<?xml version="1.0"?>
<div class="add">

<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>assume </b><b>that </b><br/><a NAME="E1:74"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>and </b><br/><a NAME="E2:74"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>

<a NAME="E3:74"/><i><font color="Green">E29</font></i>: 
(  <a href="sin_cos.html#K24">cos</a> <span class="p1">(<span class="default"><a href="complex2.html#K3" target="_self">angle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="complex2.html#K1" target="_self">.|.</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">a</font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">b</font></span><a href="complex1.html#K17">.|</a></span></span>)</span> &amp;  <a href="sin_cos.html#K21">sin</a> <span class="p1">(<span class="default"><a href="complex2.html#K3" target="_self">angle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="complex2.html#K1" target="_self">.|.</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">a</font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p3"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">b</font></span><a href="complex1.html#K17">.|</a></span></span>)</span></span>)</span> )
 
<b>by </b><i>, , <a class="ref" href="complex2.html#T20" target="_self">Th20</a></i>;<br/>
<a NAME="E4:74"/><i><font color="Green">E30</font></i>: 
 <a href="complex2.html#K3" target="_self">angle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">a</font>,<a href="complex1.html#K5">0c</a> ,<font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="complex2.html#T25" target="_self">Th25</a></i>;<br/>
<a NAME="E5:74"/>
( <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">a</font></span><a href="complex1.html#K17">.|</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">b</font></span><a href="complex1.html#K17">.|</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 
<b>by </b><i>, , <a class="ref" href="complex1.html#T131">COMPLEX1:131</a></i>;<br/>
<a NAME="E6:74"/><b>then </b><i><font color="Green">E31</font></i>: 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">a</font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">b</font></span><a href="complex1.html#K17">.|</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T6">XCMPLX_1:6</a></i>;<br/>
<a NAME="E7:74"/><i><font color="Green">E32</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">a</font>,<a href="complex1.html#K5">0c</a> ,<font color="Maroon">b</font> &amp;  <a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">a</font>,<a href="complex1.html#K5">0c</a> ,<font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;</a> 2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a>  )
 
<b>by </b><i><a class="ref" href="complex2.html#T22" target="_self">Th22</a></i>;<br/>
<a NAME="E8:74"/><i><font color="Green">E33</font></i>: 
<a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;</a> 2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> 
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="comptrig.html#T21">COMPTRIG:21</a></i>;<br/>
<a NAME="E9:74"/><i><font color="Green">E48</font></i>: 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="complex2.html#K1" target="_self">.|.</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">a</font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p3"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">b</font></span><a href="complex1.html#K17">.|</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="complex2.html#K1" target="_self">.|.</a> <font color="Maroon">b</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">a</font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">b</font></span><a href="complex1.html#K17">.|</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T188">XCMPLX_1:188</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:74_1"/><i><font color="Green">E49</font></i>: 
 <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="complex2.html#K1" target="_self">.|.</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:74_1"/><b>then </b>
(  <a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="complex2.html#K1" target="_self">.|.</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">a</font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">b</font></span><a href="complex1.html#K17">.|</a></span> or  <a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="complex2.html#K1" target="_self">.|.</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">a</font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">b</font></span><a href="complex1.html#K17">.|</a></span></span>)</span> )
 <b>by </b><i><a class="ref" href="complex2.html#T9" target="_self">Th9</a></i>;<br/><a NAME="E3:74_1"/><b>then </b>
(  <a href="sin_cos.html#K21">sin</a> <span class="p1">(<span class="default"><a href="complex2.html#K3" target="_self">angle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> 1 or  <a href="sin_cos.html#K21">sin</a> <span class="p1">(<span class="default"><a href="complex2.html#K3" target="_self">angle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 )
 <b>by </b><i>, , <a class="ref" href="complex2.html#T34" target="_self">Th34</a>, <a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a></i>;<br/><b>hence </b><a NAME="E4:74_1"/>
(  <a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">a</font>,0,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2 or  <a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">a</font>,0,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a>  )
 <b>by </b><i>, <a class="ref" href="complex2.html#T27" target="_self">Th27</a>, , , <a class="ref" href="complex2.html#T35" target="_self">Th35</a>, , <a class="ref" href="comptrig.html#T21">COMPTRIG:21</a>, <a class="ref" href="sin_cos.html#T82">SIN_COS:82</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E11:74"/>
(  <a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">a</font>,0,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2 or  <a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">a</font>,0,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a>  )
 ;<br/>

<a NAME="E12:74"/><b>then </b>
0 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">a</font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">b</font></span><a href="complex1.html#K17">.|</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="complex2.html#K1" target="_self">.|.</a> <font color="Maroon">b</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="complex2.html#T27" target="_self">Th27</a>, , <a class="ref" href="sin_cos.html#T82">SIN_COS:82</a>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/>
<b>hence </b><a NAME="E13:74"/>
 <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="complex2.html#K1" target="_self">.|.</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/>


</div>
