<?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><a NAME="E1:92"/><i><font color="Green">E85</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> &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> 0 )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="92_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:92_1"/>
(  <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#NR2" target="_self">&lt;&gt;</a> 0 or  <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#NR2" target="_self">&lt;&gt;</a> 0 )
 </span><b>by </b><i><a class="txt" href="complex2.html#E1:92"><i><font color="Green">E85</font></i></a>, <a class="ref" href="complex2.html#T97" target="_self">Th97</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="92_1_1"><b>suppose </b></a><a NAME="E1:92_1_1"/><i><font color="Green">E86</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#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:92_1_1"/><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>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="ref" href="complex2.html#T84" target="_self">Th84</a></i>;<br/><b>thus </b><a NAME="E3:92_1_1"/>
( (  <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>  <a href="sin_cos.html#K33">PI</a>  ) or (  <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>  <a href="sin_cos.html#K33">PI</a>  &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 ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="92_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="E1:92_1_1_1_1"/>
(  <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>  or  <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>  <a href="sin_cos.html#K33">PI</a>  or  <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#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="92_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:92_1_1_1_1_1"/>
 <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/><div class="add"><b>hence </b><a NAME="E2:92_1_1_1_1_1"/>
( (  <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>  <a href="sin_cos.html#K33">PI</a>  ) or (  <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>  <a href="sin_cos.html#K33">PI</a>  &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#E1:92"><i><font color="Green">E85</font></i></a>, <a class="txt" href="complex2.html#E1:92_1_1"><i><font color="Green">E86</font></i></a>, <a class="txt" href="complex2.html#E2:92_1_1"><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="92_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:92_1_1_1_1_2"/>
 <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>  <a href="sin_cos.html#K33">PI</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:92_1_1_1_1_2"/>
( (  <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>  <a href="sin_cos.html#K33">PI</a>  ) or (  <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>  <a href="sin_cos.html#K33">PI</a>  &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#E1:92"><i><font color="Green">E85</font></i></a>, <a class="ref" href="complex2.html#T100" target="_self">Th100</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="92_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:92_1_1_1_1_3"/>
 <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#NR4" target="_self">&gt;</a>  <a href="sin_cos.html#K33">PI</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:92_1_1_1_1_3"/>
( (  <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>  <a href="sin_cos.html#K33">PI</a>  ) or (  <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>  <a href="sin_cos.html#K33">PI</a>  &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#E1:92"><i><font color="Green">E85</font></i></a>, <a class="ref" href="complex2.html#T99" target="_self">Th99</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="92_1_2"><b>suppose </b></a><a NAME="E1:92_1_2"/><i><font color="Green">E86</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#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:92_1_2"/><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>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="complex2.html#T84" target="_self">Th84</a></i>;<br/><b>thus </b><a NAME="E3:92_1_2"/>
( (  <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>  <a href="sin_cos.html#K33">PI</a>  ) or (  <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>  <a href="sin_cos.html#K33">PI</a>  &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 ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="92_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="E1:92_1_2_1_1"/>
(  <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>  or  <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>  <a href="sin_cos.html#K33">PI</a>  or  <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#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="92_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:92_1_2_1_1_1"/>
 <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/><div class="add"><b>hence </b><a NAME="E2:92_1_2_1_1_1"/>
( (  <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>  <a href="sin_cos.html#K33">PI</a>  ) or (  <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>  <a href="sin_cos.html#K33">PI</a>  &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#E1:92"><i><font color="Green">E85</font></i></a>, <a class="txt" href="complex2.html#E1:92_1_2"><i><font color="Green">E86</font></i></a>, <a class="txt" href="complex2.html#E2:92_1_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="92_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:92_1_2_1_1_2"/>
 <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>  <a href="sin_cos.html#K33">PI</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:92_1_2_1_1_2"/>
( (  <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>  <a href="sin_cos.html#K33">PI</a>  ) or (  <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>  <a href="sin_cos.html#K33">PI</a>  &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#E1:92"><i><font color="Green">E85</font></i></a>, <a class="ref" href="complex2.html#T100" target="_self">Th100</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="92_1_2_1_1_3"><b>suppose </b></a><a NAME="E1:92_1_2_1_1_3"/>
 <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#NR4" target="_self">&gt;</a>  <a href="sin_cos.html#K33">PI</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:92_1_2_1_1_3"/>
( (  <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>  <a href="sin_cos.html#K33">PI</a>  ) or (  <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>  <a href="sin_cos.html#K33">PI</a>  &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#E1:92"><i><font color="Green">E85</font></i></a>, <a class="ref" href="complex2.html#T99" target="_self">Th99</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><b>end;</b></div>

</div>
