<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:87"/><i><font color="Green">E80</font></i>: 
 <a href="complex1.html#K4">Im</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>and </b><br/><a NAME="E2:87"/><i><font color="Green">E81</font></i>: 
 <a href="complex1.html#K3">Re</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>and </b><br/><a NAME="E3:87"/><i><font color="Green">E82</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a>  <a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E4:87"/><i><font color="Green">E83</font></i>: 
 <a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="sin_cos.html#K33">PI</a> 
 ;<br/>

<a NAME="E5:87"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">0 <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="complex2.html#E1:87"><i><font color="Green">E80</font></i></a>, <a class="ref" href="complex1.html#T29">COMPLEX1:29</a></i>;<br/>
<a NAME="E6:87"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,0</span><a href="hahnban1.html#K1">**]</a></span>
 
<b>by </b><i><a class="ref" href="hahnban1.html#D1">HAHNBAN1:def 1</a></i>;<br/>
<a NAME="E7:87"/><b>then </b><i><font color="Green">E84</font></i>: 
 <a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="complex2.html#E2:87"><i><font color="Green">E81</font></i></a>, <a class="ref" href="comptrig.html#T53">COMPTRIG:53</a></i>;<br/>
<b>then </b><i><font color="Green">E85</font></i>:  <a href="comptrig.html#K1">Arg</a> <span class="p1">(<span class="default"><a href="complex1.html#K10">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <a href="sin_cos.html#K33">PI</a> 
<b>by </b><i><a class="txt" href="complex2.html#E2:87"><i><font color="Green">E81</font></i></a>, <a class="ref" href="complex2.html#T22" target="_self">Th22</a>, <a class="ref" href="complex1.html#T12">COMPLEX1:12</a>, <a class="ref" href="comptrig.html#T21">COMPTRIG:21</a></i>
<br/>.= 
 <a href="sin_cos.html#K33">PI</a> 
<b>by </b><i><a class="txt" href="complex2.html#E7:87"><i><font color="Green">E84</font></i></a></i>
;<br/>
<a NAME="E9:87"/>
 <a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default">0,<a href="sin_cos.html#K33">PI</a> </span><a href="rcomp_1.html#K2">.[</a></span>
 
<b>by </b><i><a class="txt" href="complex2.html#E3:87"><i><font color="Green">E82</font></i></a>, <a class="txt" href="complex2.html#E4:87"><i><font color="Green">E83</font></i></a>, <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/>
<a NAME="E10:87"/><b>then </b><i><font color="Green">E86</font></i>: 
 <a href="complex1.html#K4">Im</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="complex2.html#T27" target="_self">Th27</a></i>;<br/>
<i><font color="Green">E87</font></i>:  <a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T48">COMPLEX1:48</a></i>
<br/>.= 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="complex2.html#E1:87"><i><font color="Green">E80</font></i></a></i>
;<br/>
<a NAME="E12:87"/>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K4">Im</a> <font color="Maroon">c<sub>2</sub></font>
 
;<br/>
<a NAME="E13:87"/><b>then </b><i><font color="Green">E88</font></i>: 
 <a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> 0
 
<b>by </b><i><a class="txt" href="complex2.html#E10:87"><i><font color="Green">E86</font></i></a>, <a class="txt" href="complex2.html#E11:87"><i><font color="Green">E87</font></i></a>, <a class="ref" href="xreal_1.html#T60">XREAL_1:60</a></i>;<br/>
<a NAME="E14:87"/><b>then </b><i><font color="Green">E89</font></i>: 
 <a href="comptrig.html#K1">Arg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="sin_cos.html#K33">PI</a> 
 
<b>by </b><i><a class="ref" href="complex2.html#T37" target="_self">Th37</a></i>;<br/>
<a NAME="E15:87"/>
<span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="complex1.html#K11">-</a> <a href="complex1.html#K5">0c</a> </span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complex1.html#K11">-</a> <a href="complex1.html#K5">0c</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="txt" href="complex2.html#E3:87"><i><font color="Green">E82</font></i></a>, <a class="txt" href="complex2.html#E7:87"><i><font color="Green">E84</font></i></a></i>;<br/>
<a NAME="E16:87"/><b>then </b><i><font color="Green">E90</font></i>: 
 <a href="complex2.html#K5" target="_self">angle</a> <font color="Maroon">c<sub>1</sub></font>,<a href="complex1.html#K5">0c</a> ,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="complex2.html#D6" target="_self">Def6</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:87_3"/><i><font color="Green">E91</font></i>: 
 <a href="complex1.html#K4">Im</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>and </b><br/><a NAME="E2:87_3"/><i><font color="Green">E92</font></i>: 
 <a href="complex1.html#K3">Re</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>and </b><br/><a NAME="E3:87_3"/><i><font color="Green">E93</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a>  <a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E4:87_3"/><i><font color="Green">E94</font></i>: 
 <a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="sin_cos.html#K33">PI</a> 
 ;<br/><a NAME="E5:87_3"/>
 <a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default">0,<a href="sin_cos.html#K33">PI</a> </span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="txt" href="complex2.html#E3:87_3"><i><font color="Green">E93</font></i></a>, <a class="txt" href="complex2.html#E4:87_3"><i><font color="Green">E94</font></i></a>, <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/><a NAME="E6:87_3"/><b>then </b><i><font color="Green">E95</font></i>: 
 <a href="complex1.html#K4">Im</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="complex2.html#T27" target="_self">Th27</a></i>;<br/><i><font color="Green">E96</font></i>:  <a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T48">COMPLEX1:48</a></i>
<br/>.= 
 <a href="complex1.html#K4">Im</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="complex2.html#E1:87_3"><i><font color="Green">E91</font></i></a></i>
;<br/><a NAME="E8:87_3"/><i><font color="Green">E97</font></i>: 
 <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="complex1.html#T48">COMPLEX1:48</a></i>;<br/><a NAME="E9:87_3"/>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K3">Re</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E10:87_3"/><b>then </b>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> 0
 <b>by </b><i><a class="txt" href="complex2.html#E2:87_3"><i><font color="Green">E92</font></i></a>, <a class="ref" href="xreal_1.html#T60">XREAL_1:60</a></i>;<br/><a NAME="E11:87_3"/><b>then </b><i><font color="Green">E98</font></i>: 
<span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K3">+</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><b>set </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="complex1.html#K3">Re</a> <font color="Maroon">c<sub>4</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>7</sub></font> =  <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>;<br/><b>set </b><font color="Maroon">c<sub>8</sub></font> =  <a href="complex1.html#K4">Im</a> <font color="Maroon">c<sub>4</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>9</sub></font> =  <a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>;<br/><b>set </b><font color="Maroon">c<sub>10</sub></font> =  <a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>4</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>11</sub></font> =  <a href="comptrig.html#K1">Arg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>;<br/><b>set </b><font color="Maroon">c<sub>12</sub></font> = <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="complex1.html#K17">.|</a></span>;<br/><b>set </b><font color="Maroon">c<sub>13</sub></font> = <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span>;<br/><a NAME="E12:87_3"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p4">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p4">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span></span><a href="arytm_0.html#K5">*]</a></span>
 <b>by </b><i><a class="ref" href="complex2.html#T19" target="_self">Th19</a></i>;<br/><font color="Maroon">c<sub>4</sub></font> = 
<span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T29">COMPLEX1:29</a></i>
<br/>.= 
<span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="arytm_0.html#K5">*]</a></span>
<b>by </b><i><a class="ref" href="complex1.html#T27">COMPLEX1:27</a></i>
;<br/><a NAME="E14:87_3"/><b>then </b><i><font color="Green">E100</font></i>: 
(  <a href="complex1.html#K3">Re</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> &amp;  <a href="complex1.html#K4">Im</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="arytm_0.html#T12">ARYTM_0:12</a>, <a class="txt" href="complex2.html#E12:87_3"><i><font color="Green">E99</font></i></a></i>;<br/><a NAME="E15:87_3"/><i><font color="Green">E101</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p4">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p4">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="arytm_0.html#K5">*]</a></span>
 <b>by </b><i><a class="ref" href="complex2.html#T19" target="_self">Th19</a></i>;<br/><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font> = 
<span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T29">COMPLEX1:29</a></i>
<br/>.= 
<span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span><a href="arytm_0.html#K5">*]</a></span>
<b>by </b><i><a class="ref" href="complex1.html#T27">COMPLEX1:27</a></i>
;<br/><a NAME="E17:87_3"/><b>then </b><i><font color="Green">E102</font></i>: 
(  <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> &amp;  <a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="arytm_0.html#T12">ARYTM_0:12</a>, <a class="txt" href="complex2.html#E15:87_3"><i><font color="Green">E101</font></i></a></i>;<br/><a NAME="E18:87_3"/><i><font color="Green">E103</font></i>: 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="complex1.html#K17">.|</a></span> <a href="hidden.html#R1">=</a>  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="complex1.html#D16">COMPLEX1:def 16</a></i>;<br/><a NAME="E19:87_3"/><i><font color="Green">E104</font></i>: 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="hidden.html#R1">=</a>  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="complex2.html#E7:87_3"><i><font color="Green">E96</font></i></a>, <a class="ref" href="complex1.html#D16">COMPLEX1:def 16</a></i>;<br/><a NAME="E20:87_3"/><i><font color="Green">E105</font></i>: 
( <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="complex1.html#K17">.|</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 <b>by </b><i><a class="txt" href="complex2.html#E6:87_3"><i><font color="Green">E95</font></i></a>, <a class="ref" href="complex1.html#T12">COMPLEX1:12</a>, <a class="ref" href="complex1.html#T133">COMPLEX1:133</a></i>;<br/><a NAME="E21:87_3"/><i><font color="Green">E106</font></i>: 
( <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 <b>by </b><i><a class="txt" href="complex2.html#E6:87_3"><i><font color="Green">E95</font></i></a>, <a class="txt" href="complex2.html#E7:87_3"><i><font color="Green">E96</font></i></a>, <a class="ref" href="complex1.html#T12">COMPLEX1:12</a>, <a class="ref" href="complex1.html#T133">COMPLEX1:133</a></i>;<br/><a NAME="E22:87_3"/>
( <span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <span class="p1">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 <b>by </b><i><a class="ref" href="xreal_1.html#T65">XREAL_1:65</a></i>;<br/><a NAME="E23:87_3"/><b>then </b><i><font color="Green">E107</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="real_1.html#K3">+</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E24:87_3"/>
( <span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <span class="p1">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 <b>by </b><i><a class="ref" href="xreal_1.html#T65">XREAL_1:65</a></i>;<br/><a NAME="E25:87_3"/><b>then </b><i><font color="Green">E108</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="real_1.html#K3">+</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E26:87_3"/><i><font color="Green">E109</font></i>: 
 <a href="sin_cos.html#K21">sin</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="complex2.html#E6:87_3"><i><font color="Green">E95</font></i></a>, <a class="txt" href="complex2.html#E7:87_3"><i><font color="Green">E96</font></i></a>, <a class="txt" href="complex2.html#E38"><i><font color="Green">Lemma27</font></i></a></i>;<br/><a NAME="E27:87_3"/><b>then </b><i><font color="Green">E110</font></i>: 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K6">/</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="complex1.html#K17">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="complex2.html#E7:87_3"><i><font color="Green">E96</font></i></a>, <a class="txt" href="complex2.html#E14:87_3"><i><font color="Green">E100</font></i></a>, <a class="txt" href="complex2.html#E17:87_3"><i><font color="Green">E102</font></i></a>, <a class="txt" href="complex2.html#E20:87_3"><i><font color="Green">E105</font></i></a>, <a class="ref" href="xcmplx_1.html#T95">XCMPLX_1:95</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="87_3_4"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:87_3_4"/>
( 0 <a href="xxreal_0.html#R1">&lt;</a>  <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> or 0 <a href="hidden.html#R1">=</a>  <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> or 0 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="87_3_4_1"><b>suppose </b></a><a NAME="E1:87_3_4_1"/><i><font color="Green">E111</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a>  <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:87_3_4_1"/><b>then </b>
 <a href="complex1.html#K3">Re</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="complex2.html#E8:87_3"><i><font color="Green">E97</font></i></a>, <a class="txt" href="complex2.html#E11:87_3"><i><font color="Green">E98</font></i></a></i>;<br/><a NAME="E3:87_3_4_1"/><b>then </b><i><font color="Green">E112</font></i>: 
 <a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default">0,<span class="p2">(<span class="default"><a href="sin_cos.html#K33">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="txt" href="complex2.html#E6:87_3"><i><font color="Green">E95</font></i></a>, <a class="txt" href="complex2.html#E36"><i><font color="Green">Lemma25</font></i></a></i>;<br/><a NAME="E4:87_3_4_1"/><i><font color="Green">E113</font></i>: 
 <a href="comptrig.html#K1">Arg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default">0,<span class="p2">(<span class="default"><a href="sin_cos.html#K33">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="txt" href="complex2.html#E6:87_3"><i><font color="Green">E95</font></i></a>, <a class="txt" href="complex2.html#E7:87_3"><i><font color="Green">E96</font></i></a>, <a class="txt" href="complex2.html#E1:87_3_4_1"><i><font color="Green">E111</font></i></a>, <a class="txt" href="complex2.html#E36"><i><font color="Green">Lemma25</font></i></a></i>;<br/><a NAME="E5:87_3_4_1"/>
<span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> 
 <b>by </b><i><a class="txt" href="complex2.html#E8:87_3"><i><font color="Green">E97</font></i></a>, <a class="txt" href="complex2.html#E11:87_3"><i><font color="Green">E98</font></i></a>, <a class="txt" href="complex2.html#E1:87_3_4_1"><i><font color="Green">E111</font></i></a>, <a class="ref" href="square_1.html#T78">SQUARE_1:78</a></i>;<br/><a NAME="E6:87_3_4_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E7:87_3_4_1"/><b>then </b>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span>
 <b>by </b><i><a class="txt" href="complex2.html#E18:87_3"><i><font color="Green">E103</font></i></a>, <a class="txt" href="complex2.html#E19:87_3"><i><font color="Green">E104</font></i></a>, <a class="txt" href="complex2.html#E23:87_3"><i><font color="Green">E107</font></i></a>, <a class="ref" href="square_1.html#T95">SQUARE_1:95</a></i>;<br/><a NAME="E8:87_3_4_1"/><b>then </b>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K6">/</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> 1
 <b>by </b><i><a class="txt" href="complex2.html#E21:87_3"><i><font color="Green">E106</font></i></a>, <a class="ref" href="real_2.html#T142">REAL_2:142</a></i>;<br/><a NAME="E9:87_3_4_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p3">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p3">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="complex2.html#E26:87_3"><i><font color="Green">E109</font></i></a>, <a class="txt" href="complex2.html#E27:87_3"><i><font color="Green">E110</font></i></a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E10:87_3_4_1"/><b>then </b>
 <a href="sin_cos.html#K21">sin</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="complex2.html#E26:87_3"><i><font color="Green">E109</font></i></a>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/><a NAME="E11:87_3_4_1"/><b>then </b>
 <a href="comptrig.html#K1">Arg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="complex2.html#E3:87_3_4_1"><i><font color="Green">E112</font></i></a>, <a class="txt" href="complex2.html#E4:87_3_4_1"><i><font color="Green">E113</font></i></a>, <a class="ref" href="complex2.html#T7" target="_self">Th7</a></i>;<br/><a NAME="E12:87_3_4_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><b>hence </b><a NAME="E13:87_3_4_1"/>
<span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="87_3_4_2"><b>suppose </b></a><a NAME="E1:87_3_4_2"/><i><font color="Green">E111</font></i>: 
0 <a href="hidden.html#R1">=</a>  <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 ;<br/><div class="add"><b>then </b><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font> = 
0 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T29">COMPLEX1:29</a></i>
<br/>.= 
<span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default">0,<span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span><a href="arytm_0.html#K5">*]</a></span>
<b>by </b><i><a class="ref" href="complex1.html#T27">COMPLEX1:27</a></i>
;<br/><a NAME="E3:87_3_4_2"/><b>then </b><i><font color="Green">E112</font></i>: 
 <a href="comptrig.html#K1">Arg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <a href="sin_cos.html#K33">PI</a>  <a href="real_1.html#K6">/</a> 2
 <b>by </b><i><a class="txt" href="complex2.html#E6:87_3"><i><font color="Green">E95</font></i></a>, <a class="txt" href="complex2.html#E7:87_3"><i><font color="Green">E96</font></i></a>, <a class="ref" href="complex2.html#T30" target="_self">Th30</a></i>;<br/><a NAME="E4:87_3_4_2"/>
 <a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default">0,<span class="p2">(<span class="default"><a href="sin_cos.html#K33">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="txt" href="complex2.html#E6:87_3"><i><font color="Green">E95</font></i></a>, <a class="txt" href="complex2.html#E8:87_3"><i><font color="Green">E97</font></i></a>, <a class="txt" href="complex2.html#E11:87_3"><i><font color="Green">E98</font></i></a>, <a class="txt" href="complex2.html#E1:87_3_4_2"><i><font color="Green">E111</font></i></a>, <a class="txt" href="complex2.html#E36"><i><font color="Green">Lemma25</font></i></a></i>;<br/><a NAME="E5:87_3_4_2"/><b>then </b>
 <a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="comptrig.html#K1">Arg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="complex2.html#E3:87_3_4_2"><i><font color="Green">E112</font></i></a>, <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/><a NAME="E6:87_3_4_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><b>hence </b><a NAME="E7:87_3_4_2"/>
<span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="87_3_4_3"><b>suppose </b></a><a NAME="E1:87_3_4_3"/><i><font color="Green">E111</font></i>: 
0 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 ;<br/><div class="add"><b>thus </b><a NAME="E2:87_3_4_3"/>
<span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_3_4_3_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="87_3_4_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:87_3_4_3_1_1"/>
( 0 <a href="xxreal_0.html#R1">&lt;</a>  <a href="complex1.html#K3">Re</a> <font color="Maroon">c<sub>4</sub></font> or 0 <a href="hidden.html#R1">=</a>  <a href="complex1.html#K3">Re</a> <font color="Maroon">c<sub>4</sub></font> or 0 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="complex1.html#K3">Re</a> <font color="Maroon">c<sub>4</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="87_3_4_3_1_1_1"><b>suppose </b></a><a NAME="E1:87_3_4_3_1_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;</a>  <a href="complex1.html#K3">Re</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><a NAME="E2:87_3_4_3_1_1_1"/><b>then </b>
 <a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default">0,<span class="p2">(<span class="default"><a href="sin_cos.html#K33">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="txt" href="complex2.html#E6:87_3"><i><font color="Green">E95</font></i></a>, <a class="txt" href="complex2.html#E36"><i><font color="Green">Lemma25</font></i></a></i>;<br/><a NAME="E3:87_3_4_3_1_1_1"/><b>then </b><i><font color="Green">E112</font></i>: 
 <a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <a href="sin_cos.html#K33">PI</a>  <a href="real_1.html#K6">/</a> 2
 <b>by </b><i><a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/><a NAME="E4:87_3_4_3_1_1_1"/>
 <a href="comptrig.html#K1">Arg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K33">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span>,<a href="sin_cos.html#K33">PI</a> </span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="txt" href="complex2.html#E6:87_3"><i><font color="Green">E95</font></i></a>, <a class="txt" href="complex2.html#E7:87_3"><i><font color="Green">E96</font></i></a>, <a class="txt" href="complex2.html#E1:87_3_4_3"><i><font color="Green">E111</font></i></a>, <a class="txt" href="complex2.html#E37"><i><font color="Green">Lemma26</font></i></a></i>;<br/><a NAME="E5:87_3_4_3_1_1_1"/><b>then </b>
 <a href="comptrig.html#K1">Arg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <a href="sin_cos.html#K33">PI</a>  <a href="real_1.html#K6">/</a> 2
 <b>by </b><i><a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/><a NAME="E6:87_3_4_3_1_1_1"/><b>then </b>
 <a href="comptrig.html#K1">Arg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="complex2.html#E3:87_3_4_3_1_1_1"><i><font color="Green">E112</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E7:87_3_4_3_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><b>hence </b><a NAME="E8:87_3_4_3_1_1_1"/>
<span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="87_3_4_3_1_1_2"><b>suppose </b></a><a NAME="E1:87_3_4_3_1_1_2"/>
0 <a href="hidden.html#R1">=</a>  <a href="complex1.html#K3">Re</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><b>then </b><font color="Maroon">c<sub>4</sub></font> = 
0 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T29">COMPLEX1:29</a></i>
<br/>.= 
<span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default">0,<span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="arytm_0.html#K5">*]</a></span>
<b>by </b><i><a class="ref" href="complex1.html#T27">COMPLEX1:27</a></i>
;<br/><a NAME="E3:87_3_4_3_1_1_2"/><b>then </b><i><font color="Green">E112</font></i>: 
 <a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <a href="sin_cos.html#K33">PI</a>  <a href="real_1.html#K6">/</a> 2
 <b>by </b><i><a class="txt" href="complex2.html#E6:87_3"><i><font color="Green">E95</font></i></a>, <a class="ref" href="complex2.html#T30" target="_self">Th30</a></i>;<br/><a NAME="E4:87_3_4_3_1_1_2"/>
 <a href="comptrig.html#K1">Arg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K33">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span>,<a href="sin_cos.html#K33">PI</a> </span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="txt" href="complex2.html#E6:87_3"><i><font color="Green">E95</font></i></a>, <a class="txt" href="complex2.html#E7:87_3"><i><font color="Green">E96</font></i></a>, <a class="txt" href="complex2.html#E1:87_3_4_3"><i><font color="Green">E111</font></i></a>, <a class="txt" href="complex2.html#E37"><i><font color="Green">Lemma26</font></i></a></i>;<br/><a NAME="E5:87_3_4_3_1_1_2"/><b>then </b>
 <a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="comptrig.html#K1">Arg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="complex2.html#E3:87_3_4_3_1_1_2"><i><font color="Green">E112</font></i></a>, <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/><a NAME="E6:87_3_4_3_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><b>hence </b><a NAME="E7:87_3_4_3_1_1_2"/>
<span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="87_3_4_3_1_1_3"><b>suppose </b></a><a NAME="E1:87_3_4_3_1_1_3"/><i><font color="Green">E112</font></i>: 
0 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="complex1.html#K3">Re</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><a NAME="E2:87_3_4_3_1_1_3"/><b>then </b><i><font color="Green">E113</font></i>: 
 <a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K33">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span>,<a href="sin_cos.html#K33">PI</a> </span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="txt" href="complex2.html#E6:87_3"><i><font color="Green">E95</font></i></a>, <a class="txt" href="complex2.html#E37"><i><font color="Green">Lemma26</font></i></a></i>;<br/><a NAME="E3:87_3_4_3_1_1_3"/><i><font color="Green">E114</font></i>: 
 <a href="comptrig.html#K1">Arg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K33">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span>,<a href="sin_cos.html#K33">PI</a> </span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="txt" href="complex2.html#E6:87_3"><i><font color="Green">E95</font></i></a>, <a class="txt" href="complex2.html#E7:87_3"><i><font color="Green">E96</font></i></a>, <a class="txt" href="complex2.html#E1:87_3_4_3"><i><font color="Green">E111</font></i></a>, <a class="txt" href="complex2.html#E37"><i><font color="Green">Lemma26</font></i></a></i>;<br/><a NAME="E4:87_3_4_3_1_1_3"/>
<span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> 
 <b>by </b><i><a class="txt" href="complex2.html#E8:87_3"><i><font color="Green">E97</font></i></a>, <a class="txt" href="complex2.html#E11:87_3"><i><font color="Green">E98</font></i></a>, <a class="txt" href="complex2.html#E1:87_3_4_3_1_1_3"><i><font color="Green">E112</font></i></a>, <a class="ref" href="jgraph_5.html#T2">JGRAPH_5:2</a></i>;<br/><a NAME="E5:87_3_4_3_1_1_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E6:87_3_4_3_1_1_3"/><b>then </b>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span>
 <b>by </b><i><a class="txt" href="complex2.html#E18:87_3"><i><font color="Green">E103</font></i></a>, <a class="txt" href="complex2.html#E19:87_3"><i><font color="Green">E104</font></i></a>, <a class="txt" href="complex2.html#E25:87_3"><i><font color="Green">E108</font></i></a>, <a class="ref" href="square_1.html#T95">SQUARE_1:95</a></i>;<br/><a NAME="E7:87_3_4_3_1_1_3"/><b>then </b>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K6">/</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 <b>by </b><i><a class="txt" href="complex2.html#E20:87_3"><i><font color="Green">E105</font></i></a>, <a class="ref" href="real_2.html#T142">REAL_2:142</a></i>;<br/><a NAME="E8:87_3_4_3_1_1_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p3">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p3">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="complex2.html#E26:87_3"><i><font color="Green">E109</font></i></a>, <a class="txt" href="complex2.html#E27:87_3"><i><font color="Green">E110</font></i></a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E9:87_3_4_3_1_1_3"/><b>then </b>
 <a href="sin_cos.html#K21">sin</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="complex2.html#E26:87_3"><i><font color="Green">E109</font></i></a>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/><a NAME="E10:87_3_4_3_1_1_3"/><b>then </b>
 <a href="comptrig.html#K1">Arg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="complex2.html#E2:87_3_4_3_1_1_3"><i><font color="Green">E113</font></i></a>, <a class="txt" href="complex2.html#E3:87_3_4_3_1_1_3"><i><font color="Green">E114</font></i></a>, <a class="ref" href="complex2.html#T8" target="_self">Th8</a></i>;<br/><a NAME="E11:87_3_4_3_1_1_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><b>hence </b><a NAME="E12:87_3_4_3_1_1_3"/>
<span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E18:87"/><b>then </b>
<span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="complex2.html#E1:87"><i><font color="Green">E80</font></i></a>, <a class="txt" href="complex2.html#E2:87"><i><font color="Green">E81</font></i></a>, <a class="txt" href="complex2.html#E3:87"><i><font color="Green">E82</font></i></a>, <a class="txt" href="complex2.html#E4:87"><i><font color="Green">E83</font></i></a></i>;<br/>
<a NAME="E19:87"/><b>then </b>
<span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><a href="complex1.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
;<br/>
<a NAME="E20:87"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="complex2.html#E13:87"><i><font color="Green">E88</font></i></a>, <a class="txt" href="complex2.html#E14:87"><i><font color="Green">E89</font></i></a>, <a class="ref" href="complex2.html#T22" target="_self">Th22</a>, <a class="ref" href="complex1.html#T12">COMPLEX1:12</a></i>;<br/>
<a NAME="E21:87"/><b>then </b>
<span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
;<br/>
<a NAME="E22:87"/><b>then </b>
<span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><a href="complex1.html#K10">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="complex2.html#E4:87"><i><font color="Green">E83</font></i></a>, <a class="txt" href="complex2.html#E10:87"><i><font color="Green">E86</font></i></a>, <a class="ref" href="complex2.html#T22" target="_self">Th22</a>, <a class="ref" href="complex1.html#T12">COMPLEX1:12</a></i>;<br/>
<a NAME="E23:87"/><b>then </b><i><font color="Green">E91</font></i>: 
<span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><a href="complex1.html#K5">0c</a>  <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
;<br/>
<a NAME="E24:87"/><b>then </b>
 <a href="complex2.html#K5" target="_self">angle</a> <a href="complex1.html#K5">0c</a> ,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><a href="complex1.html#K5">0c</a>  <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="complex2.html#D6" target="_self">Def6</a></i>;<br/>
<a NAME="E25:87"/><b>then </b><i><font color="Green">E92</font></i>: 
 <a href="complex2.html#K5" target="_self">angle</a> <a href="complex1.html#K5">0c</a> ,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><a href="complex1.html#K10">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T150">XCMPLX_1:150</a></i>;<br/>
<a NAME="E26:87"/>
 <a href="comptrig.html#K1">Arg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="sin_cos.html#K33">PI</a> 
 
<b>by </b><i><a class="txt" href="complex2.html#E13:87"><i><font color="Green">E88</font></i></a>, <a class="txt" href="complex2.html#E14:87"><i><font color="Green">E89</font></i></a>, <a class="ref" href="complex2.html#T26" target="_self">Th26</a>, <a class="ref" href="complex1.html#T12">COMPLEX1:12</a></i>;<br/>
<a NAME="E27:87"/><b>then </b>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="real_1.html#K1">-</a> <a href="sin_cos.html#K33">PI</a> 
 
<b>by </b><i><a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/>
<a NAME="E28:87"/><b>then </b>
<span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><a href="complex1.html#K10">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default">0 <a href="real_1.html#K5">-</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="real_1.html#K3">+</a> <a href="sin_cos.html#K33">PI</a> 
 
<b>by </b><i><a class="txt" href="complex2.html#E8:87"><i><font color="Green">E85</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E29:87"/><b>then </b>
<span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><a href="complex1.html#K10">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
;<br/>
<a NAME="E30:87"/><b>then </b>
<span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><a href="complex1.html#K5">0c</a>  <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
;<br/>
<a NAME="E31:87"/><b>then </b><i><font color="Green">E93</font></i>: 
 <a href="complex2.html#K5" target="_self">angle</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<a href="complex1.html#K5">0c</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><a href="complex1.html#K5">0c</a>  <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="complex2.html#D6" target="_self">Def6</a></i>;<br/>
<a NAME="E32:87"/><b>then </b><i><font color="Green">E94</font></i>: 
 <a href="complex2.html#K5" target="_self">angle</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<a href="complex1.html#K5">0c</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><a href="complex1.html#K10">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T150">XCMPLX_1:150</a></i>;<br/>
<i><font color="Green">E95</font></i>: <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><a href="complex1.html#K10">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <a href="sin_cos.html#K33">PI</a> </span>)</span>
<b>by </b><i><a class="txt" href="complex2.html#E4:87"><i><font color="Green">E83</font></i></a>, <a class="txt" href="complex2.html#E10:87"><i><font color="Green">E86</font></i></a>, <a class="ref" href="complex2.html#T22" target="_self">Th22</a>, <a class="ref" href="complex1.html#T12">COMPLEX1:12</a></i>
<br/>.= 
 <a href="real_1.html#K1">-</a> <a href="sin_cos.html#K33">PI</a> 

;<br/>
<i><font color="Green">E96</font></i>: <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><a href="complex1.html#K10">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <a href="sin_cos.html#K33">PI</a> </span>)</span>
<b>by </b><i><a class="txt" href="complex2.html#E2:87"><i><font color="Green">E81</font></i></a>, <a class="txt" href="complex2.html#E7:87"><i><font color="Green">E84</font></i></a>, <a class="ref" href="complex2.html#T22" target="_self">Th22</a>, <a class="ref" href="complex1.html#T12">COMPLEX1:12</a>, <a class="ref" href="comptrig.html#T21">COMPTRIG:21</a></i>
<br/>.= 
 <a href="real_1.html#K1">-</a> <a href="sin_cos.html#K33">PI</a> 

;<br/>
<i><font color="Green">E97</font></i>: <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><a href="complex1.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <a href="sin_cos.html#K33">PI</a> </span>)</span>
<b>by </b><i><a class="txt" href="complex2.html#E13:87"><i><font color="Green">E88</font></i></a>, <a class="txt" href="complex2.html#E14:87"><i><font color="Green">E89</font></i></a>, <a class="ref" href="complex2.html#T22" target="_self">Th22</a>, <a class="ref" href="complex1.html#T12">COMPLEX1:12</a></i>
<br/>.= 
 <a href="sin_cos.html#K33">PI</a> 

;<br/>
<b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex2.html#K5" target="_self">angle</a> <font color="Maroon">c<sub>1</sub></font>,<a href="complex1.html#K5">0c</a> ,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="complex2.html#K5" target="_self">angle</a> <a href="complex1.html#K5">0c</a> ,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="complex2.html#K5" target="_self">angle</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<a href="complex1.html#K5">0c</a> </span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p5">(<span class="default"><a href="complex1.html#K10">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p3">(<span class="default"><a href="complex1.html#K10">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="complex2.html#E16:87"><i><font color="Green">E90</font></i></a>, <a class="txt" href="complex2.html#E25:87"><i><font color="Green">E92</font></i></a>, <a class="txt" href="complex2.html#E31:87"><i><font color="Green">E93</font></i></a>, <a class="ref" href="xcmplx_1.html#T150">XCMPLX_1:150</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <a href="sin_cos.html#K33">PI</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="complex2.html#E33:87"><i><font color="Green">E95</font></i></a>, <a class="txt" href="complex2.html#E34:87"><i><font color="Green">E96</font></i></a></i>
<br/>.= 
 <a href="sin_cos.html#K33">PI</a> 
<b>by </b><i><a class="txt" href="complex2.html#E35:87"><i><font color="Green">E97</font></i></a></i>
;<br/>

<b>thus </b><a NAME="E37:87"/>
0 <a href="xxreal_0.html#R1">&lt;</a>  <a href="complex2.html#K5" target="_self">angle</a> <a href="complex1.html#K5">0c</a> ,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="complex2.html#E23:87"><i><font color="Green">E91</font></i></a>, <a class="ref" href="complex2.html#D6" target="_self">Def6</a></i>;<br/>

<a NAME="E38:87"/>
<span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <a href="sin_cos.html#K33">PI</a>  <a href="hidden.html#R1">=</a>  <a href="comptrig.html#K1">Arg</a> <span class="p1">(<span class="default"><a href="complex1.html#K10">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="complex2.html#E34:87"><i><font color="Green">E96</font></i></a></i>;<br/>
<b>then </b><i><font color="Green">E98</font></i>: <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><a href="complex1.html#K10">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="real_1.html#K3">+</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span>

;<br/>
<a NAME="E40:87"/>
2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="complex2.html#E18"><i><font color="Green">Lemma12</font></i></a></i>;<br/>
<a NAME="E41:87"/><b>then </b>
( <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 
<b>by </b><i><a class="txt" href="complex2.html#E18"><i><font color="Green">Lemma12</font></i></a>, <a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/>
<a NAME="E42:87"/><b>then </b>
<span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 <a href="real_1.html#K3">+</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<b>hence </b><a NAME="E43:87"/>
0 <a href="xxreal_0.html#R1">&lt;</a>  <a href="complex2.html#K5" target="_self">angle</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<a href="complex1.html#K5">0c</a> 
 <b>by </b><i><a class="txt" href="complex2.html#E32:87"><i><font color="Green">E94</font></i></a>, <a class="txt" href="complex2.html#E35:87"><i><font color="Green">E97</font></i></a>, <a class="txt" href="complex2.html#E39:87"><i><font color="Green">E98</font></i></a></i>;<br/>


</div>
