<?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>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/>





<b>assume </b><a NAME="E1:61"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<a NAME="E2:61"/><i><font color="Green">E26</font></i>: 
 <a href="comptrig.html#K1">Arg</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="complex1.html#K9">*</a> <span class="p2"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">r</font>,0</span><a href="arytm_0.html#K5">*]</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="comptrig.html#K1">Arg</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="complex2.html#T19" target="_self">Th19</a>, <a class="ref" href="complex2.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E3:61"/><i><font color="Green">E29</font></i>: 
 <a href="comptrig.html#K1">Arg</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="complex1.html#K9">*</a> <span class="p2"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">r</font>,0</span><a href="arytm_0.html#K5">*]</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="comptrig.html#K1">Arg</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="complex2.html#T19" target="_self">Th19</a>, <a class="ref" href="complex2.html#T3" target="_self">Th3</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:61_1"/>
( <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 or <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_1"><b>suppose </b></a><a NAME="E1:61_1_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:61_1_1"/>
<span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">r</font>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">0 <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span>
 <b>by </b><i><a class="ref" href="complex1.html#T27">COMPLEX1:27</a></i>;<br/><a NAME="E3:61_1_1"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">b</font> <a href="complex1.html#K9">*</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">r</font>,0</span><a href="arytm_0.html#K5">*]</a></span> <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>, <a class="ref" href="xcmplx_1.html#T6">XCMPLX_1:6</a>, <a class="ref" href="complex2.html#T19" target="_self">Th19</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"><font color="Maroon">b</font> <a href="complex1.html#K9">*</a> <span class="p3"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">r</font>,0</span><a href="arytm_0.html#K5">*]</a></span></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"><font color="Maroon">a</font> <a href="complex1.html#K9">*</a> <span class="p5"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">r</font>,0</span><a href="arytm_0.html#K5">*]</a></span></span>)</span></span>)</span></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#T21" target="_self">Th21</a>, <a class="ref" href="complex2.html#T22" target="_self">Th22</a>, </i>
<br/>.= 
 <a href="complex2.html#K3" target="_self">angle</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="complex1.html#K9">*</a> <span class="p2"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">r</font>,0</span><a href="arytm_0.html#K5">*]</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="complex1.html#K9">*</a> <span class="p2"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">r</font>,0</span><a href="arytm_0.html#K5">*]</a></span></span>)</span>
<b>by </b><i><a class="ref" href="complex2.html#T22" target="_self">Th22</a>, </i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_2"><b>suppose </b></a><a NAME="E1:61_1_2"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>thus </b><a NAME="E2:61_1_2"/>
 <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#K3" target="_self">angle</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="complex1.html#K9">*</a> <span class="p2"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">r</font>,0</span><a href="arytm_0.html#K5">*]</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="complex1.html#K9">*</a> <span class="p2"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">r</font>,0</span><a href="arytm_0.html#K5">*]</a></span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_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="E1:61_1_2_1_1"/>
(  <a href="comptrig.html#K1">Arg</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0 or  <a href="comptrig.html#K1">Arg</a> <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:61_1_2_1_1_1"/><i><font color="Green">E33</font></i>: 
 <a href="comptrig.html#K1">Arg</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </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/>
<br/>.= 
 <a href="comptrig.html#K1">Arg</a> <a href="complex1.html#K5">0c</a> 
<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"><font color="Maroon">b</font> <a href="complex1.html#K9">*</a> <span class="p3"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">r</font>,0</span><a href="arytm_0.html#K5">*]</a></span></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"><font color="Maroon">a</font> <a href="complex1.html#K9">*</a> <span class="p5"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">r</font>,0</span><a href="arytm_0.html#K5">*]</a></span></span>)</span></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="complex2.html#K3" target="_self">angle</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="complex1.html#K9">*</a> <span class="p2"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">r</font>,0</span><a href="arytm_0.html#K5">*]</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="complex1.html#K9">*</a> <span class="p2"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">r</font>,0</span><a href="arytm_0.html#K5">*]</a></span></span>)</span>
<b>by </b><i><a class="ref" href="complex2.html#T20" target="_self">Th20</a>, <a class="ref" href="complex2.html#T22" target="_self">Th22</a>, </i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:61_1_2_1_1_2"/><i><font color="Green">E48</font></i>: 
 <a href="comptrig.html#K1">Arg</a> <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><b>hence </b> <a href="complex2.html#K3" target="_self">angle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> = 
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="ref" href="complex2.html#T21" target="_self">Th21</a>, </i>
<br/>.= 
 <a href="complex2.html#K3" target="_self">angle</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="complex1.html#K9">*</a> <span class="p2"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">r</font>,0</span><a href="arytm_0.html#K5">*]</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="complex1.html#K9">*</a> <span class="p2"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">r</font>,0</span><a href="arytm_0.html#K5">*]</a></span></span>)</span>
<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>, <a class="ref" href="complex2.html#T22" target="_self">Th22</a>, </i>
;<br/><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>
