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<b>let </b><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/> <a href="complex2.html#K3" title="COMPLEX2:func.3">angle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex2.html#K3" title="COMPLEX2:func.3">angle</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p2"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><font color="Olive" title="b1">r</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p2"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><font color="Olive" title="b1">r</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span></span>)</span></span><br/><b>let </b><font color="Maroon" title="c3">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies  <a href="complex2.html#K3" title="COMPLEX2:func.3">angle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex2.html#K3" title="COMPLEX2:func.3">angle</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p2"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><font color="Maroon" title="c3">r</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p2"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><font color="Maroon" title="c3">r</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span></span>)</span> )</span><br/>





<b>assume </b><a NAME="E1:61"/><i><font color="Green" title="E54">A1</font></i>: 
<font color="Maroon" title="c3">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="complex2.html#K3" title="COMPLEX2:func.3">angle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex2.html#K3" title="COMPLEX2:func.3">angle</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p2"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><font color="Maroon" title="c3">r</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p2"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><font color="Maroon" title="c3">r</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span></span>)</span></span><br/>

<a NAME="E2:61"/><i><font color="Green" title="E55">A2</font></i>: 
 <a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p2"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><font color="Maroon" title="c3">r</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c1">a</font>
 
<b>by </b><i><a class="txt" href="complex2.html#E1:61"><i><font color="Green" title="E54">A1</font></i></a>, <a class="ref" href="complex2.html#T40" target="_self" title="COMPLEX2:th.40">Th40</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" title="c2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </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" title="E56">A3</font></i>: 
<font color="Maroon" title="c2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="complex2.html#K3" title="COMPLEX2:func.3">angle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex2.html#K3" title="COMPLEX2:func.3">angle</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p2"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><font color="Maroon" title="c3">r</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p2"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><font color="Maroon" title="c3">r</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span></span>)</span></span><br/><div class="add"><a NAME="E2:61_1_1"/>
<span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><font color="Maroon" title="c3">r</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> </span>)</span>
 <b>by </b><i><a class="ref" href="complex1.html#T27" title="COMPLEX1:th.27">COMPLEX1:27</a></i>;<br/><a NAME="E3:61_1_1"/><b>then </b><i><font color="Green" title="E57">A4</font></i>: 
<font color="Maroon" title="c2">b</font> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><font color="Maroon" title="c3">r</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="complex2.html#E1:61"><i><font color="Green" title="E54">A1</font></i></a>, <a class="txt" href="complex2.html#E1:61_1_1"><i><font color="Green" title="E56">A3</font></i></a>, <a class="ref" href="xcmplx_1.html#T6" title="XCMPLX_1:th.6">XCMPLX_1:6</a></i>;<br/><b>thus </b> <a href="complex2.html#K3" title="COMPLEX2:func.3">angle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> = 
 <a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p1">(<span class="default"><a href="complex2.html#K2" title="COMPLEX2:func.2">Rotate</a> <font color="Maroon" title="c2">b</font>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="complex2.html#E1:61_1_1"><i><font color="Green" title="E56">A3</font></i></a>, <a class="ref" href="complex2.html#D5" target="_self" title="COMPLEX2:def.5">Def5</a></i>
<br/>.= 
 <a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p1">(<span class="default"><a href="complex2.html#K2" title="COMPLEX2:func.2">Rotate</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p3"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><font color="Maroon" title="c3">r</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p5"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><font color="Maroon" title="c3">r</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="complex2.html#E1:61"><i><font color="Green" title="E54">A1</font></i></a>, <a class="txt" href="complex2.html#E2:61"><i><font color="Green" title="E55">A2</font></i></a>, <a class="txt" href="complex2.html#E1:61_1_1"><i><font color="Green" title="E56">A3</font></i></a>, <a class="txt" href="complex2.html#E3:61_1_1"><i><font color="Green" title="E57">A4</font></i></a>, <a class="ref" href="complex2.html#T40" target="_self" title="COMPLEX2:th.40">Th40</a>, <a class="ref" href="complex2.html#T76" target="_self" title="COMPLEX2:th.76">Th76</a></i>
<br/>.= 
 <a href="complex2.html#K3" title="COMPLEX2:func.3">angle</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p2"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><font color="Maroon" title="c3">r</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p2"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><font color="Maroon" title="c3">r</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="complex2.html#E3:61_1_1"><i><font color="Green" title="E57">A4</font></i></a>, <a class="ref" href="complex2.html#D5" target="_self" title="COMPLEX2:def.5">Def5</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><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" title="E56">A5</font></i>: 
<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="complex2.html#K3" title="COMPLEX2:func.3">angle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex2.html#K3" title="COMPLEX2:func.3">angle</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p2"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><font color="Maroon" title="c3">r</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p2"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><font color="Maroon" title="c3">r</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span></span>)</span></span><br/><div class="add"><b>thus </b><a NAME="E2:61_1_2"/>
 <a href="complex2.html#K3" title="COMPLEX2:func.3">angle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex2.html#K3" title="COMPLEX2:func.3">angle</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p2"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><font color="Maroon" title="c3">r</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p2"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><font color="Maroon" title="c3">r</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span></span>)</span>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</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" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  <a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </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" title="E57">A6</font></i>: 
 <a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="complex2.html#K3" title="COMPLEX2:func.3">angle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex2.html#K3" title="COMPLEX2:func.3">angle</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p2"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><font color="Maroon" title="c3">r</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p2"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><font color="Maroon" title="c3">r</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span></span>)</span></span><br/><div class="add"><b>hence </b> <a href="complex2.html#K3" title="COMPLEX2:func.3">angle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> = 
 <a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p1">(<span class="default"><a href="complex2.html#K2" title="COMPLEX2:func.2">Rotate</a> <font color="Maroon" title="c2">b</font>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complex2.html#D5" target="_self" title="COMPLEX2:def.5">Def5</a></i>
<br/>.= 
 <a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a> 
<b>by </b><i><a class="txt" href="complex2.html#E1:61_1_2"><i><font color="Green" title="E56">A5</font></i></a>, <a class="ref" href="complex2.html#T66" target="_self" title="COMPLEX2:th.66">Th66</a></i>
<br/>.= 
 <a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p1">(<span class="default"><a href="complex2.html#K2" title="COMPLEX2:func.2">Rotate</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p3"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><font color="Maroon" title="c3">r</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p5"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><font color="Maroon" title="c3">r</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="complex2.html#E1:61_1_2"><i><font color="Green" title="E56">A5</font></i></a>, <a class="ref" href="complex2.html#T66" target="_self" title="COMPLEX2:th.66">Th66</a></i>
<br/>.= 
 <a href="complex2.html#K3" title="COMPLEX2:func.3">angle</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p2"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><font color="Maroon" title="c3">r</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p2"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><font color="Maroon" title="c3">r</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="complex2.html#E2:61"><i><font color="Green" title="E55">A2</font></i></a>, <a class="txt" href="complex2.html#E1:61_1_2_1_1_1"><i><font color="Green" title="E57">A6</font></i></a>, <a class="ref" href="complex2.html#D5" target="_self" title="COMPLEX2:def.5">Def5</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><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" title="E57">A7</font></i>: 
 <a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="complex2.html#K3" title="COMPLEX2:func.3">angle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex2.html#K3" title="COMPLEX2:func.3">angle</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p2"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><font color="Maroon" title="c3">r</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p2"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><font color="Maroon" title="c3">r</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span></span>)</span></span><br/><div class="add"><b>hence </b> <a href="complex2.html#K3" title="COMPLEX2:func.3">angle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> = 
<span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c1">a</font></span>)</span>
<b>by </b><i><a class="txt" href="complex2.html#E1:61_1_2"><i><font color="Green" title="E56">A5</font></i></a>, <a class="ref" href="complex2.html#D5" target="_self" title="COMPLEX2:def.5">Def5</a></i>
<br/>.= 
 <a href="complex2.html#K3" title="COMPLEX2:func.3">angle</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p2"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><font color="Maroon" title="c3">r</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p2"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><font color="Maroon" title="c3">r</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="complex2.html#E2:61"><i><font color="Green" title="E55">A2</font></i></a>, <a class="txt" href="complex2.html#E1:61_1_2"><i><font color="Green" title="E56">A5</font></i></a>, <a class="txt" href="complex2.html#E1:61_1_2_1_1_2"><i><font color="Green" title="E57">A7</font></i></a>, <a class="ref" href="complex2.html#D5" target="_self" title="COMPLEX2:def.5">Def5</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><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>
