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<b>let </b><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font>, <font color="Maroon" title="c3">c</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"> ( <font color="Maroon" title="c1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">b</font> &amp; <font color="Maroon" title="c2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">c</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font> &amp;  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  implies ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c1">a</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c1">a</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> ) )</span><br/>





<b>assume </b><b>that </b><br/><a NAME="E1:86"/><i><font color="Green" title="E72">A1</font></i>: 
<font color="Maroon" title="c1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">b</font>
 <b>and </b><br/><a NAME="E2:86"/><i><font color="Green" title="E73">A2</font></i>: 
<font color="Maroon" title="c2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">c</font>
 <b>and </b><br/><a NAME="E3:86"/><i><font color="Green" title="E74">A3</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>
 <b>and </b><br/><a NAME="E4:86"/><i><font color="Green" title="E75">A4</font></i>: 
 <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c1">a</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c1">a</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> )</span><br/>

<i><font color="Green" title="E76">A5</font></i>:  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span>,<a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a> ,<span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span> = 
 <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span>

<br/>.= 
 <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>
<b>by </b><i><a class="ref" href="complex2.html#T86" target="_self" title="COMPLEX2:th.86">Th86</a></i>
;<br/>
<b>set </b><font color="Maroon" title="c4">r</font> =  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c5">A</font> =  <a href="complex2.html#K2" title="COMPLEX2:func.2">Rotate</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p4">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c6">B</font> =  <a href="complex2.html#K2" title="COMPLEX2:func.2">Rotate</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p4">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span></span>)</span>;<br/>
<a NAME="E6:86"/><i><font color="Green" title="E77">A6</font></i>: 
 <a href="complex2.html#K2" title="COMPLEX2:func.2">Rotate</a> <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a> ,<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p4">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a> 
 
<b>by </b><i><a class="ref" href="complex2.html#T66" target="_self" title="COMPLEX2:th.66">Th66</a></i>;<br/>
<a NAME="E7:86"/><i><font color="Green" title="E78">A7</font></i>: 
<font color="Maroon" title="c1">a</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <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> 
 
<b>by </b><i><a class="txt" href="complex2.html#E1:86"><i><font color="Green" title="E72">A1</font></i></a></i>;<br/>
<a NAME="E8:86"/><i><font color="Green" title="E79">A8</font></i>: 
<font color="Maroon" title="c3">c</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">a</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <font color="Maroon" title="c2">b</font></span>)</span>
 
<b>by </b><i><a class="txt" href="complex2.html#E3:86"><i><font color="Green" title="E74">A3</font></i></a>, <a class="ref" href="complex2.html#T93" target="_self" title="COMPLEX2:th.93">Th93</a></i>;<br/>
<a NAME="E9:86"/><i><font color="Green" title="E80">A9</font></i>: 
<font color="Maroon" title="c3">c</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <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> 
 
<b>by </b><i><a class="txt" href="complex2.html#E2:86"><i><font color="Green" title="E73">A2</font></i></a></i>;<br/>
<a NAME="E10:86"/><b>then </b><i><font color="Green" title="E81">A10</font></i>: 
 <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span>,<a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a> ,<span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</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="c1">a</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <font color="Maroon" title="c2">b</font></span>)</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#K8" title="COMPLEX1:func.8">+</a> <span class="p5">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<a href="complex1.html#K5" title="COMPLEX1:func.5">0c</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="c3">c</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <font color="Maroon" title="c2">b</font></span>)</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#K8" title="COMPLEX1:func.8">+</a> <span class="p5">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="complex2.html#E6:86"><i><font color="Green" title="E77">A6</font></i></a>, <a class="txt" href="complex2.html#E7:86"><i><font color="Green" title="E78">A7</font></i></a>, <a class="ref" href="complex2.html#T92" target="_self" title="COMPLEX2:th.92">Th92</a></i>;<br/>
<a NAME="E11:86"/><i><font color="Green" title="E82">A11</font></i>: 
 <a href="complex2.html#K2" title="COMPLEX2:func.2">Rotate</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p4">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <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>, <a class="ref" href="sin_cos.html#T34" title="SIN_COS:th.34">SIN_COS:34</a></i>;<br/>
<a NAME="E12:86"/><b>then </b><i><font color="Green" title="E83">A12</font></i>: 
 <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</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="c1">a</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <font color="Maroon" title="c2">b</font></span>)</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#K8" title="COMPLEX1:func.8">+</a> <span class="p5">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="complex1.html#T28" title="COMPLEX1:th.28">COMPLEX1:28</a></i>;<br/>
<a NAME="E13:86"/>
<font color="Maroon" title="c1">a</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a> 
 
<b>by </b><i><a class="txt" href="complex2.html#E1:86"><i><font color="Green" title="E72">A1</font></i></a></i>;<br/>
<a NAME="E14:86"/><b>then </b>
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="complex1.html#T133" title="COMPLEX1:th.133">COMPLEX1:133</a></i>;<br/>
<a NAME="E15:86"/><b>then </b><i><font color="Green" title="E84">A13</font></i>: 
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</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="c1">a</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <font color="Maroon" title="c2">b</font></span>)</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#K8" title="COMPLEX1:func.8">+</a> <span class="p5">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="complex2.html#E11:86"><i><font color="Green" title="E82">A11</font></i></a>, <a class="ref" href="complex1.html#T28" title="COMPLEX1:th.28">COMPLEX1:28</a></i>;<br/>
<a NAME="E16:86"/><b>then </b><i><font color="Green" title="E85">A14</font></i>: 
 <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="c1">a</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <font color="Maroon" title="c2">b</font></span>)</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#K8" title="COMPLEX1:func.8">+</a> <span class="p5">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a> 
 
<b>by </b><i><a class="txt" href="complex2.html#E12:86"><i><font color="Green" title="E83">A12</font></i></a>, <a class="ref" href="complex2.html#T34" target="_self" title="COMPLEX2:th.34">Th34</a></i>;<br/>
<a NAME="E17:86"/><b>then </b>
<span class="p1">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex2.html#K2" title="COMPLEX2:func.2">Rotate</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p5">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p5">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a> </span>)</span></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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex2.html#K2" title="COMPLEX2:func.2">Rotate</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p5">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p5">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="comptrig.html#T52" title="COMPTRIG:th.52">COMPTRIG:52</a></i>;<br/>
<a NAME="E18:86"/><b>then </b><i><font color="Green" title="E86">A15</font></i>: 
 <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c3">c</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <font color="Maroon" title="c2">b</font></span>)</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#K8" title="COMPLEX1:func.8">+</a> <span class="p5">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="complex2.html#E5:86"><i><font color="Green" title="E76">A5</font></i></a>, <a class="txt" href="complex2.html#E10:86"><i><font color="Green" title="E81">A10</font></i></a>, <a class="txt" href="complex2.html#E16:86"><i><font color="Green" title="E85">A14</font></i></a>, <a class="ref" href="complex2.html#D6" target="_self" title="COMPLEX2:def.6">Def6</a></i>;<br/>
<i><font color="Green" title="E87">A16</font></i>:  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c1">a</font> = 
 <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complex2.html#T86" target="_self" title="COMPLEX2:th.86">Th86</a></i>
<br/>.= 
 <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</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="c3">c</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <font color="Maroon" title="c2">b</font></span>)</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#K8" title="COMPLEX1:func.8">+</a> <span class="p5">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<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="c1">a</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <font color="Maroon" title="c2">b</font></span>)</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#K8" title="COMPLEX1:func.8">+</a> <span class="p5">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="complex2.html#E6:86"><i><font color="Green" title="E77">A6</font></i></a>, <a class="txt" href="complex2.html#E8:86"><i><font color="Green" title="E79">A8</font></i></a>, <a class="txt" href="complex2.html#E9:86"><i><font color="Green" title="E80">A9</font></i></a>, <a class="ref" href="complex2.html#T92" target="_self" title="COMPLEX2:th.92">Th92</a></i>
;<br/>
<i><font color="Green" title="E88">A17</font></i>:  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> = 
 <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complex2.html#T86" target="_self" title="COMPLEX2:th.86">Th86</a></i>
<br/>.= 
 <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</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="c3">c</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <font color="Maroon" title="c2">b</font></span>)</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#K8" title="COMPLEX1:func.8">+</a> <span class="p5">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<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="c1">a</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <font color="Maroon" title="c2">b</font></span>)</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#K8" title="COMPLEX1:func.8">+</a> <span class="p5">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a> 
<b>by </b><i><a class="txt" href="complex2.html#E6:86"><i><font color="Green" title="E77">A6</font></i></a>, <a class="txt" href="complex2.html#E7:86"><i><font color="Green" title="E78">A7</font></i></a>, <a class="txt" href="complex2.html#E8:86"><i><font color="Green" title="E79">A8</font></i></a>, <a class="ref" href="complex2.html#T92" target="_self" title="COMPLEX2:th.92">Th92</a></i>
;<br/>
<b>hence </b><a NAME="E21:86"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c1">a</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> 
 <b>by </b><i><a class="txt" href="complex2.html#E3:86"><i><font color="Green" title="E74">A3</font></i></a>, <a class="txt" href="complex2.html#E4:86"><i><font color="Green" title="E75">A4</font></i></a>, <a class="txt" href="complex2.html#E5:86"><i><font color="Green" title="E76">A5</font></i></a>, <a class="txt" href="complex2.html#E10:86"><i><font color="Green" title="E81">A10</font></i></a>, <a class="txt" href="complex2.html#E12:86"><i><font color="Green" title="E83">A12</font></i></a>, <a class="txt" href="complex2.html#E15:86"><i><font color="Green" title="E84">A13</font></i></a>, <a class="txt" href="complex2.html#E18:86"><i><font color="Green" title="E86">A15</font></i></a>, <a class="txt" href="complex2.html#E19:86"><i><font color="Green" title="E87">A16</font></i></a>, <a class="txt" href="complex2.html#E109"><i><font color="Green" title="E70">Lm6</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c1">a</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> )</span><br/>

<b>thus </b><a NAME="E22:86"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c1">a</font>
 <b>by </b><i><a class="txt" href="complex2.html#E3:86"><i><font color="Green" title="E74">A3</font></i></a>, <a class="txt" href="complex2.html#E4:86"><i><font color="Green" title="E75">A4</font></i></a>, <a class="txt" href="complex2.html#E12:86"><i><font color="Green" title="E83">A12</font></i></a>, <a class="txt" href="complex2.html#E15:86"><i><font color="Green" title="E84">A13</font></i></a>, <a class="txt" href="complex2.html#E18:86"><i><font color="Green" title="E86">A15</font></i></a>, <a class="txt" href="complex2.html#E19:86"><i><font color="Green" title="E87">A16</font></i></a>, <a class="txt" href="complex2.html#E109"><i><font color="Green" title="E70">Lm6</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><br/>

<b>thus </b><a NAME="E23:86"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>
 <b>by </b><i><a class="txt" href="complex2.html#E3:86"><i><font color="Green" title="E74">A3</font></i></a>, <a class="txt" href="complex2.html#E4:86"><i><font color="Green" title="E75">A4</font></i></a>, <a class="txt" href="complex2.html#E12:86"><i><font color="Green" title="E83">A12</font></i></a>, <a class="txt" href="complex2.html#E15:86"><i><font color="Green" title="E84">A13</font></i></a>, <a class="txt" href="complex2.html#E18:86"><i><font color="Green" title="E86">A15</font></i></a>, <a class="txt" href="complex2.html#E20:86"><i><font color="Green" title="E88">A17</font></i></a>, <a class="txt" href="complex2.html#E109"><i><font color="Green" title="E70">Lm6</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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