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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</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="E1:77"/><i><font color="Green">E67</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> &amp;  <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> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<a NAME="E2:77"/><i><font color="Green">E68</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</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/>
<a NAME="E3:77"/><i><font color="Green">E69</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</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>2</sub></font>
 
<b>by </b><i><a class="ref" href="complsp2.html#T48" target="_self">Th48</a></i>;<br/>
<a NAME="E4:77"/><i><font color="Green">E70</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="complsp2.html#T48" target="_self">Th48</a></i>;<br/>
<a NAME="E5:77"/><i><font color="Green">E71</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/>
<a NAME="E6:77"/><i><font color="Green">E72</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>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="complsp2.html#T48" target="_self">Th48</a></i>;<br/>
<a NAME="E7:77"/><i><font color="Green">E73</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>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="complsp2.html#T48" target="_self">Th48</a></i>;<br/>
<span class="p1"><a href="complsp2.html#K8" target="_self">|(</a><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>,<font color="Maroon">c<sub>3</sub></font></span><a href="complsp2.html#K8" target="_self">)|</a></span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid_2.html#K1">|(</a><span class="default"><span class="p4">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <span class="p5">(<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></span>)</span>,<span class="p4">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="euclid_2.html#K1">)|</a></span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="xcmplx_0.html#K3">*</a> <span class="p4"><a href="euclid_2.html#K1">|(</a><span class="default"><span class="p5">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <span class="p0">(<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></span>)</span>,<span class="p5">(<span class="default"><a href="complsp2.html#K7" target="_self">Im</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="euclid_2.html#K1">)|</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="xcmplx_0.html#K3">*</a> <span class="p3"><a href="euclid_2.html#K1">|(</a><span class="default"><span class="p4">(<span class="default"><a href="complsp2.html#K7" target="_self">Im</a> <span class="p5">(<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></span>)</span>,<span class="p4">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="euclid_2.html#K1">)|</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1"><a href="euclid_2.html#K1">|(</a><span class="default"><span class="p2">(<span class="default"><a href="complsp2.html#K7" target="_self">Im</a> <span class="p3">(<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></span>)</span>,<span class="p2">(<span class="default"><a href="complsp2.html#K7" target="_self">Im</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="euclid_2.html#K1">)|</a></span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid_2.html#K1">|(</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p5">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="euclid_2.html#K1">)|</a></span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="xcmplx_0.html#K3">*</a> <span class="p4"><a href="euclid_2.html#K1">|(</a><span class="default"><span class="p5">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <span class="p0">(<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></span>)</span>,<span class="p5">(<span class="default"><a href="complsp2.html#K7" target="_self">Im</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="euclid_2.html#K1">)|</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="xcmplx_0.html#K3">*</a> <span class="p3"><a href="euclid_2.html#K1">|(</a><span class="default"><span class="p4">(<span class="default"><a href="complsp2.html#K7" target="_self">Im</a> <span class="p5">(<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></span>)</span>,<span class="p4">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="euclid_2.html#K1">)|</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1"><a href="euclid_2.html#K1">|(</a><span class="default"><span class="p2">(<span class="default"><a href="complsp2.html#K7" target="_self">Im</a> <span class="p3">(<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></span>)</span>,<span class="p2">(<span class="default"><a href="complsp2.html#K7" target="_self">Im</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="euclid_2.html#K1">)|</a></span>
<b>by </b><i><a class="ref" href="complsp2.html#T49" target="_self">Th49</a>, <a class="txt" href="complsp2.html#E1:77"><i><font color="Green">E67</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid_2.html#K1">|(</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p5">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="euclid_2.html#K1">)|</a></span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="xcmplx_0.html#K3">*</a> <span class="p4"><a href="euclid_2.html#K1">|(</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p0">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="complsp2.html#K7" target="_self">Im</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="euclid_2.html#K1">)|</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="xcmplx_0.html#K3">*</a> <span class="p3"><a href="euclid_2.html#K1">|(</a><span class="default"><span class="p4">(<span class="default"><a href="complsp2.html#K7" target="_self">Im</a> <span class="p5">(<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></span>)</span>,<span class="p4">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="euclid_2.html#K1">)|</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1"><a href="euclid_2.html#K1">|(</a><span class="default"><span class="p2">(<span class="default"><a href="complsp2.html#K7" target="_self">Im</a> <span class="p3">(<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></span>)</span>,<span class="p2">(<span class="default"><a href="complsp2.html#K7" target="_self">Im</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="euclid_2.html#K1">)|</a></span>
<b>by </b><i><a class="ref" href="complsp2.html#T49" target="_self">Th49</a>, <a class="txt" href="complsp2.html#E1:77"><i><font color="Green">E67</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid_2.html#K1">|(</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p5">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="euclid_2.html#K1">)|</a></span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="xcmplx_0.html#K3">*</a> <span class="p4"><a href="euclid_2.html#K1">|(</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p0">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="complsp2.html#K7" target="_self">Im</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="euclid_2.html#K1">)|</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="xcmplx_0.html#K3">*</a> <span class="p3"><a href="euclid_2.html#K1">|(</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="complsp2.html#K7" target="_self">Im</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p5">(<span class="default"><a href="complsp2.html#K7" target="_self">Im</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="euclid_2.html#K1">)|</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1"><a href="euclid_2.html#K1">|(</a><span class="default"><span class="p2">(<span class="default"><a href="complsp2.html#K7" target="_self">Im</a> <span class="p3">(<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></span>)</span>,<span class="p2">(<span class="default"><a href="complsp2.html#K7" target="_self">Im</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="euclid_2.html#K1">)|</a></span>
<b>by </b><i><a class="ref" href="complsp2.html#T49" target="_self">Th49</a>, <a class="txt" href="complsp2.html#E1:77"><i><font color="Green">E67</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid_2.html#K1">|(</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p5">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="euclid_2.html#K1">)|</a></span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="xcmplx_0.html#K3">*</a> <span class="p4"><a href="euclid_2.html#K1">|(</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p0">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="complsp2.html#K7" target="_self">Im</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="euclid_2.html#K1">)|</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="xcmplx_0.html#K3">*</a> <span class="p3"><a href="euclid_2.html#K1">|(</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="complsp2.html#K7" target="_self">Im</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p5">(<span class="default"><a href="complsp2.html#K7" target="_self">Im</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="euclid_2.html#K1">)|</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1"><a href="euclid_2.html#K1">|(</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complsp2.html#K7" target="_self">Im</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="complsp2.html#K7" target="_self">Im</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="complsp2.html#K7" target="_self">Im</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="euclid_2.html#K1">)|</a></span>
<b>by </b><i><a class="ref" href="complsp2.html#T49" target="_self">Th49</a>, <a class="txt" href="complsp2.html#E1:77"><i><font color="Green">E67</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="euclid_2.html#K1">|(</a><span class="default"><span class="p5">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p5">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="euclid_2.html#K1">)|</a></span> <a href="xcmplx_0.html#K2">+</a> <span class="p4"><a href="euclid_2.html#K1">|(</a><span class="default"><span class="p5">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p5">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="euclid_2.html#K1">)|</a></span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="xcmplx_0.html#K3">*</a> <span class="p4"><a href="euclid_2.html#K1">|(</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p0">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="complsp2.html#K7" target="_self">Im</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="euclid_2.html#K1">)|</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="xcmplx_0.html#K3">*</a> <span class="p3"><a href="euclid_2.html#K1">|(</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="complsp2.html#K7" target="_self">Im</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p5">(<span class="default"><a href="complsp2.html#K7" target="_self">Im</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="euclid_2.html#K1">)|</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1"><a href="euclid_2.html#K1">|(</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complsp2.html#K7" target="_self">Im</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="complsp2.html#K7" target="_self">Im</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="complsp2.html#K7" target="_self">Im</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="euclid_2.html#K1">)|</a></span>
<b>by </b><i><a class="ref" href="euclid_2.html#T19">EUCLID_2:19</a>, <a class="txt" href="complsp2.html#E1:77"><i><font color="Green">E67</font></i></a>, <a class="txt" href="complsp2.html#E2:77"><i><font color="Green">E68</font></i></a>, <a class="txt" href="complsp2.html#E3:77"><i><font color="Green">E69</font></i></a>, <a class="txt" href="complsp2.html#E4:77"><i><font color="Green">E70</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="euclid_2.html#K1">|(</a><span class="default"><span class="p5">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p5">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="euclid_2.html#K1">)|</a></span> <a href="xcmplx_0.html#K2">+</a> <span class="p4"><a href="euclid_2.html#K1">|(</a><span class="default"><span class="p5">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p5">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="euclid_2.html#K1">)|</a></span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5"><a href="euclid_2.html#K1">|(</a><span class="default"><span class="p0">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p0">(<span class="default"><a href="complsp2.html#K7" target="_self">Im</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="euclid_2.html#K1">)|</a></span> <a href="xcmplx_0.html#K2">+</a> <span class="p5"><a href="euclid_2.html#K1">|(</a><span class="default"><span class="p0">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p0">(<span class="default"><a href="complsp2.html#K7" target="_self">Im</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="euclid_2.html#K1">)|</a></span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="xcmplx_0.html#K3">*</a> <span class="p3"><a href="euclid_2.html#K1">|(</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="complsp2.html#K7" target="_self">Im</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p5">(<span class="default"><a href="complsp2.html#K7" target="_self">Im</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="euclid_2.html#K1">)|</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1"><a href="euclid_2.html#K1">|(</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complsp2.html#K7" target="_self">Im</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="complsp2.html#K7" target="_self">Im</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="complsp2.html#K7" target="_self">Im</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="euclid_2.html#K1">)|</a></span>
<b>by </b><i><a class="ref" href="euclid_2.html#T19">EUCLID_2:19</a>, <a class="txt" href="complsp2.html#E1:77"><i><font color="Green">E67</font></i></a>, <a class="txt" href="complsp2.html#E2:77"><i><font color="Green">E68</font></i></a>, <a class="txt" href="complsp2.html#E3:77"><i><font color="Green">E69</font></i></a>, <a class="txt" href="complsp2.html#E7:77"><i><font color="Green">E73</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="euclid_2.html#K1">|(</a><span class="default"><span class="p5">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p5">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="euclid_2.html#K1">)|</a></span> <a href="xcmplx_0.html#K2">+</a> <span class="p4"><a href="euclid_2.html#K1">|(</a><span class="default"><span class="p5">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p5">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="euclid_2.html#K1">)|</a></span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5"><a href="euclid_2.html#K1">|(</a><span class="default"><span class="p0">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p0">(<span class="default"><a href="complsp2.html#K7" target="_self">Im</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="euclid_2.html#K1">)|</a></span> <a href="xcmplx_0.html#K2">+</a> <span class="p5"><a href="euclid_2.html#K1">|(</a><span class="default"><span class="p0">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p0">(<span class="default"><a href="complsp2.html#K7" target="_self">Im</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="euclid_2.html#K1">)|</a></span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid_2.html#K1">|(</a><span class="default"><span class="p5">(<span class="default"><a href="complsp2.html#K7" target="_self">Im</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p5">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="euclid_2.html#K1">)|</a></span> <a href="xcmplx_0.html#K2">+</a> <span class="p4"><a href="euclid_2.html#K1">|(</a><span class="default"><span class="p5">(<span class="default"><a href="complsp2.html#K7" target="_self">Im</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p5">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="euclid_2.html#K1">)|</a></span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1"><a href="euclid_2.html#K1">|(</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complsp2.html#K7" target="_self">Im</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="complsp2.html#K7" target="_self">Im</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="complsp2.html#K7" target="_self">Im</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="euclid_2.html#K1">)|</a></span>
<b>by </b><i><a class="ref" href="euclid_2.html#T19">EUCLID_2:19</a>, <a class="txt" href="complsp2.html#E1:77"><i><font color="Green">E67</font></i></a>, <a class="txt" href="complsp2.html#E4:77"><i><font color="Green">E70</font></i></a>, <a class="txt" href="complsp2.html#E5:77"><i><font color="Green">E71</font></i></a>, <a class="txt" href="complsp2.html#E6:77"><i><font color="Green">E72</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="euclid_2.html#K1">|(</a><span class="default"><span class="p5">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p5">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="euclid_2.html#K1">)|</a></span> <a href="xcmplx_0.html#K2">+</a> <span class="p4"><a href="euclid_2.html#K1">|(</a><span class="default"><span class="p5">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p5">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="euclid_2.html#K1">)|</a></span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5"><a href="euclid_2.html#K1">|(</a><span class="default"><span class="p0">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p0">(<span class="default"><a href="complsp2.html#K7" target="_self">Im</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="euclid_2.html#K1">)|</a></span> <a href="xcmplx_0.html#K2">+</a> <span class="p5"><a href="euclid_2.html#K1">|(</a><span class="default"><span class="p0">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p0">(<span class="default"><a href="complsp2.html#K7" target="_self">Im</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="euclid_2.html#K1">)|</a></span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid_2.html#K1">|(</a><span class="default"><span class="p5">(<span class="default"><a href="complsp2.html#K7" target="_self">Im</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p5">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="euclid_2.html#K1">)|</a></span> <a href="xcmplx_0.html#K2">+</a> <span class="p4"><a href="euclid_2.html#K1">|(</a><span class="default"><span class="p5">(<span class="default"><a href="complsp2.html#K7" target="_self">Im</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p5">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="euclid_2.html#K1">)|</a></span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid_2.html#K1">|(</a><span class="default"><span class="p3">(<span class="default"><a href="complsp2.html#K7" target="_self">Im</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="complsp2.html#K7" target="_self">Im</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="euclid_2.html#K1">)|</a></span> <a href="xcmplx_0.html#K2">+</a> <span class="p2"><a href="euclid_2.html#K1">|(</a><span class="default"><span class="p3">(<span class="default"><a href="complsp2.html#K7" target="_self">Im</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="complsp2.html#K7" target="_self">Im</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="euclid_2.html#K1">)|</a></span></span>)</span>
<b>by </b><i><a class="ref" href="euclid_2.html#T19">EUCLID_2:19</a>, <a class="txt" href="complsp2.html#E1:77"><i><font color="Green">E67</font></i></a>, <a class="txt" href="complsp2.html#E5:77"><i><font color="Green">E71</font></i></a>, <a class="txt" href="complsp2.html#E6:77"><i><font color="Green">E72</font></i></a>, <a class="txt" href="complsp2.html#E7:77"><i><font color="Green">E73</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="euclid_2.html#K1">|(</a><span class="default"><span class="p5">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p5">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="euclid_2.html#K1">)|</a></span> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="xcmplx_0.html#K3">*</a> <span class="p5"><a href="euclid_2.html#K1">|(</a><span class="default"><span class="p0">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p0">(<span class="default"><a href="complsp2.html#K7" target="_self">Im</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="euclid_2.html#K1">)|</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="xcmplx_0.html#K3">*</a> <span class="p4"><a href="euclid_2.html#K1">|(</a><span class="default"><span class="p5">(<span class="default"><a href="complsp2.html#K7" target="_self">Im</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p5">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="euclid_2.html#K1">)|</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2"><a href="euclid_2.html#K1">|(</a><span class="default"><span class="p3">(<span class="default"><a href="complsp2.html#K7" target="_self">Im</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="complsp2.html#K7" target="_self">Im</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="euclid_2.html#K1">)|</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="euclid_2.html#K1">|(</a><span class="default"><span class="p5">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p5">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="euclid_2.html#K1">)|</a></span> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="xcmplx_0.html#K3">*</a> <span class="p5"><a href="euclid_2.html#K1">|(</a><span class="default"><span class="p0">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p0">(<span class="default"><a href="complsp2.html#K7" target="_self">Im</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="euclid_2.html#K1">)|</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="xcmplx_0.html#K3">*</a> <span class="p4"><a href="euclid_2.html#K1">|(</a><span class="default"><span class="p5">(<span class="default"><a href="complsp2.html#K7" target="_self">Im</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p5">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="euclid_2.html#K1">)|</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2"><a href="euclid_2.html#K1">|(</a><span class="default"><span class="p3">(<span class="default"><a href="complsp2.html#K7" target="_self">Im</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="complsp2.html#K7" target="_self">Im</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="euclid_2.html#K1">)|</a></span></span>)</span>

<br/>.= 
<span class="p1"><a href="complsp2.html#K8" target="_self">|(</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="complsp2.html#K8" target="_self">)|</a></span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="euclid_2.html#K1">|(</a><span class="default"><span class="p5">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p5">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="euclid_2.html#K1">)|</a></span> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="xcmplx_0.html#K3">*</a> <span class="p5"><a href="euclid_2.html#K1">|(</a><span class="default"><span class="p0">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p0">(<span class="default"><a href="complsp2.html#K7" target="_self">Im</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="euclid_2.html#K1">)|</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="xcmplx_0.html#K3">*</a> <span class="p4"><a href="euclid_2.html#K1">|(</a><span class="default"><span class="p5">(<span class="default"><a href="complsp2.html#K7" target="_self">Im</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p5">(<span class="default"><a href="complsp2.html#K6" target="_self">Re</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="euclid_2.html#K1">)|</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2"><a href="euclid_2.html#K1">|(</a><span class="default"><span class="p3">(<span class="default"><a href="complsp2.html#K7" target="_self">Im</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="complsp2.html#K7" target="_self">Im</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="euclid_2.html#K1">)|</a></span></span>)</span>

<br/>.= 
<span class="p1"><a href="complsp2.html#K8" target="_self">|(</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="complsp2.html#K8" target="_self">)|</a></span> <a href="binop_2.html#K3">+</a> <span class="p1"><a href="complsp2.html#K8" target="_self">|(</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="complsp2.html#K8" target="_self">)|</a></span>

;<br/>
<b>hence </b><a NAME="E9:77"/>
<span class="p1"><a href="complsp2.html#K8" target="_self">|(</a><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>,<font color="Maroon">c<sub>3</sub></font></span><a href="complsp2.html#K8" target="_self">)|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complsp2.html#K8" target="_self">|(</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="complsp2.html#K8" target="_self">)|</a></span> <a href="binop_2.html#K3">+</a> <span class="p1"><a href="complsp2.html#K8" target="_self">|(</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="complsp2.html#K8" target="_self">)|</a></span>
 ;<br/>


</div>
