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

<b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a> ;<br/><b>let </b><font color="Maroon">b</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="E1:85"/><i><font color="Green">E23</font></i>: 
 <a href="complex1.html#K4">Im</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0
 <b>and </b><br/><a NAME="E2:85"/><i><font color="Green">E26</font></i>: 
 <a href="complex1.html#K3">Re</a> <font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>and </b><br/><a NAME="E3:85"/><i><font color="Green">E29</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a>  <a href="comptrig.html#K1">Arg</a> <font color="Maroon">b</font>
 <b>and </b><br/><a NAME="E4:85"/><i><font color="Green">E30</font></i>: 
 <a href="comptrig.html#K1">Arg</a> <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="sin_cos.html#K35">PI</a> 
 ;<br/>

<a NAME="E5:85"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">a</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="ref" href="complex2.html#T34" target="_self">Th34</a>, <a class="ref" href="complex1.html#T29">COMPLEX1:29</a></i>;<br/>
<a NAME="E6:85"/><b>then </b><i><font color="Green">E31</font></i>: 
 <a href="comptrig.html#K1">Arg</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="complex2.html#T35" target="_self">Th35</a>, <a class="ref" href="comptrig.html#T53">COMPTRIG:53</a></i>;<br/>
<b>then </b><i><font color="Green">E32</font></i>:  <a href="comptrig.html#K1">Arg</a> <span class="p1">(<span class="default"><a href="complex1.html#K10">-</a> <font color="Maroon">a</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K3">+</a> <a href="sin_cos.html#K35">PI</a> 
<b>by </b><i><a class="ref" href="complex2.html#T35" target="_self">Th35</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#K35">PI</a> 
<b>by </b><i><a class="ref" href="complex2.html#T38" target="_self">Th38</a></i>
;<br/>
<a NAME="E8:85"/>
 <a href="comptrig.html#K1">Arg</a> <font color="Maroon">b</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#K35">PI</a> </span><a href="rcomp_1.html#K2">.[</a></span>
 
<b>by </b><i><a class="ref" href="complex2.html#T36" target="_self">Th36</a>, <a class="ref" href="complex2.html#T37" target="_self">Th37</a>, <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/>
<a NAME="E9:85"/><b>then </b><i><font color="Green">E33</font></i>: 
 <a href="complex1.html#K4">Im</a> <font color="Maroon">b</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i/>;<br/>
<i><font color="Green">E48</font></i>:  <a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="complex1.html#K11">-</a> <font color="Maroon">b</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">a</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">b</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">b</font></span>)</span>
<b>by </b><i><a class="ref" href="complex2.html#T34" target="_self">Th34</a></i>
;<br/>
<a NAME="E11:85"/>
 <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">b</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K4">Im</a> <font color="Maroon">b</font>
 
;<br/>
<a NAME="E12:85"/><b>then </b><i><font color="Green">E49</font></i>: 
 <a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="complex1.html#K11">-</a> <font color="Maroon">b</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> 0
 
<b>by </b><i><a class="ref" href="complex2.html#T40" target="_self">Th40</a>, <a class="ref" href="complex2.html#T41" target="_self">Th41</a>, <a class="ref" href="xreal_1.html#T60">XREAL_1:60</a></i>;<br/>
<a NAME="E13:85"/><b>then </b><i><font color="Green">E50</font></i>: 
 <a href="comptrig.html#K1">Arg</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="complex1.html#K11">-</a> <font color="Maroon">b</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="sin_cos.html#K35">PI</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E14:85"/>
<span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon">b</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">a</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="ref" href="complex2.html#T36" target="_self">Th36</a>, <a class="ref" href="complex2.html#T38" target="_self">Th38</a></i>;<br/>
<a NAME="E15:85"/><b>then </b><i><font color="Green">E51</font></i>: 
 <a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">a</font>,<a href="complex1.html#K5">0c</a> ,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">b</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">a</font></span>)</span>
 
<b>by </b><i><a class="ref" href="complex2.html#T21" target="_self">Th21</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a> , <font color="Maroon">b</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="E1:85_3"/><i><font color="Green">E52</font></i>: 
 <a href="complex1.html#K4">Im</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0
 <b>and </b><br/><a NAME="E2:85_3"/><i><font color="Green">E68</font></i>: 
 <a href="complex1.html#K3">Re</a> <font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>and </b><br/><a NAME="E3:85_3"/><i><font color="Green">E69</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a>  <a href="comptrig.html#K1">Arg</a> <font color="Maroon">b</font>
 <b>and </b><br/><a NAME="E4:85_3"/><i><font color="Green">E70</font></i>: 
 <a href="comptrig.html#K1">Arg</a> <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="sin_cos.html#K35">PI</a> 
 ;<br/><a NAME="E5:85_3"/>
 <a href="comptrig.html#K1">Arg</a> <font color="Maroon">b</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#K35">PI</a> </span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="ref" href="complex2.html#T47" target="_self">Th47</a>, <a class="ref" href="complex2.html#T49" target="_self">Th49</a>, <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/><a NAME="E6:85_3"/><b>then </b><i><font color="Green">E120</font></i>: 
 <a href="complex1.html#K4">Im</a> <font color="Maroon">b</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i/>;<br/><i><font color="Green">E121</font></i>:  <a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">b</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">a</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">b</font>
<b>by </b><i><a class="ref" href="complex2.html#T44" target="_self">Th44</a></i>
;<br/><a NAME="E8:85_3"/><i><font color="Green">E122</font></i>: 
 <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</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">b</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">a</font></span>)</span>
 <b>by </b><i><a class="ref" href="complex1.html#T48">COMPLEX1:48</a></i>;<br/><a NAME="E9:85_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">a</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K3">Re</a> <font color="Maroon">a</font>
 ;<br/><a NAME="E10:85_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">a</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> 0
 <b>by </b><i><a class="ref" href="complex2.html#T45" target="_self">Th45</a>, <a class="ref" href="xreal_1.html#T60">XREAL_1:60</a></i>;<br/><a NAME="E11:85_3"/><b>then </b><i><font color="Green">E123</font></i>: 
<span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">b</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">a</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">b</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">ba</font> = <font color="Maroon">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</font>;<br/><b>set </b><font color="Maroon">Rb</font> =  <a href="complex1.html#K3">Re</a> <font color="Maroon">b</font>;<br/><b>set </b><font color="Maroon">Rba</font> =  <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</font></span>)</span>;<br/><b>set </b><font color="Maroon">Ib</font> =  <a href="complex1.html#K4">Im</a> <font color="Maroon">b</font>;<br/><b>set </b><font color="Maroon">Iba</font> =  <a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</font></span>)</span>;<br/><b>set </b><font color="Maroon">B</font> =  <a href="comptrig.html#K1">Arg</a> <font color="Maroon">b</font>;<br/><b>set </b><font color="Maroon">BA</font> =  <a href="comptrig.html#K1">Arg</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</font></span>)</span>;<br/><b>set </b><font color="Maroon">mb</font> = <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">b</font></span><a href="complex1.html#K17">.|</a></span>;<br/><b>set </b><font color="Maroon">mba</font> = <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</font></span>)</span></span><a href="complex1.html#K17">.|</a></span>;<br/><a NAME="E12:85_3"/><i><font color="Green">E124</font></i>: 
<font color="Maroon">b</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">b</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">b</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">b</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">b</font></span>)</span></span>)</span></span>)</span></span><a href="arytm_0.html#K5">*]</a></span>
 <b>by </b><i/>;<br/><font color="Maroon">b</font> = 
<span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">b</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">b</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">b</font></span>)</span>,<span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">b</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="E14:85_3"/><b>then </b><i><font color="Green">E125</font></i>: 
(  <a href="complex1.html#K3">Re</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">b</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">b</font></span>)</span></span>)</span> &amp;  <a href="complex1.html#K4">Im</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">b</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">b</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="arytm_0.html#T12">ARYTM_0:12</a>, <a class="ref" href="complex2.html#T63" target="_self">Th63</a></i>;<br/><a NAME="E15:85_3"/><i><font color="Green">E126</font></i>: 
<font color="Maroon">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</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">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</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">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</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">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</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">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="arytm_0.html#K5">*]</a></span>
 <b>by </b><i/>;<br/><font color="Maroon">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</font> = 
<span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</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">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</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">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</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">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</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="E17:85_3"/><b>then </b><i><font color="Green">E127</font></i>: 
(  <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</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">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</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">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> &amp;  <a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</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">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</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">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</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="ref" href="complex2.html#T64" target="_self">Th64</a></i>;<br/><a NAME="E18:85_3"/><i><font color="Green">E128</font></i>: 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">b</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">b</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">b</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="E19:85_3"/><i><font color="Green">E129</font></i>: 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</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">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</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">b</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="complex2.html#T55" target="_self">Th55</a>, <a class="ref" href="complex1.html#D16">COMPLEX1:def 16</a></i>;<br/><a NAME="E20:85_3"/><i><font color="Green">E160</font></i>: 
( <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">b</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">b</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="ref" href="complex2.html#T51" target="_self">Th51</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="E21:85_3"/><i><font color="Green">E161</font></i>: 
( <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</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">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</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="ref" href="complex2.html#T51" target="_self">Th51</a>, <a class="ref" href="complex2.html#T55" target="_self">Th55</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="E22:85_3"/>
( <span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</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">b</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="E23:85_3"/><b>then </b><i><font color="Green">E162</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">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</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">b</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="E24:85_3"/>
( <span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">b</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">b</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="E25:85_3"/><b>then </b><i><font color="Green">E163</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">b</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">b</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="E26:85_3"/><i><font color="Green">E164</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">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="complex2.html#T51" target="_self">Th51</a>, <a class="ref" href="complex2.html#T55" target="_self">Th55</a>, <a class="ref" href="comptrig.html#T63">COMPTRIG:63</a></i>;<br/><a NAME="E27:85_3"/><b>then </b><i><font color="Green">E165</font></i>: 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</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">b</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">b</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">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="complex2.html#T55" target="_self">Th55</a>, <a class="txt" href="complex2.html#E68"><i><font color="Green">Lemma91</font></i></a>, <a class="ref" href="complex2.html#T66" target="_self">Th66</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="85_3_4"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:85_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">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</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">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</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">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</font></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="85_3_4_1"><b>suppose </b></a><a NAME="E1:85_3_4_1"/><i><font color="Green">E166</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">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:85_3_4_1"/><b>then </b><i><font color="Green">E167</font></i>: 
 <a href="comptrig.html#K1">Arg</a> <font color="Maroon">b</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#K35">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="ref" href="complex2.html#T51" target="_self">Th51</a>, <a class="ref" href="comptrig.html#T59">COMPTRIG:59</a>, <a class="ref" href="complex2.html#T57" target="_self">Th57</a>, <a class="ref" href="complex2.html#T60" target="_self">Th60</a></i>;<br/><a NAME="E3:85_3_4_1"/><i><font color="Green">E168</font></i>: 
 <a href="comptrig.html#K1">Arg</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</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#K35">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="ref" href="complex2.html#T51" target="_self">Th51</a>, <a class="ref" href="complex2.html#T55" target="_self">Th55</a>, , <a class="ref" href="comptrig.html#T59">COMPTRIG:59</a></i>;<br/><a NAME="E4:85_3_4_1"/>
<span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">b</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">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> 
 <b>by </b><i><a class="ref" href="complex2.html#T57" target="_self">Th57</a>, <a class="ref" href="complex2.html#T60" target="_self">Th60</a>, , <a class="ref" href="square_1.html#T78">SQUARE_1:78</a></i>;<br/><a NAME="E5:85_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">b</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">b</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">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</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">b</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="E6:85_3_4_1"/><b>then </b>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">b</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">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</font></span>)</span></span><a href="complex1.html#K17">.|</a></span>
 <b>by </b><i><a class="ref" href="complex2.html#T67" target="_self">Th67</a>, <a class="ref" href="complex2.html#T68" target="_self">Th68</a>, <a class="ref" href="complex2.html#T70" target="_self">Th70</a>, <a class="ref" href="square_1.html#T95">SQUARE_1:95</a></i>;<br/><a NAME="E7:85_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">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</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">b</font></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> 1
 <b>by </b><i>, <a class="ref" href="real_2.html#T142">REAL_2:142</a></i>;<br/><a NAME="E8:85_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">b</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">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</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">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</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">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="complex2.html#T71" target="_self">Th71</a>, , <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E9:85_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">b</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">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="complex2.html#T71" target="_self">Th71</a>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/><a NAME="E10:85_3_4_1"/><b>then </b>
 <a href="comptrig.html#K1">Arg</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="comptrig.html#K1">Arg</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="complex2.html#T72" target="_self">Th72</a>, <a class="ref" href="complex2.html#T73" target="_self">Th73</a>, </i>;<br/><a NAME="E11:85_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">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</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">b</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">b</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">b</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="E12:85_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">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</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">b</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="85_3_4_2"><b>suppose </b></a><a NAME="E1:85_3_4_2"/><i><font color="Green">E169</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">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:85_3_4_2"/><b>then </b>
<font color="Maroon">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 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">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</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/><a NAME="E3:85_3_4_2"/><b>then </b><i><font color="Green">E170</font></i>: 
 <a href="comptrig.html#K1">Arg</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> <a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2
 <b>by </b><i><a class="ref" href="complex2.html#T51" target="_self">Th51</a>, <a class="ref" href="complex2.html#T55" target="_self">Th55</a>, <a class="ref" href="comptrig.html#T55">COMPTRIG:55</a></i>;<br/><a NAME="E4:85_3_4_2"/>
 <a href="comptrig.html#K1">Arg</a> <font color="Maroon">b</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#K35">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="ref" href="complex2.html#T51" target="_self">Th51</a>, <a class="ref" href="complex2.html#T57" target="_self">Th57</a>, <a class="ref" href="complex2.html#T60" target="_self">Th60</a>, , <a class="ref" href="comptrig.html#T59">COMPTRIG:59</a></i>;<br/><a NAME="E5:85_3_4_2"/><b>then </b>
 <a href="comptrig.html#K1">Arg</a> <font color="Maroon">b</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">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</font></span>)</span>
 <b>by </b><i><a class="ref" href="complex2.html#T72" target="_self">Th72</a>, <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/><a NAME="E6:85_3_4_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">b</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">b</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">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</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">b</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="E7:85_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">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</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">b</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="85_3_4_3"><b>suppose </b></a><a NAME="E1:85_3_4_3"/><i><font color="Green">E171</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">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</font></span>)</span>
 ;<br/><div class="add"><b>thus </b><a NAME="E2:85_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">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</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">b</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_3_4_3_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="85_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="E1:85_3_4_3_1_1"/>
( 0 <a href="xxreal_0.html#R1">&lt;</a>  <a href="complex1.html#K3">Re</a> <font color="Maroon">b</font> or 0 <a href="hidden.html#R1">=</a>  <a href="complex1.html#K3">Re</a> <font color="Maroon">b</font> or 0 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="complex1.html#K3">Re</a> <font color="Maroon">b</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="85_3_4_3_1_1_1"><b>suppose </b></a><a NAME="E1:85_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">b</font>
 ;<br/><div class="add"><a NAME="E2:85_3_4_3_1_1_1"/><b>then </b>
 <a href="comptrig.html#K1">Arg</a> <font color="Maroon">b</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#K35">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="ref" href="complex2.html#T51" target="_self">Th51</a>, <a class="ref" href="comptrig.html#T59">COMPTRIG:59</a></i>;<br/><a NAME="E3:85_3_4_3_1_1_1"/><b>then </b><i><font color="Green">E172</font></i>: 
 <a href="comptrig.html#K1">Arg</a> <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;</a> <a href="sin_cos.html#K35">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="E4:85_3_4_3_1_1_1"/>
 <a href="comptrig.html#K1">Arg</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</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#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span>,<a href="sin_cos.html#K35">PI</a> </span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="ref" href="complex2.html#T51" target="_self">Th51</a>, <a class="ref" href="complex2.html#T55" target="_self">Th55</a>, , <a class="ref" href="comptrig.html#T60">COMPTRIG:60</a></i>;<br/><a NAME="E5:85_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">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <a href="sin_cos.html#K35">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="E6:85_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">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="comptrig.html#K1">Arg</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="complex2.html#T72" target="_self">Th72</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E7:85_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">b</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">b</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">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</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">b</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="E8:85_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">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</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">b</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="85_3_4_3_1_1_2"><b>suppose </b></a><a NAME="E1:85_3_4_3_1_1_2"/>
0 <a href="hidden.html#R1">=</a>  <a href="complex1.html#K3">Re</a> <font color="Maroon">b</font>
 ;<br/><div class="add"><a NAME="E2:85_3_4_3_1_1_2"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 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">b</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/><a NAME="E3:85_3_4_3_1_1_2"/><b>then </b><i><font color="Green">E173</font></i>: 
 <a href="comptrig.html#K1">Arg</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2
 <b>by </b><i><a class="ref" href="complex2.html#T51" target="_self">Th51</a>, <a class="ref" href="comptrig.html#T55">COMPTRIG:55</a></i>;<br/><a NAME="E4:85_3_4_3_1_1_2"/>
 <a href="comptrig.html#K1">Arg</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</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#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span>,<a href="sin_cos.html#K35">PI</a> </span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="ref" href="complex2.html#T51" target="_self">Th51</a>, <a class="ref" href="complex2.html#T55" target="_self">Th55</a>, , <a class="ref" href="comptrig.html#T60">COMPTRIG:60</a></i>;<br/><a NAME="E5:85_3_4_3_1_1_2"/><b>then </b>
 <a href="comptrig.html#K1">Arg</a> <font color="Maroon">b</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">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</font></span>)</span>
 <b>by </b><i><a class="ref" href="complex2.html#T72" target="_self">Th72</a>, <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/><a NAME="E6:85_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">b</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">b</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">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</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">b</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="E7:85_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">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</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">b</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="85_3_4_3_1_1_3"><b>suppose </b></a><a NAME="E1:85_3_4_3_1_1_3"/><i><font color="Green">E174</font></i>: 
0 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="complex1.html#K3">Re</a> <font color="Maroon">b</font>
 ;<br/><div class="add"><a NAME="E2:85_3_4_3_1_1_3"/><b>then </b><i><font color="Green">E175</font></i>: 
 <a href="comptrig.html#K1">Arg</a> <font color="Maroon">b</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#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span>,<a href="sin_cos.html#K35">PI</a> </span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="ref" href="complex2.html#T51" target="_self">Th51</a>, <a class="ref" href="comptrig.html#T60">COMPTRIG:60</a></i>;<br/><a NAME="E3:85_3_4_3_1_1_3"/><i><font color="Green">E176</font></i>: 
 <a href="comptrig.html#K1">Arg</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</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#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span>,<a href="sin_cos.html#K35">PI</a> </span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="ref" href="complex2.html#T51" target="_self">Th51</a>, <a class="ref" href="complex2.html#T55" target="_self">Th55</a>, , <a class="ref" href="comptrig.html#T60">COMPTRIG:60</a></i>;<br/><a NAME="E4:85_3_4_3_1_1_3"/>
<span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">b</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">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> 
 <b>by </b><i><a class="ref" href="complex2.html#T57" target="_self">Th57</a>, <a class="ref" href="complex2.html#T60" target="_self">Th60</a>, <a class="ref" href="complex2.html#T72" target="_self">Th72</a>, <a class="ref" href="square_1.html#T114">SQUARE_1:114</a></i>;<br/><a NAME="E5:85_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">b</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">b</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">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</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">b</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="E6:85_3_4_3_1_1_3"/><b>then </b>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">b</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">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</font></span>)</span></span><a href="complex1.html#K17">.|</a></span>
 <b>by </b><i><a class="ref" href="complex2.html#T67" target="_self">Th67</a>, <a class="ref" href="complex2.html#T68" target="_self">Th68</a>, , <a class="ref" href="square_1.html#T95">SQUARE_1:95</a></i>;<br/><a NAME="E7:85_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">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</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">b</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="ref" href="real_2.html#T142">REAL_2:142</a></i>;<br/><a NAME="E8:85_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">b</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">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</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">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</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">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="complex2.html#T71" target="_self">Th71</a>, , <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E9:85_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">b</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">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="complex2.html#T71" target="_self">Th71</a>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/><a NAME="E10:85_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">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="comptrig.html#K1">Arg</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="complex2.html#T73" target="_self">Th73</a>, <a class="ref" href="complex2.html#T74" target="_self">Th74</a>, </i>;<br/><a NAME="E11:85_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">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</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">b</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">b</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">b</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="E12:85_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">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</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">b</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="E17:85"/><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">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</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">b</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="complex2.html#T34" target="_self">Th34</a>, <a class="ref" href="complex2.html#T35" target="_self">Th35</a>, <a class="ref" href="complex2.html#T36" target="_self">Th36</a>, <a class="ref" href="complex2.html#T37" target="_self">Th37</a></i>;<br/>
<a NAME="E18:85"/><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">a</font> <a href="complex1.html#K11">-</a> <font color="Maroon">b</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">b</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
;<br/>
<a NAME="E19:85"/><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">a</font> <a href="complex1.html#K11">-</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <a href="sin_cos.html#K35">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">b</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i>, <a class="ref" href="complex2.html#T42" target="_self">Th42</a>, , <a class="ref" href="complex1.html#T12">COMPLEX1:12</a></i>;<br/>
<a NAME="E20:85"/><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">a</font> <a href="complex1.html#K11">-</a> <font color="Maroon">b</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">b</font></span>)</span> <a href="real_1.html#K3">+</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
;<br/>
<a NAME="E21:85"/><b>then </b><i><font color="Green">E177</font></i>: 
<span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="complex1.html#K11">-</a> <font color="Maroon">b</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">b</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="complex2.html#T37" target="_self">Th37</a>, <a class="ref" href="complex2.html#T40" target="_self">Th40</a>, , <a class="ref" href="complex1.html#T12">COMPLEX1:12</a></i>;<br/>
<a NAME="E22:85"/><b>then </b><i><font color="Green">E178</font></i>: 
<span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="complex1.html#K11">-</a> <font color="Maroon">b</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">b</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
;<br/>
<a NAME="E23:85"/><i><font color="Green">E179</font></i>: 
 <a href="complex2.html#K4" target="_self">angle</a> <a href="complex1.html#K5">0c</a> ,<font color="Maroon">b</font>,<font color="Maroon">a</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">a</font> <a href="complex1.html#K11">-</a> <font color="Maroon">b</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">b</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="complex2.html#T44" target="_self">Th44</a>, <a class="ref" href="complex2.html#T21" target="_self">Th21</a></i>;<br/>
<a NAME="E24:85"/>
 <a href="comptrig.html#K1">Arg</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="sin_cos.html#K35">PI</a> 
 
<b>by </b><i>, <a class="ref" href="complex2.html#T42" target="_self">Th42</a>, , <a class="ref" href="complex1.html#T12">COMPLEX1:12</a></i>;<br/>
<a NAME="E25:85"/><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">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</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#K35">PI</a> 
 
<b>by </b><i><a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/>
<a NAME="E26:85"/><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">a</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">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</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#K35">PI</a> </span>)</span> <a href="real_1.html#K3">+</a> <a href="sin_cos.html#K35">PI</a> 
 
<b>by </b><i><a class="ref" href="complex2.html#T39" target="_self">Th39</a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E27:85"/><b>then </b><i><font color="Green">E180</font></i>: 
 <a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">b</font>,<font color="Maroon">a</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">a</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">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="complex2.html#T21" target="_self">Th21</a></i>;<br/>
<i><font color="Green">E181</font></i>: <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">b</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">b</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">b</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">b</font></span>)</span> <a href="real_1.html#K3">+</a> <a href="sin_cos.html#K35">PI</a> </span>)</span>
<b>by </b><i><a class="ref" href="complex2.html#T37" target="_self">Th37</a>, <a class="ref" href="complex2.html#T40" target="_self">Th40</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#K35">PI</a> 

;<br/>
<i><font color="Green">E182</font></i>: <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">a</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">a</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">a</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">a</font></span>)</span> <a href="real_1.html#K3">+</a> <a href="sin_cos.html#K35">PI</a> </span>)</span>
<b>by </b><i><a class="ref" href="complex2.html#T35" target="_self">Th35</a>, <a class="ref" href="complex2.html#T38" target="_self">Th38</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#K35">PI</a> 

;<br/>
<i><font color="Green">E183</font></i>: <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="complex1.html#K11">-</a> <font color="Maroon">b</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">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</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">a</font> <a href="complex1.html#K11">-</a> <font color="Maroon">b</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">a</font> <a href="complex1.html#K11">-</a> <font color="Maroon">b</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">a</font> <a href="complex1.html#K11">-</a> <font color="Maroon">b</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">a</font> <a href="complex1.html#K11">-</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <a href="sin_cos.html#K35">PI</a> </span>)</span>
<b>by </b><i>, <a class="ref" href="complex2.html#T42" target="_self">Th42</a>, , <a class="ref" href="complex1.html#T12">COMPLEX1:12</a></i>
<br/>.= 
 <a href="sin_cos.html#K35">PI</a> 

;<br/>
<b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">a</font>,<a href="complex1.html#K5">0c</a> ,<font color="Maroon">b</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="complex2.html#K4" target="_self">angle</a> <a href="complex1.html#K5">0c</a> ,<font color="Maroon">b</font>,<font color="Maroon">a</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">b</font>,<font color="Maroon">a</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">b</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">b</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">a</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">a</font> <a href="complex1.html#K11">-</a> <font color="Maroon">b</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">a</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">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complex2.html#T43" target="_self">Th43</a>, <a class="ref" href="complex2.html#T47" target="_self">Th47</a>, <a class="ref" href="complex2.html#T49" target="_self">Th49</a></i>
<br/>.= 
 <a href="sin_cos.html#K35">PI</a> 
<b>by </b><i><a class="ref" href="complex2.html#T57" target="_self">Th57</a>, <a class="ref" href="complex2.html#T51" target="_self">Th51</a>, <a class="ref" href="complex2.html#T55" target="_self">Th55</a></i>
;<br/>

<b>thus </b><a NAME="E32:85"/>
0 <a href="xxreal_0.html#R1">&lt;</a>  <a href="complex2.html#K4" target="_self">angle</a> <a href="complex1.html#K5">0c</a> ,<font color="Maroon">b</font>,<font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="complex2.html#T45" target="_self">Th45</a>, <a class="ref" href="complex2.html#T21" target="_self">Th21</a></i>;<br/>

<a NAME="E33:85"/><i><font color="Green">E184</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">a</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">a</font> <a href="complex1.html#K11">-</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">a</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#K35">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">a</font> <a href="complex1.html#K11">-</a> <font color="Maroon">b</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="complex2.html#T55" target="_self">Th55</a></i>;<br/>
<a NAME="E34:85"/>
2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">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">a</font> <a href="complex1.html#K11">-</a> <font color="Maroon">b</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="comptrig.html#T52">COMPTRIG:52</a></i>;<br/>
<a NAME="E35:85"/><b>then </b>
( <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">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">a</font> <a href="complex1.html#K11">-</a> <font color="Maroon">b</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">a</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 
<b>by </b><i><a class="ref" href="comptrig.html#T52">COMPTRIG:52</a>, <a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/>
<a NAME="E36:85"/><b>then </b>
<span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">a</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#K35">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">a</font> <a href="complex1.html#K11">-</a> <font color="Maroon">b</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="E37:85"/>
0 <a href="xxreal_0.html#R1">&lt;</a>  <a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">b</font>,<font color="Maroon">a</font>,<a href="complex1.html#K5">0c</a> 
 <b>by </b><i><a class="ref" href="complex2.html#T49" target="_self">Th49</a>, <a class="ref" href="complex2.html#T57" target="_self">Th57</a>, <a class="ref" href="complex2.html#T60" target="_self">Th60</a></i>;<br/>


</div>
