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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">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/>



<b>assume </b><a NAME="E1:52"/>
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>

<a NAME="E2:52"/><b>then </b><i><font color="Green">E23</font></i>: 
 <a href="complex2.html#K2" target="_self">Rotate</a> <font color="Maroon">a</font>,<font color="Maroon">r</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="complex1.html#K5">0c</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E3:52"/><i><font color="Green">E26</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="hidden.html#R1">=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="complex2.html#K2" target="_self">Rotate</a> <font color="Maroon">a</font>,<font color="Maroon">r</font></span>)</span></span><a href="complex1.html#K17">.|</a></span>
 
<b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">AR</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E5:52"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">AR</font> <a href="hidden.html#R1">=</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#K35">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p3"><a href="int_1.html#K1">[\</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <span class="p0">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">a</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E6:52"/><i><font color="Green">E30</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">AR</font>
 <b>and </b><br/><a NAME="E7:52"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">AR</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="comptrig.html#T21">COMPTRIG:21</a></i>;<br/>
<b>reconsider </b><font color="Maroon">ar</font> = <font color="Maroon">AR</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E9:52"/><i><font color="Green">E32</font></i>: 
 <a href="sin_cos.html#K24">cos</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="sin_cos.html#K24">cos</a> <font color="Maroon">ar</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E10:52"/><i><font color="Green">E33</font></i>: 
 <a href="sin_cos.html#K21">sin</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="sin_cos.html#K21">sin</a> <font color="Maroon">ar</font>
 
<b>by </b><i>, </i>;<br/>
<b>take </b><font color="Maroon">i</font> =  <a href="xcmplx_0.html#K4">-</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span>;<br/>

<b>thus </b><a NAME="E11:52"/>
 <a href="comptrig.html#K1">Arg</a> <span class="p1">(<span class="default"><a href="complex2.html#K2" target="_self">Rotate</a> <font color="Maroon">a</font>,<font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</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#K35">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">a</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="complex2.html#T11" target="_self">Th11</a>, <a class="ref" href="complex2.html#T12" target="_self">Th12</a>, , , , , , </i>;<br/>


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