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<div class="add">

<span class="kw">let </span><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:()"><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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <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>  implies (  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complex2.html#K1" title="COMPLEX2:func.1">.|.</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  iff (  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> (<font color="Maroon" title="c1">a</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<font color="Maroon" title="c2">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> 2 or  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> (<font color="Maroon" title="c1">a</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<font color="Maroon" title="c2">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">3 <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> 2</span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a> ) ) )</span><br/>



<a NAME="E1:73"/><span class="lab"><font color="Green" title="E62">A1</font></span>: 
 <a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complex2.html#K1" title="COMPLEX2:func.1">.|.</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">a</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">b</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complex2.html#K1" title="COMPLEX2:func.1">.|.</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">a</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">b</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span></span>)</span>
 
;<br/>
<span class="kw">assume </span><a NAME="E2:73"/><span class="lab"><font color="Green" title="E63">A2</font></span>: 
( <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>  &amp; <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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complex2.html#K1" title="COMPLEX2:func.1">.|.</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  iff (  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> (<font color="Maroon" title="c1">a</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<font color="Maroon" title="c2">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> 2 or  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> (<font color="Maroon" title="c1">a</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<font color="Maroon" title="c2">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">3 <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> 2</span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a> ) )</span><br/>

<a NAME="E3:73"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A3</font></span>: 
( <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">a</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</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>  &amp; <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">b</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</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>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T45" title="COMPLEX1:th.45">COMPLEX1:45</a></span>;<br/>
<a NAME="E4:73"/><span class="lab"><font color="Green" title="E65">A4</font></span>: 
(  <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#K4" title="COMPLEX2:func.4">angle</a> (<font color="Maroon" title="c1">a</font>,<a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a>,<font color="Maroon" title="c2">b</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>,<a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a>,<font color="Maroon" title="c2">b</font>) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex2.html#T70" target="_self" title="COMPLEX2:th.70">Th84</a>, <a class="ref" href="complex2.html#T73" target="_self" title="COMPLEX2:th.73">Th87</a></span>;<br/>
<a NAME="E5:73"/><span class="lab"><font color="Green" title="E66">A5</font></span>: 
(  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> (<font color="Maroon" title="c1">a</font>,<a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a>,<font color="Maroon" title="c2">b</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a> &amp; <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex2.html#T70" target="_self" title="COMPLEX2:th.70">Th84</a>, <a class="ref" href="comptrig.html#T5" title="COMPTRIG:th.5">COMPTRIG:5</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E6:73"/><span class="lab"><font color="Green" title="E67">A6</font></span>: 
 <a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a> <span class="p1">(<span class="default"><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>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complex2.html#K1" title="COMPLEX2:func.1">.|.</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">a</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">b</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:73"><span class="lab"><font color="Green" title="E63">A2</font></span></a>, <a class="ref" href="complex2.html#T69" target="_self" title="COMPLEX2:th.69">Th83</a></span>;<br/>
<a NAME="E7:73"/><span class="lab"><font color="Green" title="E68">A7</font></span>: 
 <a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a> <span class="p1">(<span class="default"><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>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complex2.html#K1" title="COMPLEX2:func.1">.|.</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">a</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">b</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:73"><span class="lab"><font color="Green" title="E63">A2</font></span></a>, <a class="ref" href="complex2.html#T69" target="_self" title="COMPLEX2:th.69">Th83</a></span>;<br/>
<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> (<font color="Maroon" title="c1">a</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<font color="Maroon" title="c2">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> 2 or  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> (<font color="Maroon" title="c1">a</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<font color="Maroon" title="c2">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">3 <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> 2</span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a> ) implies  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complex2.html#K1" title="COMPLEX2:func.1">.|.</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:73_1"/><span class="lab"><font color="Green" title="E69">A8</font></span>: 
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complex2.html#K1" title="COMPLEX2:func.1">.|.</a> <font color="Maroon" title="c2">b</font></span>)</span> <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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> (<font color="Maroon" title="c1">a</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<font color="Maroon" title="c2">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> 2 or  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> (<font color="Maroon" title="c1">a</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<font color="Maroon" title="c2">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">3 <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> 2</span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a> )</span><br/><a NAME="E2:73_1"/><span class="kw">then </span>
(  <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complex2.html#K1" title="COMPLEX2:func.1">.|.</a> <font color="Maroon" title="c2">b</font></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"><font color="Maroon" title="c1">a</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">b</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> or  <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complex2.html#K1" title="COMPLEX2:func.1">.|.</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">a</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">b</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complex2.html#T50" target="_self" title="COMPLEX2:th.50">Th64</a></span>;<br/><a NAME="E3:73_1"/><span class="kw">then </span>
(  <a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a> <span class="p1">(<span class="default"><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>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or  <a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a> <span class="p1">(<span class="default"><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>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:73"><span class="lab"><font color="Green" title="E68">A7</font></span></a>, <a class="txt" href="#E3:73"><span class="lab"><font color="Green" title="E64">A3</font></span></a>, <a class="txt" href="#E1:73"><span class="lab"><font color="Green" title="E62">A1</font></span></a>, <a class="ref" href="xcmplx_1.html#T60" title="XCMPLX_1:th.60">XCMPLX_1:60</a></span>;<br/><span class="kw">hence </span><a NAME="E4:73_1"/>
(  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> (<font color="Maroon" title="c1">a</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<font color="Maroon" title="c2">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> 2 or  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> (<font color="Maroon" title="c1">a</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<font color="Maroon" title="c2">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">3 <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> 2</span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</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="#E6:73"><span class="lab"><font color="Green" title="E67">A6</font></span></a>, <a class="txt" href="#E4:73"><span class="lab"><font color="Green" title="E65">A4</font></span></a>, <a class="txt" href="#E5:73"><span class="lab"><font color="Green" title="E66">A5</font></span></a>, <a class="txt" href="#E1:73_1"><span class="lab"><font color="Green" title="E69">A8</font></span></a>, <a class="ref" href="complex2.html#T11" target="_self" title="COMPLEX2:th.11">Th12</a>, <a class="ref" href="comptrig.html#T5" title="COMPTRIG:th.5">COMPTRIG:5</a>, <a class="ref" href="sin_cos.html#T77" title="SIN_COS:th.77">SIN_COS:77</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/></div>
<span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E9:73"/>
(  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> (<font color="Maroon" title="c1">a</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<font color="Maroon" title="c2">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> 2 or  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> (<font color="Maroon" title="c1">a</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<font color="Maroon" title="c2">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">3 <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> 2</span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</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">  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complex2.html#K1" title="COMPLEX2:func.1">.|.</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/>

<span class="kw">hence </span><a NAME="E10:73"/>
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complex2.html#K1" title="COMPLEX2:func.1">.|.</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:73"><span class="lab"><font color="Green" title="E67">A6</font></span></a>, <a class="txt" href="#E3:73"><span class="lab"><font color="Green" title="E64">A3</font></span></a>, <a class="ref" href="complex2.html#T73" target="_self" title="COMPLEX2:th.73">Th87</a>, <a class="ref" href="sin_cos.html#T77" title="SIN_COS:th.77">SIN_COS:77</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/>


</div>
