<?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/>





<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:94_1"/><i><font color="Green">E86</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> <a href="hidden.html#R1">=</a>  <a href="sin_cos.html#K33">PI</a>  or <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> 5 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a>  )
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="94_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:94_1_1"/>
( <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>  or <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> 5 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a>  )
 </span><b>by </b><i><a class="txt" href="complex2.html#E1:94_1"><i><font color="Green">E86</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="94_1_1_1"><b>suppose </b></a><a NAME="E1:94_1_1_1"/><i><font color="Green">E87</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> <a href="hidden.html#R1">=</a>  <a href="sin_cos.html#K33">PI</a> 
 ;<br/><div class="add"><b>thus </b><a NAME="E2:94_1_1_1"/>
( <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> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> &amp; <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> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="94_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:94_1_1_1_1"/><i><font color="Green">E88</font></i>: 
(  not <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> or  not <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> or  not <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> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="94_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:94_1_1_1_1_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> or <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> )
 </span><b>by </b><i><a class="txt" href="complex2.html#E1:94_1_1_1_1"><i><font color="Green">E88</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="94_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:94_1_1_1_1_1_1"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><a NAME="E2:94_1_1_1_1_1_1"/><i><font color="Green">E90</font></i>: 
 <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>1</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="complex2.html#T93" target="_self">Th93</a></i>;<br/><a NAME="E3:94_1_1_1_1_1_1"/>
(  not  <a href="complex2.html#K5" target="_self">angle</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0 or  not  <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>1</sub></font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="txt" href="complex2.html#E1:94_1_1_1"><i><font color="Green">E87</font></i></a>, <a class="txt" href="complex2.html#E1:94_1_1_1_1_1_1"><i><font color="Green">E89</font></i></a>, <a class="ref" href="complex2.html#T93" target="_self">Th93</a>, <a class="ref" href="comptrig.html#T21">COMPTRIG:21</a></i>;<br/><b>hence </b><a NAME="E4:94_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="complex2.html#E1:94_1_1_1"><i><font color="Green">E87</font></i></a>, <a class="txt" href="complex2.html#E1:94_1_1_1_1_1_1"><i><font color="Green">E89</font></i></a>, <a class="txt" href="complex2.html#E2:94_1_1_1_1_1_1"><i><font color="Green">E90</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="94_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:94_1_1_1_1_1_2"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:94_1_1_1_1_1_2"/><i><font color="Green">E90</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>1</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="complex2.html#T93" target="_self">Th93</a></i>;<br/><a NAME="E3:94_1_1_1_1_1_2"/>
(  not  <a href="complex2.html#K5" target="_self">angle</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 or  not  <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>1</sub></font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="txt" href="complex2.html#E1:94_1_1_1"><i><font color="Green">E87</font></i></a>, <a class="txt" href="complex2.html#E1:94_1_1_1_1_1_2"><i><font color="Green">E89</font></i></a>, <a class="ref" href="complex2.html#T93" target="_self">Th93</a>, <a class="ref" href="comptrig.html#T21">COMPTRIG:21</a></i>;<br/><b>hence </b><a NAME="E4:94_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="complex2.html#E1:94_1_1_1"><i><font color="Green">E87</font></i></a>, <a class="txt" href="complex2.html#E1:94_1_1_1_1_1_2"><i><font color="Green">E89</font></i></a>, <a class="txt" href="complex2.html#E2:94_1_1_1_1_1_2"><i><font color="Green">E90</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="94_1_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:94_1_1_1_1_1_3"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:94_1_1_1_1_1_3"/><i><font color="Green">E90</font></i>: 
 <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>2</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="complex2.html#T93" target="_self">Th93</a></i>;<br/><a NAME="E3:94_1_1_1_1_1_3"/>
(  not  <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>2</sub></font> <a href="hidden.html#R1">=</a> 0 or  not  <a href="complex2.html#K5" target="_self">angle</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="txt" href="complex2.html#E1:94_1_1_1"><i><font color="Green">E87</font></i></a>, <a class="txt" href="complex2.html#E1:94_1_1_1_1_1_3"><i><font color="Green">E89</font></i></a>, <a class="ref" href="complex2.html#T93" target="_self">Th93</a>, <a class="ref" href="comptrig.html#T21">COMPTRIG:21</a></i>;<br/><b>hence </b><a NAME="E4:94_1_1_1_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="complex2.html#E1:94_1_1_1"><i><font color="Green">E87</font></i></a>, <a class="txt" href="complex2.html#E1:94_1_1_1_1_1_3"><i><font color="Green">E89</font></i></a>, <a class="txt" href="complex2.html#E2:94_1_1_1_1_1_3"><i><font color="Green">E90</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></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="94_1_1_2"><b>suppose </b></a><a NAME="E1:94_1_1_2"/><i><font color="Green">E87</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> <a href="hidden.html#R1">=</a> 5 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> 
 ;<br/><div class="add"><a NAME="E2:94_1_1_2"/><i><font color="Green">E88</font></i>: 
<span class="p1">(<span class="default">2 <a href="real_1.html#K3">+</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a>  <a href="xxreal_0.html#R1">&lt;</a> 5 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> 
 <b>by </b><i><a class="ref" href="comptrig.html#T21">COMPTRIG:21</a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E3:94_1_1_2"/><b>then </b><i><font color="Green">E89</font></i>: 
<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> <a href="xxreal_0.html#R1">&lt;</a> 5 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> 
 ;<br/><b>thus </b><a NAME="E4:94_1_1_2"/>
( <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> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> &amp; <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> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="94_1_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:94_1_1_2_1"/><i><font color="Green">E90</font></i>: 
(  not <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> or  not <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> or  not <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> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="94_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:94_1_1_2_1_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> )
 </span><b>by </b><i><a class="txt" href="complex2.html#E1:94_1_1_2_1"><i><font color="Green">E90</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="94_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:94_1_1_2_1_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><a NAME="E2:94_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E91</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="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="complex2.html#T93" target="_self">Th93</a></i>;<br/><a NAME="E3:94_1_1_2_1_1_1"/><i><font color="Green">E92</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="E4:94_1_1_2_1_1_1"/>
<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> 2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> 
 <b>by </b><i><a class="txt" href="complex2.html#E2:94_1_1_2_1_1_1"><i><font color="Green">E91</font></i></a>, <a class="ref" href="complex2.html#T84" target="_self">Th84</a></i>;<br/><b>hence </b><a NAME="E5:94_1_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="complex2.html#E1:94_1_1_2"><i><font color="Green">E87</font></i></a>, <a class="txt" href="complex2.html#E3:94_1_1_2"><i><font color="Green">E89</font></i></a>, <a class="txt" href="complex2.html#E3:94_1_1_2_1_1_1"><i><font color="Green">E92</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="94_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:94_1_1_2_1_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:94_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E91</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="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="complex2.html#T93" target="_self">Th93</a></i>;<br/><a NAME="E3:94_1_1_2_1_1_2"/>
(  <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>  &amp;  <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="E4:94_1_1_2_1_1_2"/><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="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><b>hence </b><a NAME="E5:94_1_1_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="complex2.html#E1:94_1_1_2"><i><font color="Green">E87</font></i></a>, <a class="txt" href="complex2.html#E2:94_1_1_2"><i><font color="Green">E88</font></i></a>, <a class="txt" href="complex2.html#E2:94_1_1_2_1_1_2"><i><font color="Green">E91</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="94_1_1_2_1_1_3"><b>suppose </b></a><a NAME="E1:94_1_1_2_1_1_3"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:94_1_1_2_1_1_3"/><b>then </b><i><font color="Green">E91</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> 0
 <b>by </b><i><a class="ref" href="complex2.html#T93" target="_self">Th93</a></i>;<br/><a NAME="E3:94_1_1_2_1_1_3"/><i><font color="Green">E92</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="E4:94_1_1_2_1_1_3"/>
<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> 2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> 
 <b>by </b><i><a class="txt" href="complex2.html#E2:94_1_1_2_1_1_3"><i><font color="Green">E91</font></i></a>, <a class="ref" href="complex2.html#T84" target="_self">Th84</a></i>;<br/><b>hence </b><a NAME="E5:94_1_1_2_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="complex2.html#E1:94_1_1_2"><i><font color="Green">E87</font></i></a>, <a class="txt" href="complex2.html#E3:94_1_1_2"><i><font color="Green">E89</font></i></a>, <a class="txt" href="complex2.html#E3:94_1_1_2_1_1_3"><i><font color="Green">E92</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:94"/><i><font color="Green">E86</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> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> &amp; <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> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="94_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:94_2"/>
(  <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> 0 or ( 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</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> &amp;  <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>  ) or ( 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</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> &amp;  <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="sin_cos.html#K33">PI</a>  ) or ( 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</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> &amp;  <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>  ) )
 </span><b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="94_2_1"><b>suppose </b></a><a NAME="E1:94_2_1"/>
 <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> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:94_2_1"/>
( <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>  or <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> 5 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a>  )
 <b>by </b><i><a class="txt" href="complex2.html#E2:94"><i><font color="Green">E86</font></i></a>, <a class="txt" href="complex2.html#E120"><i><font color="Green">Lemma85</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="94_2_2"><b>suppose </b></a><a NAME="E1:94_2_2"/><i><font color="Green">E87</font></i>: 
( 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</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> &amp;  <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/><div class="add"><a NAME="E2:94_2_2"/>
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>by </b><i><a class="ref" href="complex2.html#T84" target="_self">Th84</a></i>;<br/><b>hence </b><a NAME="E3:94_2_2"/>
( <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>  or <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> 5 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a>  )
 <b>by </b><i><a class="txt" href="complex2.html#E2:94"><i><font color="Green">E86</font></i></a>, <a class="txt" href="complex2.html#E1:94_2_2"><i><font color="Green">E87</font></i></a>, <a class="ref" href="complex2.html#T98" target="_self">Th98</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="94_2_3"><b>suppose </b></a><a NAME="E1:94_2_3"/><i><font color="Green">E87</font></i>: 
( 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</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> &amp;  <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="sin_cos.html#K33">PI</a>  )
 ;<br/><div class="add"><a NAME="E2:94_2_3"/><b>then </b>
(  <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="hidden.html#R1">=</a> 0 &amp;  <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="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="txt" href="complex2.html#E2:94"><i><font color="Green">E86</font></i></a>, <a class="ref" href="complex2.html#T100" target="_self">Th100</a></i>;<br/><b>hence </b><a NAME="E3:94_2_3"/>
( <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>  or <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> 5 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a>  )
 <b>by </b><i><a class="txt" href="complex2.html#E1:94_2_3"><i><font color="Green">E87</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="94_2_4"><b>suppose </b></a><a NAME="E1:94_2_4"/>
( 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</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> &amp;  <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/><div class="add"><b>hence </b><a NAME="E2:94_2_4"/>
( <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>  or <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> 5 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a>  )
 <b>by </b><i><a class="txt" href="complex2.html#E2:94"><i><font color="Green">E86</font></i></a>, <a class="ref" href="complex2.html#T99" target="_self">Th99</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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