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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K2">COMPLEX</a> ;<br/>

<a NAME="E1:69"/><i><font color="Green">E59</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="complsp2.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="complsp2.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E2:69"/><i><font color="Green">E60</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="complsp1.html#K6">*</a> <font color="Maroon">c<sub>1</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#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E3:69"/><i><font color="Green">E61</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="complsp1.html#K6">*</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="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#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E4:69"/><i><font color="Green">E62</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="complsp1.html#K5">-</a> <span class="p2">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="complsp1.html#K6">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complsp2.html#K1" target="_self">*'</a> </span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="complsp1.html#K6">*</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>
 
<b>by </b><i><a class="ref" href="complsp2.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E5:69"/><i><font color="Green">E63</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="complsp2.html#K7" target="_self">Im</a> <font color="Maroon">c<sub>1</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#T48" target="_self">Th48</a></i>;<br/>
<i><font color="Green">E64</font></i>:  <a href="complsp2.html#K6" target="_self">Re</a> <span class="p1">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="complsp1.html#K6">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> = 
<span class="p1">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="complsp1.html#K6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="complsp1.html#K6">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="complsp1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="complsp1.html#K6">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="complsp2.html#K1" target="_self">*'</a> </span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="complsp1.html#K6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="complsp1.html#K6">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="complsp1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K1">-</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span> <a href="complsp1.html#K6">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complsp2.html#K1" target="_self">*'</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T116">COMPLEX1:116</a>, <a class="ref" href="complsp2.html#T17" target="_self">Th17</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="complsp1.html#K6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="complsp1.html#K6">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="complsp1.html#K3">+</a> <span class="p2">(<span class="default"><a href="complsp1.html#K5">-</a> <span class="p3">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="complsp1.html#K6">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complsp2.html#K1" target="_self">*'</a> </span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complsp2.html#T54" target="_self">Th54</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="complsp1.html#K6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="complsp1.html#K6">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="complsp1.html#K4">-</a> <span class="p2">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="complsp1.html#K6">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complsp2.html#K1" target="_self">*'</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complsp2.html#T31" target="_self">Th31</a>, <a class="txt" href="complsp2.html#E1:69"><i><font color="Green">E59</font></i></a>, <a class="txt" href="complsp2.html#E2:69"><i><font color="Green">E60</font></i></a>, <a class="txt" href="complsp2.html#E3:69"><i><font color="Green">E61</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="complsp1.html#K6">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="complsp1.html#K6">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complsp1.html#K4">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complsp2.html#K1" target="_self">*'</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complsp2.html#T43" target="_self">Th43</a>, <a class="txt" href="complsp2.html#E1:69"><i><font color="Green">E59</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span> <a href="complsp1.html#K6">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complsp1.html#K4">-</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>
<b>by </b><i><a class="ref" href="complsp2.html#T53" target="_self">Th53</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K7">-</a> <span class="p4">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span></span>)</span> <a href="complsp1.html#K6">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complsp1.html#K4">-</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>

<br/>.= 
<span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="complsp1.html#K6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K7">-</a> <span class="p4">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span> <a href="complsp1.html#K6">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complsp1.html#K4">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complsp2.html#K1" target="_self">*'</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complsp2.html#T53" target="_self">Th53</a></i>
<br/>.= 
 <a href="complsp1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K7">-</a> <span class="p4">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span> <a href="complsp1.html#K6">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complsp1.html#K4">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complsp2.html#K1" target="_self">*'</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complsp2.html#T12" target="_self">Th12</a></i>
<br/>.= 
 <a href="rvsum_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="complsp2.html#K7" target="_self">Im</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="complsp2.html#T57" target="_self">Th57</a>, <a class="txt" href="complsp2.html#E5:69"><i><font color="Green">E63</font></i></a></i>
;<br/>
 <a href="complsp2.html#K7" target="_self">Im</a> <span class="p1">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="complsp1.html#K6">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K7">-</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span> <a href="complsp1.html#K6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="complsp1.html#K6">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="complsp1.html#K4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="complsp1.html#K6">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="complsp2.html#K1" target="_self">*'</a> </span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K7">-</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span> <a href="complsp1.html#K6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="complsp1.html#K6">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="complsp1.html#K4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K1">-</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span> <a href="complsp1.html#K6">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complsp2.html#K1" target="_self">*'</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T116">COMPLEX1:116</a>, <a class="ref" href="complsp2.html#T17" target="_self">Th17</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K7">-</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span> <a href="complsp1.html#K6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="complsp1.html#K6">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="complsp1.html#K4">-</a> <span class="p2">(<span class="default"><a href="complsp1.html#K5">-</a> <span class="p3">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="complsp1.html#K6">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complsp2.html#K1" target="_self">*'</a> </span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complsp2.html#T54" target="_self">Th54</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K7">-</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span> <a href="complsp1.html#K6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="complsp1.html#K6">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="complsp1.html#K3">+</a> <span class="p2">(<span class="default"><a href="complsp1.html#K5">-</a> <span class="p3">(<span class="default"><a href="complsp1.html#K5">-</a> <span class="p4">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="complsp1.html#K6">*</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complsp2.html#K1" target="_self">*'</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complsp2.html#T31" target="_self">Th31</a>, <a class="txt" href="complsp2.html#E1:69"><i><font color="Green">E59</font></i></a>, <a class="txt" href="complsp2.html#E2:69"><i><font color="Green">E60</font></i></a>, <a class="txt" href="complsp2.html#E3:69"><i><font color="Green">E61</font></i></a>, <a class="txt" href="complsp2.html#E4:69"><i><font color="Green">E62</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K7">-</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span> <a href="complsp1.html#K6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="complsp1.html#K6">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="complsp1.html#K3">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="complsp1.html#K6">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complsp2.html#K1" target="_self">*'</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complsp2.html#T38" target="_self">Th38</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K7">-</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span> <a href="complsp1.html#K6">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="complsp1.html#K6">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complsp1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complsp2.html#K1" target="_self">*'</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complsp2.html#T30" target="_self">Th30</a>, <a class="txt" href="complsp2.html#E1:69"><i><font color="Green">E59</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K7">-</a> <span class="p4">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span> <a href="complsp1.html#K6">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complsp1.html#K3">+</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>
<b>by </b><i><a class="ref" href="complsp2.html#T53" target="_self">Th53</a></i>
<br/>.= 
 <a href="complsp2.html#K6" target="_self">Re</a> <font color="Maroon">c<sub>1</sub></font>

;<br/>
<b>hence </b><a NAME="E8:69"/>
(  <a href="complsp2.html#K6" target="_self">Re</a> <span class="p1">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="complsp1.html#K6">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rvsum_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="complsp2.html#K7" target="_self">Im</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp;  <a href="complsp2.html#K7" target="_self">Im</a> <span class="p1">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="complsp1.html#K6">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="complsp2.html#K6" target="_self">Re</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="complsp2.html#E6:69"><i><font color="Green">E64</font></i></a></i>;<br/>


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