="_self">*'</a> </span>)</span> <a href="matrix_1.html#K4">@</a> </span>)</span> = 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="matrixc1.html#K1" target="_self">*'</a> </span>)</span>
<b>by </b><i><a class="ref" href="matrix_2.html#T12">MATRIX_2:12</a>, , </i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">M</font>
<b>by </b><i/>
;<br/>
<a NAME="E9:71"/>
for <font color="Olive">i</font>, <font color="Olive">j</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">i</font>,<font color="Olive">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"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span> <a href="matrixc1.html#K1" target="_self">*'</a> </span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span> <a href="matrixc1.html#K1" target="_self">*'</a> </span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="matrixc1.html#K1" target="_self">*'</a> </span>)</span> <a href="matrix_1.html#K4">@</a> </span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">j</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>



<b>assume </b><a NAME="E1:71_5"/><i><font color="Green">E53</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"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span> <a href="matrixc1.html#K1" target="_self">*'</a> </span>)</span>
 ;<br/>

<a NAME="E2:71_5"/><i><font color="Green">E54</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> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span> &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> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span> )
 
<b>by </b><i><a class="ref" href="matrixc1.html#T6" target="_self">Th6</a>, <a class="ref" href="matrixc1.html#T7" target="_self">Th7</a>, , </i>;<br/>
<a NAME="E3:71_5"/><i><font color="Green">E55</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span> <a href="matrixc1.html#K1" target="_self">*'</a> </span>)</span> &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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span> <a href="matrixc1.html#K1" target="_self">*'</a> </span>)</span> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:71_5"/><i><font color="Green">E56</font></i>: 
( 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="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="matrixc1.html#K1" target="_self">*'</a> </span>)</span> &amp; 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="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="matrixc1.html#K1" target="_self">*'</a> </span>)</span> )
 
<b>by </b><i><a class="ref" href="matrixc1.html#T9" target="_self">Th9</a>, , , <a class="txt" href="matrixc1.html#E5:71_5"><i><font color="Green">E61</font></i></a></i>;<br/>
<a NAME="E5:71_5"/><i><font color="Green">E61</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"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:71_5"/><i><font color="Green">E62</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">j</font>,<font color="Maroon">i</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">M</font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D7">MATRIX_1:def 7</a>, <a class="ref" href="matrixc1.html#T11" target="_self">Th11</a></i>;<br/>
<a NAME="E7:71_5"/><i><font color="Green">E63</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">j</font>,<font color="Maroon">i</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">M</font> <a href="matrixc1.html#K1" target="_self">*'</a> </span>)</span>
 
<b>by </b><i>, <a class="ref" href="matrixc1.html#T10" target="_self">Th10</a></i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span> <a href="matrixc1.html#K1" target="_self">*'</a> </span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span> <a href="complex1.html#K15">*'</a> 
<b>by </b><i><a class="ref" href="matrixc1.html#T11" target="_self">Th11</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,<font color="Maroon">i</font></span>)</span> <a href="complex1.html#K15">*'</a> 
<b>by </b><i><a class="ref" href="matrix_1.html#D7">MATRIX_1:def 7</a>, <a class="ref" href="matrixc1.html#T13" target="_self">Th13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="matrixc1.html#K1" target="_self">*'</a> </span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,<font color="Maroon">i</font>
<b>by </b><i>, <a class="ref" href="matrixc1.html#T13" target="_self">Th13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="matrixc1.html#K1" target="_self">*'</a> </span>)</span> <a href="matrix_1.html#K4">@</a> </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_1.html#D7">MATRIX_1:def 7</a>, <a class="ref" href="matrixc1.html#T14" target="_self">Th14</a></i>
;<br/>
<b>hence </b><a NAME="E9:71_5"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span> <a href="matrixc1.html#K1" target="_self">*'</a> </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"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="matrixc1.html#K1" target="_self">*'</a> </span>)</span> <a href="matrix_1.html#K4">@</a> </span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:71"/>
<span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </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">M</font> <a href="matrixc1.html#K1" target="_self">*'</a> </span>)</span> <a href="matrix_1.html#K4">@</a> 
 <b>by </b><i><a class="ref" href="matrix_1.html#T21">MATRIX_1:21</a>, , , , </i>;<br/>


</div>
