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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="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K2">COMPLEX</a> ;<br/>

<b>assume </b><a NAME="E1:24"/><i><font color="Green">E21</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>
 ;<br/>

<a NAME="E2:24"/><b>then </b><i><font color="Green">E22</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="complsp1.html#K3">+</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="complsp2.html#T6" target="_self">Th6</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>1</sub></font> as    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K2">COMPLEX</a>  <b>by </b><i><a class="ref" href="finseq_2.html#T110">FINSEQ_2:110</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>2</sub></font> as    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K2">COMPLEX</a>  <b>by </b><i><a class="ref" href="finseq_2.html#T110">FINSEQ_2:110</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="complsp2.html#K1" target="_self">*'</a>  as    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complsp2.html#K1" target="_self">*'</a> </span>)</span></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K2">COMPLEX</a>  <b>by </b><i><a class="ref" href="finseq_2.html#T110">FINSEQ_2:110</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="complsp2.html#K1" target="_self">*'</a>  as    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="complsp2.html#K1" target="_self">*'</a> </span>)</span></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K2">COMPLEX</a>  <b>by </b><i><a class="ref" href="finseq_2.html#T110">FINSEQ_2:110</a></i>;<br/>
<a NAME="E7:24"/><i><font color="Green">E23</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> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complsp2.html#K1" target="_self">*'</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="complsp2.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E8:24"/><i><font color="Green">E24</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> <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="complsp2.html#K1" target="_self">*'</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="complsp2.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E9:24"/><b>then </b>
 <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="complsp2.html#K1" target="_self">*'</a> </span>)</span> <a href="complsp1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="complsp2.html#K1" target="_self">*'</a> </span>)</span></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="txt" href="complsp2.html#E7:24"><i><font color="Green">E23</font></i></a>, <a class="ref" href="complsp2.html#T6" target="_self">Th6</a>, <a class="txt" href="complsp2.html#E1:24"><i><font color="Green">E21</font></i></a></i>;<br/>
<a NAME="E10:24"/><b>then </b><i><font color="Green">E25</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="complsp1.html#K3">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="complsp2.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"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complsp2.html#K1" target="_self">*'</a> </span>)</span> <a href="complsp1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="complsp2.html#K1" target="_self">*'</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="complsp2.html#E2:24"><i><font color="Green">E22</font></i></a>, <a class="ref" href="complsp2.html#D1" target="_self">Def1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:24_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></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">c<sub>1</sub></font> <a href="complsp1.html#K3">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="complsp2.html#K1" target="_self">*'</a> </span>)</span> )
 ;<br/><a NAME="E2:24_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>7</sub></font> &amp; <font color="Maroon">c<sub>7</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="complsp1.html#K3">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="complsp2.html#D1" target="_self">Def1</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="complsp1.html#K3">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="complsp2.html#K1" target="_self">*'</a> </span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">c<sub>7</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complsp1.html#K3">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="complex1.html#K15">*'</a> 
<b>by </b><i><a class="ref" href="complsp2.html#D1" target="_self">Def1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrix_5.html#K9">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="binop_2.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="matrix_5.html#K9">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="complex1.html#K15">*'</a> 
<b>by </b><i><a class="ref" href="complsp2.html#T18" target="_self">Th18</a>, <a class="txt" href="complsp2.html#E1:24"><i><font color="Green">E21</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_5.html#K9">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="complex1.html#K15">*'</a> </span>)</span> <a href="binop_2.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_5.html#K9">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="complex1.html#K15">*'</a> </span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T118">COMPLEX1:118</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="complsp2.html#K1" target="_self">*'</a> </span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="binop_2.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_5.html#K9">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="complex1.html#K15">*'</a> </span>)</span>
<b>by </b><i><a class="ref" href="complsp2.html#D1" target="_self">Def1</a>, <a class="txt" href="complsp2.html#E2:24_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="complsp2.html#E2:24"><i><font color="Green">E22</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="complsp2.html#K1" target="_self">*'</a> </span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="binop_2.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="complsp2.html#K1" target="_self">*'</a> </span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="complsp2.html#D1" target="_self">Def1</a>, <a class="txt" href="complsp2.html#E2:24_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="complsp2.html#E2:24"><i><font color="Green">E22</font></i></a>, <a class="txt" href="complsp2.html#E1:24"><i><font color="Green">E21</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="complsp1.html#K3">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="ref" href="complsp2.html#T18" target="_self">Th18</a>, <a class="txt" href="complsp2.html#E1:24"><i><font color="Green">E21</font></i></a>, <a class="txt" href="complsp2.html#E7:24"><i><font color="Green">E23</font></i></a>, <a class="txt" href="complsp2.html#E8:24"><i><font color="Green">E24</font></i></a></i>
;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E12:24"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complsp1.html#K3">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="complsp2.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="complsp2.html#K1" target="_self">*'</a> </span>)</span> <a href="complsp1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="complsp2.html#K1" target="_self">*'</a> </span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a>, <a class="txt" href="complsp2.html#E10:24"><i><font color="Green">E25</font></i></a></i>;<br/>


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