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





<b>assume </b><b>that </b><br/><a NAME="E1:63"/><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="E2:63"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>

<b>consider </b><font color="Maroon">i</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E4:63"/><i><font color="Green">E29</font></i>: 
 <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">b</font>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">a</font></span>)</span></span>)</span></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"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">b</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="complex2.html#T21" target="_self">Th21</a>, <a class="ref" href="complex2.html#T10" target="_self">Th10</a></i>;<br/>
<a NAME="E5:63"/><i><font color="Green">E30</font></i>: 
 <a href="complex2.html#K2" target="_self">Rotate</a> <font color="Maroon">b</font>,<font color="Maroon">r</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="complex2.html#T21" target="_self">Th21</a>, </i>;<br/>
<b>then consider </b><font color="Maroon">j</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E7:63"/><i><font color="Green">E31</font></i>: 
 <a href="comptrig.html#K1">Arg</a> <span class="p1">(<span class="default"><a href="complex2.html#K2" target="_self">Rotate</a> <span class="p2">(<span class="default"><a href="complex2.html#K2" target="_self">Rotate</a> <font color="Maroon">b</font>,<font color="Maroon">r</font></span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p4">(<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>)</span></span>)</span></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">j</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p4">(<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>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p3">(<span class="default"><a href="complex2.html#K2" target="_self">Rotate</a> <font color="Maroon">b</font>,<font color="Maroon">r</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="complex2.html#T10" target="_self">Th10</a></i>;<br/>
<b>consider </b><font color="Maroon">k</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E9:63"/><i><font color="Green">E32</font></i>: 
 <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">k</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#T20" target="_self">Th20</a>, <a class="ref" href="complex2.html#T10" target="_self">Th10</a></i>;<br/>
<b>consider </b><font color="Maroon">l</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E11:63"/><i><font color="Green">E33</font></i>: 
 <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">b</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">l</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">b</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="complex2.html#T21" target="_self">Th21</a>, <a class="ref" href="complex2.html#T10" target="_self">Th10</a></i>;<br/>
<a NAME="E12:63"/><i><font color="Green">E48</font></i>: 
 <a href="comptrig.html#K1">Arg</a> <span class="p1">(<span class="default"><a href="complex2.html#K2" target="_self">Rotate</a> <span class="p2">(<span class="default"><a href="complex2.html#K2" target="_self">Rotate</a> <font color="Maroon">b</font>,<font color="Maroon">r</font></span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p4">(<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>)</span></span>)</span></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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">l</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><a href="complex2.html#K2" target="_self">Rotate</a> <font color="Maroon">b</font>,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p4">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="complex2.html#T22" target="_self">Th22</a>, <a class="ref" href="complex2.html#T24" target="_self">Th24</a>, <a class="ref" href="complex2.html#T25" target="_self">Th25</a>, <a class="ref" href="complex2.html#T26" target="_self">Th26</a></i>;<br/>
<a NAME="E13:63"/><i><font color="Green">E49</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a>  <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">b</font>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> &amp;  <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">b</font>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> 2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a>  &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="comptrig.html#K1">Arg</a> <span class="p1">(<span class="default"><a href="complex2.html#K2" target="_self">Rotate</a> <span class="p2">(<span class="default"><a href="complex2.html#K2" target="_self">Rotate</a> <font color="Maroon">b</font>,<font color="Maroon">r</font></span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p4">(<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>)</span></span>)</span></span>)</span> &amp;  <a href="comptrig.html#K1">Arg</a> <span class="p1">(<span class="default"><a href="complex2.html#K2" target="_self">Rotate</a> <span class="p2">(<span class="default"><a href="complex2.html#K2" target="_self">Rotate</a> <font color="Maroon">b</font>,<font color="Maroon">r</font></span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p4">(<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>)</span></span>)</span></span>)</span> <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#T52">COMPTRIG:52</a></i>;<br/>
<b>thus </b> <a href="complex2.html#K3" target="_self">angle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> = 
 <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">b</font>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complex2.html#T21" target="_self">Th21</a>, </i>
<br/>.= 
 <a href="comptrig.html#K1">Arg</a> <span class="p1">(<span class="default"><a href="complex2.html#K2" target="_self">Rotate</a> <span class="p2">(<span class="default"><a href="complex2.html#K2" target="_self">Rotate</a> <font color="Maroon">b</font>,<font color="Maroon">r</font></span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p4">(<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>)</span></span>)</span></span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
 <a href="complex2.html#K3" target="_self">angle</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>,<span class="p1">(<span class="default"><a href="complex2.html#K2" target="_self">Rotate</a> <font color="Maroon">b</font>,<font color="Maroon">r</font></span>)</span>
<b>by </b><i><a class="ref" href="complex2.html#T23" target="_self">Th23</a>, </i>
;<br/>


</div>
