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<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">a</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="E1:66"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>and </b><br/><a NAME="E2:66"/><i><font color="Green">E26</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> 0
 ;<br/>

<a NAME="E3:66"/><i><font color="Green">E29</font></i>: 
 <a href="complex1.html#K10">-</a> <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="complex2.html#T20" target="_self">Th20</a></i>;<br/>
<a NAME="E4:66"/><i><font color="Green">E30</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> 0
 
<b>by </b><i><a class="ref" href="complex2.html#T20" target="_self">Th20</a>, <a class="ref" href="complex2.html#T21" target="_self">Th21</a>, </i>;<br/>
<a NAME="E5:66"/><i><font color="Green">E31</font></i>: 
 <a href="complex2.html#K2" target="_self">Rotate</a> <font color="Maroon">b</font>,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</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#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="complex2.html#T20" target="_self">Th20</a>, </i>;<br/>
<a NAME="E6:66"/><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> <span class="p2">(<span class="default"><a href="complex1.html#K10">-</a> <font color="Maroon">b</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> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="comptrig.html#K1">Arg</a> <span class="p1">(<span class="default"><a href="complex1.html#K10">-</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/>;<br/>
 <a href="comptrig.html#K1">Arg</a> <span class="p1">(<span class="default"><a href="complex1.html#K10">-</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> = 
<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> <a href="real_1.html#K3">+</a> <a href="sin_cos.html#K35">PI</a> 
<b>by </b><i><a class="ref" href="complex2.html#T23" target="_self">Th23</a>, <a class="ref" href="complex2.html#T24" target="_self">Th24</a>, , <a class="ref" href="comptrig.html#T21">COMPTRIG:21</a></i>
<br/>.= 
 <a href="sin_cos.html#K35">PI</a> 
<b>by </b><i><a class="ref" href="complex2.html#T23" target="_self">Th23</a></i>
;<br/>
<b>hence </b><a NAME="E8:66"/>
 <a href="complex2.html#K3" target="_self">angle</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><a href="complex1.html#K10">-</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="sin_cos.html#K35">PI</a> 
 <b>by </b><i><a class="ref" href="complex2.html#T22" target="_self">Th22</a>, <a class="ref" href="complex2.html#T25" target="_self">Th25</a>, </i>;<br/>


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