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<b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a> ;<br/><b>let </b><font color="Maroon">b</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a> ;<br/><b>let </b><font color="Maroon">c</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a> ;<br/>





<b>assume </b><a NAME="E1:90"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font> &amp;  <a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:90_1"/>
(  <a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 or  <a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">c</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span><b>by </b><i><a class="ref" href="complex2.html#T39" target="_self">Th39</a>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_1_1"><b>suppose </b></a><a NAME="E1:90_1_1"/><i><font color="Green">E26</font></i>: 
 <a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:90_1_1"/><i><font color="Green">E29</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="complex2.html#T22" target="_self">Th22</a></i>;<br/><b>thus </b><a NAME="E3:90_1_1"/>
( (  <a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0 &amp;  <a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">c</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="sin_cos.html#K35">PI</a>  ) or (  <a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="sin_cos.html#K35">PI</a>  &amp;  <a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">c</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0 ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="90_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_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="E1:90_1_1_1_1"/>
(  <a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="sin_cos.html#K35">PI</a>  or  <a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="sin_cos.html#K35">PI</a>  or  <a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="sin_cos.html#K35">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="90_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:90_1_1_1_1_1"/>
 <a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="sin_cos.html#K35">PI</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:90_1_1_1_1_1"/>
( (  <a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0 &amp;  <a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">c</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="sin_cos.html#K35">PI</a>  ) or (  <a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="sin_cos.html#K35">PI</a>  &amp;  <a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">c</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0 ) )
 <b>by </b><i><a class="ref" href="complex2.html#T39" target="_self">Th39</a>, <a class="ref" href="complex2.html#T40" target="_self">Th40</a>, <a class="ref" href="complex2.html#T41" target="_self">Th41</a>, <a class="ref" href="complex2.html#T34" target="_self">Th34</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:90_1_1_1_1_2"/>
 <a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="sin_cos.html#K35">PI</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:90_1_1_1_1_2"/>
( (  <a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0 &amp;  <a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">c</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="sin_cos.html#K35">PI</a>  ) or (  <a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="sin_cos.html#K35">PI</a>  &amp;  <a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">c</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0 ) )
 <b>by </b><i><a class="ref" href="complex2.html#T39" target="_self">Th39</a>, <a class="ref" href="complex2.html#T37" target="_self">Th37</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:90_1_1_1_1_3"/>
 <a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="sin_cos.html#K35">PI</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:90_1_1_1_1_3"/>
( (  <a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0 &amp;  <a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">c</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="sin_cos.html#K35">PI</a>  ) or (  <a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="sin_cos.html#K35">PI</a>  &amp;  <a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">c</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0 ) )
 <b>by </b><i><a class="ref" href="complex2.html#T39" target="_self">Th39</a>, <a class="ref" href="complex2.html#T35" target="_self">Th35</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="90_1_2"><b>suppose </b></a><a NAME="E1:90_1_2"/><i><font color="Green">E30</font></i>: 
 <a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">c</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:90_1_2"/><i><font color="Green">E31</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">c</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="complex2.html#T22" target="_self">Th22</a></i>;<br/><b>thus </b><a NAME="E3:90_1_2"/>
( (  <a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0 &amp;  <a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">c</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="sin_cos.html#K35">PI</a>  ) or (  <a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="sin_cos.html#K35">PI</a>  &amp;  <a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">c</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0 ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="90_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_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="E1:90_1_2_1_1"/>
(  <a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">c</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="sin_cos.html#K35">PI</a>  or  <a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">c</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="sin_cos.html#K35">PI</a>  or  <a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">c</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="sin_cos.html#K35">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="90_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:90_1_2_1_1_1"/>
 <a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">c</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="sin_cos.html#K35">PI</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:90_1_2_1_1_1"/>
( (  <a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0 &amp;  <a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">c</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="sin_cos.html#K35">PI</a>  ) or (  <a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="sin_cos.html#K35">PI</a>  &amp;  <a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">c</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0 ) )
 <b>by </b><i><a class="ref" href="complex2.html#T39" target="_self">Th39</a>, <a class="ref" href="complex2.html#T40" target="_self">Th40</a>, <a class="ref" href="complex2.html#T41" target="_self">Th41</a>, <a class="ref" href="complex2.html#T34" target="_self">Th34</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:90_1_2_1_1_2"/>
 <a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">c</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="sin_cos.html#K35">PI</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:90_1_2_1_1_2"/>
( (  <a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0 &amp;  <a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">c</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="sin_cos.html#K35">PI</a>  ) or (  <a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="sin_cos.html#K35">PI</a>  &amp;  <a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">c</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0 ) )
 <b>by </b><i><a class="ref" href="complex2.html#T39" target="_self">Th39</a>, <a class="ref" href="complex2.html#T37" target="_self">Th37</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_1_2_1_1_3"><b>suppose </b></a><a NAME="E1:90_1_2_1_1_3"/>
 <a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">c</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="sin_cos.html#K35">PI</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:90_1_2_1_1_3"/>
( (  <a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0 &amp;  <a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">c</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="sin_cos.html#K35">PI</a>  ) or (  <a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="sin_cos.html#K35">PI</a>  &amp;  <a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">c</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0 ) )
 <b>by </b><i><a class="ref" href="complex2.html#T39" target="_self">Th39</a>, <a class="ref" href="complex2.html#T35" target="_self">Th35</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>
