<?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="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a> ;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:88"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E2:88"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E3:88"/><i><font color="Green">E83</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a>  <a href="complex2.html#K5" target="_self">angle</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E4:88"/><i><font color="Green">E84</font></i>: 
 <a href="complex2.html#K5" target="_self">angle</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="sin_cos.html#K33">PI</a> 
 ;<br/>

<i><font color="Green">E85</font></i>:  <a href="complex2.html#K5" target="_self">angle</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complex1.html#K8">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K10">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<a href="complex1.html#K5">0c</a> ,<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="complex1.html#K8">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K10">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> = 
 <a href="complex2.html#K5" target="_self">angle</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complex1.html#K8">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K10">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="complex1.html#K8">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K10">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="complex1.html#K8">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K10">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_0.html#D6">XCMPLX_0:def 6</a></i>
<br/>.= 
 <a href="complex2.html#K5" target="_self">angle</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="complex2.html#T86" target="_self">Th86</a></i>
;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complex1.html#K8">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K10">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="complex2.html#K3" target="_self">Rotate</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complex1.html#K8">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K10">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complex1.html#K8">+</a> <span class="p4">(<span class="default"><a href="complex1.html#K10">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="complex2.html#K3" target="_self">Rotate</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="complex1.html#K8">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K10">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complex1.html#K8">+</a> <span class="p4">(<span class="default"><a href="complex1.html#K10">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>;<br/>
<a NAME="E6:88"/><i><font color="Green">E86</font></i>: 
 <a href="complex2.html#K3" target="_self">Rotate</a> <a href="complex1.html#K5">0c</a> ,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complex1.html#K8">+</a> <span class="p4">(<span class="default"><a href="complex1.html#K10">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K5">0c</a> 
 
<b>by </b><i><a class="ref" href="complex2.html#T66" target="_self">Th66</a></i>;<br/>
<a NAME="E7:88"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="complex2.html#E1:88"><i><font color="Green">E81</font></i></a></i>;<br/>
<a NAME="E8:88"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="complex1.html#K8">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K10">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="complex1.html#K8">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K10">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="complex2.html#E3:88"><i><font color="Green">E83</font></i></a>, <a class="ref" href="complex2.html#T93" target="_self">Th93</a></i>;<br/>
<a NAME="E9:88"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="complex2.html#E2:88"><i><font color="Green">E82</font></i></a></i>;<br/>
<a NAME="E10:88"/><b>then </b><i><font color="Green">E90</font></i>: 
 <a href="complex2.html#K5" target="_self">angle</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complex1.html#K8">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K10">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<a href="complex1.html#K5">0c</a> ,<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="complex1.html#K8">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K10">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="complex2.html#K5" target="_self">angle</a> <span class="p1">(<span class="default"><a href="complex2.html#K3" target="_self">Rotate</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complex1.html#K8">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K10">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complex1.html#K8">+</a> <span class="p5">(<span class="default"><a href="complex1.html#K10">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<a href="complex1.html#K5">0c</a> ,<span class="p1">(<span class="default"><a href="complex2.html#K3" target="_self">Rotate</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="complex1.html#K8">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K10">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complex1.html#K8">+</a> <span class="p5">(<span class="default"><a href="complex1.html#K10">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="complex2.html#E6:88"><i><font color="Green">E86</font></i></a>, <a class="txt" href="complex2.html#E7:88"><i><font color="Green">E87</font></i></a>, <a class="ref" href="complex2.html#T92" target="_self">Th92</a></i>;<br/>
<i><font color="Green">E91</font></i>:  <a href="complex2.html#K3" target="_self">Rotate</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complex1.html#K8">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K10">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complex1.html#K8">+</a> <span class="p4">(<span class="default"><a href="complex1.html#K10">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> = 
<span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complex1.html#K8">+</a> <span class="p4">(<span class="default"><a href="complex1.html#K10">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span>,0</span><a href="arytm_0.html#K5">*]</a></span>
<b>by </b><i><a class="ref" href="complex2.html#T69" target="_self">Th69</a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complex1.html#K8">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K10">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">0 <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T27">COMPLEX1:27</a></i>
;<br/>
<a NAME="E12:88"/><b>then </b><i><font color="Green">E92</font></i>: 
 <a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><a href="complex2.html#K3" target="_self">Rotate</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complex1.html#K8">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K10">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complex1.html#K8">+</a> <span class="p5">(<span class="default"><a href="complex1.html#K10">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="complex1.html#T28">COMPLEX1:28</a></i>;<br/>
<a NAME="E13:88"/>
<font color="Maroon">c<sub>1</sub></font> <a href="complex1.html#K8">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K10">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="complex1.html#K5">0c</a> 
 
<b>by </b><i><a class="txt" href="complex2.html#E1:88"><i><font color="Green">E81</font></i></a>, <a class="ref" href="xcmplx_1.html#T15">XCMPLX_1:15</a></i>;<br/>
<a NAME="E14:88"/><b>then </b>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complex1.html#K8">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K10">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="complex1.html#T133">COMPLEX1:133</a></i>;<br/>
<a NAME="E15:88"/><b>then </b><i><font color="Green">E93</font></i>: 
 <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><a href="complex2.html#K3" target="_self">Rotate</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complex1.html#K8">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K10">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complex1.html#K8">+</a> <span class="p5">(<span class="default"><a href="complex1.html#K10">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="complex2.html#E11:88"><i><font color="Green">E91</font></i></a>, <a class="ref" href="complex1.html#T28">COMPLEX1:28</a></i>;<br/>
<a NAME="E16:88"/><b>then </b><i><font color="Green">E94</font></i>: 
 <a href="comptrig.html#K1">Arg</a> <span class="p1">(<span class="default"><a href="complex2.html#K3" target="_self">Rotate</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complex1.html#K8">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K10">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complex1.html#K8">+</a> <span class="p5">(<span class="default"><a href="complex1.html#K10">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K5">0c</a> 
 
<b>by </b><i><a class="txt" href="complex2.html#E12:88"><i><font color="Green">E92</font></i></a>, <a class="ref" href="complex2.html#T34" target="_self">Th34</a></i>;<br/>
<a NAME="E17:88"/><b>then </b>
<span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex2.html#K3" target="_self">Rotate</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="complex1.html#K8">+</a> <span class="p5">(<span class="default"><a href="complex1.html#K10">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p5">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complex1.html#K8">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K10">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="complex1.html#K11">-</a> <a href="complex1.html#K5">0c</a> </span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex2.html#K3" target="_self">Rotate</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complex1.html#K8">+</a> <span class="p5">(<span class="default"><a href="complex1.html#K10">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p5">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complex1.html#K8">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K10">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="complex1.html#K11">-</a> <a href="complex1.html#K5">0c</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="txt" href="complex2.html#E18"><i><font color="Green">Lemma12</font></i></a></i>;<br/>
<a NAME="E18:88"/><b>then </b><i><font color="Green">E95</font></i>: 
 <a href="complex2.html#K5" target="_self">angle</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="comptrig.html#K1">Arg</a> <span class="p1">(<span class="default"><a href="complex2.html#K3" target="_self">Rotate</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="complex1.html#K8">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K10">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complex1.html#K8">+</a> <span class="p5">(<span class="default"><a href="complex1.html#K10">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="complex2.html#E5:88"><i><font color="Green">E85</font></i></a>, <a class="txt" href="complex2.html#E10:88"><i><font color="Green">E90</font></i></a>, <a class="txt" href="complex2.html#E16:88"><i><font color="Green">E94</font></i></a>, <a class="ref" href="complex2.html#D6" target="_self">Def6</a></i>;<br/>
<i><font color="Green">E96</font></i>:  <a href="complex2.html#K5" target="_self">angle</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> = 
 <a href="complex2.html#K5" target="_self">angle</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="complex1.html#K8">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K10">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="complex1.html#K8">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K10">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complex1.html#K8">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K10">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complex2.html#T86" target="_self">Th86</a></i>
<br/>.= 
 <a href="complex2.html#K5" target="_self">angle</a> <a href="complex1.html#K5">0c</a> ,<span class="p1">(<span class="default"><a href="complex2.html#K3" target="_self">Rotate</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="complex1.html#K8">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K10">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complex1.html#K8">+</a> <span class="p5">(<span class="default"><a href="complex1.html#K10">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="complex2.html#K3" target="_self">Rotate</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complex1.html#K8">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K10">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complex1.html#K8">+</a> <span class="p5">(<span class="default"><a href="complex1.html#K10">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="complex2.html#E6:88"><i><font color="Green">E86</font></i></a>, <a class="txt" href="complex2.html#E8:88"><i><font color="Green">E88</font></i></a>, <a class="txt" href="complex2.html#E9:88"><i><font color="Green">E89</font></i></a>, <a class="ref" href="complex2.html#T92" target="_self">Th92</a></i>
;<br/>
<i><font color="Green">E97</font></i>:  <a href="complex2.html#K5" target="_self">angle</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> = 
 <a href="complex2.html#K5" target="_self">angle</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="complex1.html#K8">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K10">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complex1.html#K8">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K10">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="complex1.html#K8">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K10">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complex2.html#T86" target="_self">Th86</a></i>
<br/>.= 
 <a href="complex2.html#K5" target="_self">angle</a> <span class="p1">(<span class="default"><a href="complex2.html#K3" target="_self">Rotate</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="complex1.html#K8">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K10">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complex1.html#K8">+</a> <span class="p5">(<span class="default"><a href="complex1.html#K10">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="complex2.html#K3" target="_self">Rotate</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complex1.html#K8">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K10">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complex1.html#K8">+</a> <span class="p5">(<span class="default"><a href="complex1.html#K10">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<a href="complex1.html#K5">0c</a> 
<b>by </b><i><a class="txt" href="complex2.html#E6:88"><i><font color="Green">E86</font></i></a>, <a class="txt" href="complex2.html#E7:88"><i><font color="Green">E87</font></i></a>, <a class="txt" href="complex2.html#E8:88"><i><font color="Green">E88</font></i></a>, <a class="ref" href="complex2.html#T92" target="_self">Th92</a></i>
;<br/>
<b>hence </b><a NAME="E21:88"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex2.html#K5" target="_self">angle</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="complex2.html#K5" target="_self">angle</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="complex2.html#K5" target="_self">angle</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="sin_cos.html#K33">PI</a> 
 <b>by </b><i><a class="txt" href="complex2.html#E3:88"><i><font color="Green">E83</font></i></a>, <a class="txt" href="complex2.html#E4:88"><i><font color="Green">E84</font></i></a>, <a class="txt" href="complex2.html#E5:88"><i><font color="Green">E85</font></i></a>, <a class="txt" href="complex2.html#E10:88"><i><font color="Green">E90</font></i></a>, <a class="txt" href="complex2.html#E12:88"><i><font color="Green">E92</font></i></a>, <a class="txt" href="complex2.html#E15:88"><i><font color="Green">E93</font></i></a>, <a class="txt" href="complex2.html#E18:88"><i><font color="Green">E95</font></i></a>, <a class="txt" href="complex2.html#E19:88"><i><font color="Green">E96</font></i></a>, <a class="txt" href="complex2.html#E114"><i><font color="Green">Lemma79</font></i></a></i>;<br/>

<b>thus </b><a NAME="E22:88"/>
0 <a href="xxreal_0.html#R1">&lt;</a>  <a href="complex2.html#K5" target="_self">angle</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="complex2.html#E3:88"><i><font color="Green">E83</font></i></a>, <a class="txt" href="complex2.html#E4:88"><i><font color="Green">E84</font></i></a>, <a class="txt" href="complex2.html#E12:88"><i><font color="Green">E92</font></i></a>, <a class="txt" href="complex2.html#E15:88"><i><font color="Green">E93</font></i></a>, <a class="txt" href="complex2.html#E18:88"><i><font color="Green">E95</font></i></a>, <a class="txt" href="complex2.html#E19:88"><i><font color="Green">E96</font></i></a>, <a class="txt" href="complex2.html#E114"><i><font color="Green">Lemma79</font></i></a></i>;<br/>

<b>thus </b><a NAME="E23:88"/>
0 <a href="xxreal_0.html#R1">&lt;</a>  <a href="complex2.html#K5" target="_self">angle</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="complex2.html#E3:88"><i><font color="Green">E83</font></i></a>, <a class="txt" href="complex2.html#E4:88"><i><font color="Green">E84</font></i></a>, <a class="txt" href="complex2.html#E12:88"><i><font color="Green">E92</font></i></a>, <a class="txt" href="complex2.html#E15:88"><i><font color="Green">E93</font></i></a>, <a class="txt" href="complex2.html#E18:88"><i><font color="Green">E95</font></i></a>, <a class="txt" href="complex2.html#E20:88"><i><font color="Green">E97</font></i></a>, <a class="txt" href="complex2.html#E114"><i><font color="Green">Lemma79</font></i></a></i>;<br/>


</div>
