<?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">c</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>set </b><font color="Maroon">cb</font> = <font color="Maroon">c</font> <a href="complex1.html#K11">-</a> <font color="Maroon">b</font>;<br/>
<b>set </b><font color="Maroon">ab</font> = <font color="Maroon">a</font> <a href="complex1.html#K11">-</a> <font color="Maroon">b</font>;<br/>
<b>set </b><font color="Maroon">rc</font> =  <a href="complex2.html#K2" target="_self">Rotate</a> <font color="Maroon">c</font>,<font color="Maroon">r</font>;<br/>
<b>set </b><font color="Maroon">rb</font> =  <a href="complex2.html#K2" target="_self">Rotate</a> <font color="Maroon">b</font>,<font color="Maroon">r</font>;<br/>
<b>set </b><font color="Maroon">ra</font> =  <a href="complex2.html#K2" target="_self">Rotate</a> <font color="Maroon">a</font>,<font color="Maroon">r</font>;<br/>
<b>set </b><font color="Maroon">rcb</font> = <span class="p1">(<span class="default"><a href="complex2.html#K2" target="_self">Rotate</a> <font color="Maroon">c</font>,<font color="Maroon">r</font></span>)</span> <a href="complex1.html#K11">-</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>;<br/>
<b>set </b><font color="Maroon">rab</font> = <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="complex1.html#K11">-</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>;<br/>
<a NAME="E1:78"/><i><font color="Green">E23</font></i>: 
<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="complex1.html#K11">-</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>  <a href="complex2.html#K2" target="_self">Rotate</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="complex1.html#K11">-</a> <font color="Maroon">b</font></span>)</span>,<font color="Maroon">r</font>
 
<b>by </b><i/>;<br/>
<b>assume </b><a NAME="E2:78"/>
( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font> )
 ;<br/>

<a NAME="E3:78"/><b>then </b><i><font color="Green">E26</font></i>: 
( <font color="Maroon">a</font> <a href="complex1.html#K11">-</a> <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">c</font> <a href="complex1.html#K11">-</a> <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 
;<br/>
<b>then consider </b><font color="Maroon">abi</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E5:78"/><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> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="complex1.html#K11">-</a> <font color="Maroon">b</font></span>)</span>,<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">abi</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> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="complex1.html#K11">-</a> <font color="Maroon">b</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">cbi</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E7:78"/><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> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="complex1.html#K11">-</a> <font color="Maroon">b</font></span>)</span>,<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">cbi</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> <span class="p3">(<span class="default"><font color="Maroon">c</font> <a href="complex1.html#K11">-</a> <font color="Maroon">b</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="complex2.html#T27" target="_self">Th27</a>, <a class="ref" href="complex2.html#T10" target="_self">Th10</a></i>;<br/>
<a NAME="E8:78"/><i><font color="Green">E31</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>,<font color="Maroon">b</font>,<font color="Maroon">c</font> &amp;  <a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</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="E9:78"/><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> <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>,<span class="p1">(<span class="default"><a href="complex2.html#K2" target="_self">Rotate</a> <font color="Maroon">c</font>,<font color="Maroon">r</font></span>)</span> &amp;  <a href="complex2.html#K4" 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>,<span class="p1">(<span class="default"><a href="complex2.html#K2" target="_self">Rotate</a> <font color="Maroon">c</font>,<font color="Maroon">r</font></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="complex2.html#T22" target="_self">Th22</a></i>;<br/>
<i><font color="Green">E33</font></i>: <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex2.html#K2" target="_self">Rotate</a> <font color="Maroon">c</font>,<font color="Maroon">r</font></span>)</span> <a href="complex1.html#K11">-</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> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><span class="p3">(<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="complex1.html#K11">-</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> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<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">cbi</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">r</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p3">(<span class="default"><font color="Maroon">c</font> <a href="complex1.html#K11">-</a> <font color="Maroon">b</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<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">abi</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">r</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="complex1.html#K11">-</a> <font color="Maroon">b</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, , , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p3">(<span class="default"><font color="Maroon">c</font> <a href="complex1.html#K11">-</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="complex1.html#K11">-</a> <font color="Maroon">b</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</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"><font color="Maroon">cbi</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">abi</font></span>)</span></span>)</span>

;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:78_2"/>
( <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="complex1.html#K11">-</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="complex1.html#K11">-</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 or <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="complex1.html#K11">-</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="complex1.html#K11">-</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_2_1"><b>suppose </b></a><a NAME="E1:78_2_1"/>
<span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="complex1.html#K11">-</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="complex1.html#K11">-</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/><div class="add"><a NAME="E2:78_2_1"/><b>then </b><i><font color="Green">E48</font></i>: 
 <a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="complex1.html#K11">-</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="complex1.html#K11">-</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></i>;<br/><b>thus </b><a NAME="E3:78_2_1"/>
 <a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="complex2.html#K4" 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>,<span class="p1">(<span class="default"><a href="complex2.html#K2" target="_self">Rotate</a> <font color="Maroon">c</font>,<font color="Maroon">r</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_2_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:78_2_1_1_1"/>
( <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex2.html#K2" target="_self">Rotate</a> <font color="Maroon">c</font>,<font color="Maroon">r</font></span>)</span> <a href="complex1.html#K11">-</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> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><span class="p3">(<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="complex1.html#K11">-</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> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 or <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex2.html#K2" target="_self">Rotate</a> <font color="Maroon">c</font>,<font color="Maroon">r</font></span>)</span> <a href="complex1.html#K11">-</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> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><span class="p3">(<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="complex1.html#K11">-</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> <a href="xxreal_0.html#R1">&lt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:78_2_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex2.html#K2" target="_self">Rotate</a> <font color="Maroon">c</font>,<font color="Maroon">r</font></span>)</span> <a href="complex1.html#K11">-</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> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><span class="p3">(<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="complex1.html#K11">-</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> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/><div class="add"><a NAME="E2:78_2_1_1_1_1"/><b>then </b>
 <a href="complex2.html#K4" 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>,<span class="p1">(<span class="default"><a href="complex2.html#K2" target="_self">Rotate</a> <font color="Maroon">c</font>,<font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex2.html#K2" target="_self">Rotate</a> <font color="Maroon">c</font>,<font color="Maroon">r</font></span>)</span> <a href="complex1.html#K11">-</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> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><span class="p3">(<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="complex1.html#K11">-</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#T21" target="_self">Th21</a></i>;<br/><b>hence </b><a NAME="E3:78_2_1_1_1_1"/>
 <a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="complex2.html#K4" 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>,<span class="p1">(<span class="default"><a href="complex2.html#K2" target="_self">Rotate</a> <font color="Maroon">c</font>,<font color="Maroon">r</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="complex2.html#T34" target="_self">Th34</a>, <a class="ref" href="complex2.html#T35" target="_self">Th35</a>, <a class="ref" href="complex2.html#T36" target="_self">Th36</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:78_2_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex2.html#K2" target="_self">Rotate</a> <font color="Maroon">c</font>,<font color="Maroon">r</font></span>)</span> <a href="complex1.html#K11">-</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> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><span class="p3">(<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="complex1.html#K11">-</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> <a href="xxreal_0.html#R1">&lt;</a> 0
 ;<br/><div class="add"><b>then </b> <a href="complex2.html#K4" 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>,<span class="p1">(<span class="default"><a href="complex2.html#K2" target="_self">Rotate</a> <font color="Maroon">c</font>,<font color="Maroon">r</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p4">(<span class="default"><font color="Maroon">c</font> <a href="complex1.html#K11">-</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</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#K11">-</a> <font color="Maroon">b</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<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="p3">(<span class="default"><font color="Maroon">cbi</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">abi</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span>
<b>by </b><i><a class="ref" href="complex2.html#T35" target="_self">Th35</a>, <a class="ref" href="complex2.html#T21" target="_self">Th21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p3">(<span class="default"><font color="Maroon">c</font> <a href="complex1.html#K11">-</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="complex1.html#K11">-</a> <font color="Maroon">b</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</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"><font color="Maroon">cbi</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">abi</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span>

;<br/><b>hence </b><a NAME="E3:78_2_1_1_1_2"/>
 <a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="complex2.html#K4" 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>,<span class="p1">(<span class="default"><a href="complex2.html#K2" target="_self">Rotate</a> <font color="Maroon">c</font>,<font color="Maroon">r</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="complex2.html#T34" target="_self">Th34</a>, <a class="ref" href="complex2.html#T36" target="_self">Th36</a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_2_2"><b>suppose </b></a><a NAME="E1:78_2_2"/>
<span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="complex1.html#K11">-</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="complex1.html#K11">-</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:78_2_2"/><b>then </b><i><font color="Green">E49</font></i>: 
 <a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <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#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p3">(<span class="default"><font color="Maroon">c</font> <a href="complex1.html#K11">-</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="complex1.html#K11">-</a> <font color="Maroon">b</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="complex2.html#T21" target="_self">Th21</a></i>;<br/><b>thus </b><a NAME="E3:78_2_2"/>
 <a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="complex2.html#K4" 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>,<span class="p1">(<span class="default"><a href="complex2.html#K2" target="_self">Rotate</a> <font color="Maroon">c</font>,<font color="Maroon">r</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_2_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_2_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:78_2_2_1_1"/>
( <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex2.html#K2" target="_self">Rotate</a> <font color="Maroon">c</font>,<font color="Maroon">r</font></span>)</span> <a href="complex1.html#K11">-</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> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><span class="p3">(<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="complex1.html#K11">-</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> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 or <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex2.html#K2" target="_self">Rotate</a> <font color="Maroon">c</font>,<font color="Maroon">r</font></span>)</span> <a href="complex1.html#K11">-</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> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><span class="p3">(<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="complex1.html#K11">-</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> <a href="xxreal_0.html#R1">&lt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_2_2_1_1_1"><b>suppose </b></a><a NAME="E1:78_2_2_1_1_1"/>
<span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex2.html#K2" target="_self">Rotate</a> <font color="Maroon">c</font>,<font color="Maroon">r</font></span>)</span> <a href="complex1.html#K11">-</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> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><span class="p3">(<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="complex1.html#K11">-</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> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/><div class="add"><a NAME="E2:78_2_2_1_1_1"/><b>then </b>
 <a href="complex2.html#K4" 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>,<span class="p1">(<span class="default"><a href="complex2.html#K2" target="_self">Rotate</a> <font color="Maroon">c</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="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p4">(<span class="default"><font color="Maroon">c</font> <a href="complex1.html#K11">-</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</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#K11">-</a> <font color="Maroon">b</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</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"><font color="Maroon">cbi</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">abi</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="complex2.html#T35" target="_self">Th35</a>, <a class="ref" href="complex2.html#T21" target="_self">Th21</a></i>;<br/><b>hence </b><a NAME="E3:78_2_2_1_1_1"/>
 <a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="complex2.html#K4" 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>,<span class="p1">(<span class="default"><a href="complex2.html#K2" target="_self">Rotate</a> <font color="Maroon">c</font>,<font color="Maroon">r</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="complex2.html#T34" target="_self">Th34</a>, <a class="ref" href="complex2.html#T36" target="_self">Th36</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_2_2_1_1_2"><b>suppose </b></a><a NAME="E1:78_2_2_1_1_2"/>
<span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex2.html#K2" target="_self">Rotate</a> <font color="Maroon">c</font>,<font color="Maroon">r</font></span>)</span> <a href="complex1.html#K11">-</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> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><span class="p3">(<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="complex1.html#K11">-</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> <a href="xxreal_0.html#R1">&lt;</a> 0
 ;<br/><div class="add"><b>then </b> <a href="complex2.html#K4" 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>,<span class="p1">(<span class="default"><a href="complex2.html#K2" target="_self">Rotate</a> <font color="Maroon">c</font>,<font color="Maroon">r</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p4">(<span class="default"><font color="Maroon">c</font> <a href="complex1.html#K11">-</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</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#K11">-</a> <font color="Maroon">b</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<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="p3">(<span class="default"><font color="Maroon">cbi</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">abi</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span>
<b>by </b><i><a class="ref" href="complex2.html#T35" target="_self">Th35</a>, <a class="ref" href="complex2.html#T21" target="_self">Th21</a></i>
<br/>.= 
<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="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p4">(<span class="default"><font color="Maroon">c</font> <a href="complex1.html#K11">-</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</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#K11">-</a> <font color="Maroon">b</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</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"><font color="Maroon">cbi</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">abi</font></span>)</span></span>)</span>

;<br/><b>hence </b><a NAME="E3:78_2_2_1_1_2"/>
 <a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="complex2.html#K4" 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>,<span class="p1">(<span class="default"><a href="complex2.html#K2" target="_self">Rotate</a> <font color="Maroon">c</font>,<font color="Maroon">r</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="complex2.html#T34" target="_self">Th34</a>, <a class="ref" href="complex2.html#T36" target="_self">Th36</a>, </i>;<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>
