<?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="E1:89"/><i><font color="Green">E82</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="E2:89"/><i><font color="Green">E83</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="E3:89"/><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#NR4" target="_self">&gt;</a>  <a href="sin_cos.html#K33">PI</a> 
 ;<br/>

<a NAME="E4:89"/><i><font color="Green">E85</font></i>: 
<span class="p1">(<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="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>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="real_1.html#K3">+</a> 0
 
<b>by </b><i><a class="txt" href="complex2.html#E3:89"><i><font color="Green">E84</font></i></a>, <a class="ref" href="complex2.html#T94" target="_self">Th94</a>, <a class="ref" href="comptrig.html#T21">COMPTRIG:21</a></i>;<br/>
<a NAME="E5:89"/><b>then </b><i><font color="Green">E86</font></i>: 
<span class="p1">(<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#K6">-</a> 0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="xcmplx_0.html#K6">-</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>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 
;<br/>
<a NAME="E6:89"/><i><font color="Green">E87</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>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="complex2.html#T84" target="_self">Th84</a></i>;<br/>
<a NAME="E7:89"/><i><font color="Green">E88</font></i>: 
 <a href="complex2.html#K5" target="_self">angle</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="complex2.html#E4:89"><i><font color="Green">E85</font></i></a>, <a class="ref" href="complex2.html#T84" target="_self">Th84</a></i>;<br/>
<a NAME="E8:89"/><i><font color="Green">E89</font></i>: 
 <a href="complex2.html#K5" target="_self">angle</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="sin_cos.html#K33">PI</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:89_1"/>
not  <a href="complex2.html#K5" target="_self">angle</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="sin_cos.html#K33">PI</a> 
 ;<br/>

<a NAME="E2:89_1"/><b>then </b>
<span class="p1">(<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="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>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <a href="sin_cos.html#K33">PI</a>  <a href="real_1.html#K3">+</a> <a href="sin_cos.html#K33">PI</a> 
 
<b>by </b><i><a class="txt" href="complex2.html#E3:89"><i><font color="Green">E84</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<b>hence </b><a NAME="E3:89_1"/>
contradiction
 <b>by </b><i><a class="txt" href="complex2.html#E3:89"><i><font color="Green">E84</font></i></a>, <a class="ref" href="complex2.html#T94" target="_self">Th94</a>, <a class="ref" href="comptrig.html#T21">COMPTRIG:21</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:89"/><b>then </b>
 <a href="complex2.html#K5" target="_self">angle</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="complex2.html#E1:89"><i><font color="Green">E82</font></i></a>, <a class="txt" href="complex2.html#E2:89"><i><font color="Green">E83</font></i></a>, <a class="txt" href="complex2.html#E6:89"><i><font color="Green">E87</font></i></a>, <a class="txt" href="complex2.html#E7:89"><i><font color="Green">E88</font></i></a>, <a class="ref" href="complex2.html#T98" target="_self">Th98</a></i>;<br/>
<a NAME="E10:89"/><b>then </b>
<span class="p1">(<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> <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>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="real_1.html#K3">+</a> 0
 
<b>by </b><i><a class="ref" href="complex2.html#T94" target="_self">Th94</a></i>;<br/>
<a NAME="E11:89"/><b>then </b><i><font color="Green">E90</font></i>: 
<span class="p1">(<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> <a href="xcmplx_0.html#K6">-</a> 0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<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>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 
;<br/>
<a NAME="E12:89"/>
 <a href="complex2.html#K5" target="_self">angle</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="complex2.html#E1:89"><i><font color="Green">E82</font></i></a>, <a class="txt" href="complex2.html#E2:89"><i><font color="Green">E83</font></i></a>, <a class="txt" href="complex2.html#E6:89"><i><font color="Green">E87</font></i></a>, <a class="txt" href="complex2.html#E7:89"><i><font color="Green">E88</font></i></a>, <a class="txt" href="complex2.html#E8:89"><i><font color="Green">E89</font></i></a>, <a class="ref" href="complex2.html#T98" target="_self">Th98</a></i>;<br/>
<a NAME="E13:89"/><b>then </b>
<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="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>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="real_1.html#K3">+</a> 0
 
<b>by </b><i><a class="ref" href="complex2.html#T94" target="_self">Th94</a></i>;<br/>
<a NAME="E14:89"/><b>then </b>
<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="xcmplx_0.html#K6">-</a> 0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<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>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
 
;<br/>
<b>then </b><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> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<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>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <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>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="xcmplx_0.html#K6">-</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>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="complex2.html#E5:89"><i><font color="Green">E86</font></i></a>, <a class="txt" href="complex2.html#E11:89"><i><font color="Green">E90</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<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>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></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>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>

;<br/>
<b>hence </b><i><font color="Green">E91</font></i>: <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> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="real_1.html#K3">+</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="real_1.html#K5">-</a> <a href="sin_cos.html#K33">PI</a> 
<b>by </b><i><a class="txt" href="complex2.html#E1:89"><i><font color="Green">E82</font></i></a>, <a class="txt" href="complex2.html#E2:89"><i><font color="Green">E83</font></i></a>, <a class="txt" href="complex2.html#E6:89"><i><font color="Green">E87</font></i></a>, <a class="txt" href="complex2.html#E7:89"><i><font color="Green">E88</font></i></a>, <a class="txt" href="complex2.html#E8:89"><i><font color="Green">E89</font></i></a>, <a class="ref" href="complex2.html#T98" target="_self">Th98</a></i>
<br/>.= 
5 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> 

;<br/><br/>

<a NAME="E17:89"/><i><font color="Green">E92</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="real_1.html#K3">+</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="hidden.html#R1">=</a> 5 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> 
 
;<br/>
<a NAME="E18:89"/><i><font color="Green">E93</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <a href="sin_cos.html#K33">PI</a>  <a href="hidden.html#R1">=</a> 5 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> 
 
;<br/>
<a NAME="E19:89"/><i><font color="Green">E94</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> 2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> 
 
<b>by </b><i><a class="ref" href="complex2.html#T84" target="_self">Th84</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:89_4"/><i><font color="Green">E95</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="xxreal_0.html#R1">&lt;=</a>  <a href="sin_cos.html#K33">PI</a> 
 ;<br/><a NAME="E2:89_4"/><i><font color="Green">E96</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="xxreal_0.html#R1">&lt;</a> 2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> 
 <b>by </b><i><a class="ref" href="complex2.html#T84" target="_self">Th84</a></i>;<br/><a NAME="E3:89_4"/>
<span class="p1">(<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="p1">(<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> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="real_1.html#K3">+</a> <a href="sin_cos.html#K33">PI</a> 
 <b>by </b><i><a class="txt" href="complex2.html#E19:89"><i><font color="Green">E94</font></i></a>, <a class="txt" href="complex2.html#E1:89_4"><i><font color="Green">E95</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><b>hence </b><a NAME="E4:89_4"/>
contradiction
 <b>by </b><i><a class="txt" href="complex2.html#E16:89"><i><font color="Green">E91</font></i></a>, <a class="txt" href="complex2.html#E17:89"><i><font color="Green">E92</font></i></a>, <a class="txt" href="complex2.html#E2:89_4"><i><font color="Green">E96</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E21:89"/><i><font color="Green">E95</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="xxreal_0.html#R1">&lt;=</a>  <a href="sin_cos.html#K33">PI</a> 
 ;<br/>

<a NAME="E22:89"/>
 <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="xxreal_0.html#R1">&lt;</a> 2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> 
 
<b>by </b><i><a class="ref" href="complex2.html#T84" target="_self">Th84</a></i>;<br/>
<a NAME="E23:89"/><b>then </b>
<span class="p1">(<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="p1">(<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> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="complex2.html#E19:89"><i><font color="Green">E94</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<b>hence </b><a NAME="E24:89"/>
contradiction
 <b>by </b><i><a class="txt" href="complex2.html#E16:89"><i><font color="Green">E91</font></i></a>, <a class="txt" href="complex2.html#E18:89"><i><font color="Green">E93</font></i></a>, <a class="txt" href="complex2.html#E21:89"><i><font color="Green">E95</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>


</div>
