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<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#NR3" title="XXREAL_0:NR.3">&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#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="sin_cos.html#K33" title="SIN_COS:func.33">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#K33" title="SIN_COS:func.33">PI</a>  &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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#NR3" title="XXREAL_0:NR.3">&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/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:85"/><span class="lab"><font color="Green" title="E70">A1</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E2:85"/><span class="lab"><font color="Green" title="E71">A2</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E3:85"/><span class="lab"><font color="Green" title="E72">A3</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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>)
 <span class="kw">and </span><br/><a NAME="E4:85"/><span class="lab"><font color="Green" title="E73">A4</font></span>: 
 <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#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K33" title="SIN_COS:func.33">PI</a>  &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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#NR3" title="XXREAL_0:NR.3">&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/>

<a NAME="E5:85"/><span class="lab"><font color="Green" title="E74">A5</font></span>: 
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:85"><span class="lab"><font color="Green" title="E71">A2</font></span></a></span>;<br/>
<span class="kw">set </span><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/>
<span class="kw">set </span><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/>
<span class="kw">set </span><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="E6:85"/><span class="lab"><font color="Green" title="E75">A6</font></span>: 
 <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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex2.html#T52" target="_self" title="COMPLEX2:th.52">Th66</a></span>;<br/>
<a NAME="E7:85"/><span class="lab"><font color="Green" title="E76">A7</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:85"><span class="lab"><font color="Green" title="E72">A3</font></span></a>, <a class="ref" href="complex2.html#T79" target="_self" title="COMPLEX2:th.79">Th93</a></span>;<br/>
<span class="lab"><font color="Green" title="E77">A8</font></span>:  <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>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex2.html#T72" target="_self" title="COMPLEX2:th.72">Th86</a></span>
<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>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:85"><span class="lab"><font color="Green" title="E75">A6</font></span></a>, <a class="txt" href="#E7:85"><span class="lab"><font color="Green" title="E76">A7</font></span></a>, <a class="txt" href="#E5:85"><span class="lab"><font color="Green" title="E74">A5</font></span></a>, <a class="ref" href="complex2.html#T78" target="_self" title="COMPLEX2:th.78">Th92</a></span>
;<br/>
<span class="lab"><font color="Green" title="E78">A9</font></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>,<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>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex2.html#T72" target="_self" title="COMPLEX2:th.72">Th86</a></span>
;<br/>
<a NAME="E10:85"/><span class="lab"><font color="Green" title="E79">A10</font></span>: 
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:85"><span class="lab"><font color="Green" title="E70">A1</font></span></a></span>;<br/>
<a NAME="E11:85"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A11</font></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>,<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>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:85"><span class="lab"><font color="Green" title="E75">A6</font></span></a>, <a class="txt" href="#E5:85"><span class="lab"><font color="Green" title="E74">A5</font></span></a>, <a class="ref" href="complex2.html#T78" target="_self" title="COMPLEX2:th.78">Th92</a></span>;<br/>
<a NAME="E12:85"/>
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:85"><span class="lab"><font color="Green" title="E70">A1</font></span></a></span>;<br/>
<a NAME="E13:85"/><span class="kw">then </span>
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T47" title="COMPLEX1:th.47">COMPLEX1:47</a></span>;<br/>
<a NAME="E14:85"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A12</font></span>: 
(  <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>  &amp;  <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>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T12" title="COMPLEX1:th.12">COMPLEX1:12</a>, <a class="ref" href="sin_cos.html#T31" title="SIN_COS:th.31">SIN_COS:31</a></span>;<br/>
<a NAME="E15:85"/><span class="kw">then </span><span class="lab"><font color="Green" title="E82">A13</font></span>: 
 <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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex2.html#T21" target="_self" title="COMPLEX2:th.21">Th34</a></span>;<br/>
<a NAME="E16:85"/><span class="kw">then </span>
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="comptrig.html#T34" title="COMPTRIG:th.34">COMPTRIG:34</a></span>;<br/>
<a NAME="E17:85"/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">A14</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:85"><span class="lab"><font color="Green" title="E78">A9</font></span></a>, <a class="txt" href="#E11:85"><span class="lab"><font color="Green" title="E80">A11</font></span></a>, <a class="txt" href="#E15:85"><span class="lab"><font color="Green" title="E82">A13</font></span></a>, <a class="ref" href="complex2.html#D4" target="_self" title="COMPLEX2:def.4">Def6</a></span>;<br/>
<span class="lab"><font color="Green" title="E84">A15</font></span>:  <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>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex2.html#T72" target="_self" title="COMPLEX2:th.72">Th86</a></span>
<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>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:85"><span class="lab"><font color="Green" title="E75">A6</font></span></a>, <a class="txt" href="#E10:85"><span class="lab"><font color="Green" title="E79">A10</font></span></a>, <a class="txt" href="#E7:85"><span class="lab"><font color="Green" title="E76">A7</font></span></a>, <a class="ref" href="complex2.html#T78" target="_self" title="COMPLEX2:th.78">Th92</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E19:85"/>
<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#K33" title="SIN_COS:func.33">PI</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:85"><span class="lab"><font color="Green" title="E72">A3</font></span></a>, <a class="txt" href="#E4:85"><span class="lab"><font color="Green" title="E73">A4</font></span></a>, <a class="txt" href="#E9:85"><span class="lab"><font color="Green" title="E78">A9</font></span></a>, <a class="txt" href="#E11:85"><span class="lab"><font color="Green" title="E80">A11</font></span></a>, <a class="txt" href="#E14:85"><span class="lab"><font color="Green" title="E81">A12</font></span></a>, <a class="txt" href="#E17:85"><span class="lab"><font color="Green" title="E83">A14</font></span></a>, <a class="txt" href="#E8:85"><span class="lab"><font color="Green" title="E77">A8</font></span></a>, <a class="txt" href="complex2.html#E92"><span class="lab"><font color="Green" title="E68">Lm5</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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#NR3" title="XXREAL_0:NR.3">&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/>

<span class="kw">thus </span><a NAME="E20:85"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:85"><span class="lab"><font color="Green" title="E72">A3</font></span></a>, <a class="txt" href="#E4:85"><span class="lab"><font color="Green" title="E73">A4</font></span></a>, <a class="txt" href="#E14:85"><span class="lab"><font color="Green" title="E81">A12</font></span></a>, <a class="txt" href="#E17:85"><span class="lab"><font color="Green" title="E83">A14</font></span></a>, <a class="txt" href="#E8:85"><span class="lab"><font color="Green" title="E77">A8</font></span></a>, <a class="txt" href="complex2.html#E92"><span class="lab"><font color="Green" title="E68">Lm5</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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/>

<span class="kw">thus </span><a NAME="E21:85"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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 class="kw">by</span> <span class="lab"><a class="txt" href="#E3:85"><span class="lab"><font color="Green" title="E72">A3</font></span></a>, <a class="txt" href="#E4:85"><span class="lab"><font color="Green" title="E73">A4</font></span></a>, <a class="txt" href="#E14:85"><span class="lab"><font color="Green" title="E81">A12</font></span></a>, <a class="txt" href="#E17:85"><span class="lab"><font color="Green" title="E83">A14</font></span></a>, <a class="txt" href="#E18:85"><span class="lab"><font color="Green" title="E84">A15</font></span></a>, <a class="txt" href="complex2.html#E92"><span class="lab"><font color="Green" title="E68">Lm5</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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