<?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="xcmplx_0.html#V1">complex</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:22"/><i><font color="Green">E24</font></i>: 
(  <a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="sin_cos.html#K33">PI</a>  &amp;  <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="E2:22"/><i><font color="Green">E25</font></i>: 
<span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="complex1.html#K17">.|</a></span>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="complex1.html#K17">.|</a></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="complex1.html#T27">COMPLEX1:27</a></i>;<br/>
<a NAME="E3:22"/><i><font color="Green">E26</font></i>: 
<span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="complex1.html#K17">.|</a></span>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="complex1.html#K17">.|</a></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="complex1.html#T27">COMPLEX1:27</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:22_1_1"/>
(  <a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 or  <a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span><b>by </b><i><a class="txt" href="complex2.html#E18"><i><font color="Green">Lemma12</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_1_1"><b>case </b></a><a NAME="E1:22_1_1_1"/>
 <a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:22_1_1_1"/><b>then </b><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="complex1.html#K17">.|</a></span>,0</span><a href="arytm_0.html#K5">*]</a></span>
 <b>by </b><i><a class="ref" href="complex2.html#T24" target="_self">Th24</a></i>;<br/><a NAME="E3:22_1_1_1"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="complex1.html#K17">.|</a></span>,0</span><a href="arytm_0.html#K5">*]</a></span> &amp;  <a href="complex1.html#K4">Im</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="ref" href="complex1.html#T28">COMPLEX1:28</a>, <a class="txt" href="complex2.html#E2:22_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="complex2.html#E3:22"><i><font color="Green">E26</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:22_1_1_1_1_1"/>
(  <a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 or  <a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span><b>by </b><i><a class="txt" href="complex2.html#E18"><i><font color="Green">Lemma12</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:22_1_1_1_1_1_1"/>
 <a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:22_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="complex1.html#K17">.|</a></span>,0</span><a href="arytm_0.html#K5">*]</a></span>
 <b>by </b><i><a class="ref" href="complex2.html#T24" target="_self">Th24</a></i>;<br/><a NAME="E3:22_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E30</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="complex1.html#K17">.|</a></span>,0</span><a href="arytm_0.html#K5">*]</a></span> &amp;  <a href="complex1.html#K4">Im</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="ref" href="complex1.html#T28">COMPLEX1:28</a>, <a class="txt" href="complex2.html#E2:22"><i><font color="Green">E25</font></i></a></i>;<br/><a NAME="E4:22_1_1_1_1_1_1"/><i><font color="Green">E31</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="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <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#K3">+</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#T19">COMPLEX1:19</a></i>;<br/><a NAME="E5:22_1_1_1_1_1_1"/><i><font color="Green">E32</font></i>: 
 <a href="complex1.html#K3">Re</a> <font color="Maroon">c<sub>2</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>2</sub></font></span><a href="complex1.html#K17">.|</a></span>
 <b>by </b><i><a class="txt" href="complex2.html#E2:22_1_1_1_1_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="complex1.html#T28">COMPLEX1:28</a>, <a class="txt" href="complex2.html#E2:22"><i><font color="Green">E25</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_1_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:22_1_1_1_1_1_1_1_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K5">0c</a>  or <font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="complex1.html#K5">0c</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_1_1_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:22_1_1_1_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K5">0c</a> 
 ;<br/><div class="add"><a NAME="E2:22_1_1_1_1_1_1_1_1_1"/><b>then </b>
 <a href="comptrig.html#K1">Arg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="comptrig.html#D1">COMPTRIG:def 1</a></i>;<br/><b>hence </b><a NAME="E3:22_1_1_1_1_1_1_1_1_1"/>
 <a href="comptrig.html#K1">Arg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>2</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="ref" href="comptrig.html#T21">COMPTRIG:21</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_1_1_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:22_1_1_1_1_1_1_1_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="complex1.html#K5">0c</a> 
 ;<br/><div class="add"> <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> = 
<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="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T19">COMPLEX1:19</a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="complex1.html#K17">.|</a></span>
<b>by </b><i><a class="txt" href="complex2.html#E2:22_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="complex2.html#E5:22_1_1_1_1_1_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="complex2.html#E3:22"><i><font color="Green">E26</font></i></a>, <a class="ref" href="complex1.html#T28">COMPLEX1:28</a></i>
;<br/><a NAME="E3:22_1_1_1_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K3">+</a> <span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="complex1.html#K17">.|</a></span></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#E3:22_1_1_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="complex2.html#E3:22_1_1_1_1_1_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="complex2.html#E4:22_1_1_1_1_1_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="complex1.html#T29">COMPLEX1:29</a></i>;<br/><a NAME="E4:22_1_1_1_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>2</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"><span class="p3"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K3">+</a> <span class="p3"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="complex1.html#K17">.|</a></span></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="E5:22_1_1_1_1_1_1_1_1_2"/><i><font color="Green">E34</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="complex1.html#K17">.|</a></span>
 <b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/><a NAME="E6:22_1_1_1_1_1_1_1_1_2"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="complex1.html#K17">.|</a></span>
 <b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/><a NAME="E7:22_1_1_1_1_1_1_1_1_2"/><b>then </b>
0 <a href="real_1.html#K3">+</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="complex1.html#K17">.|</a></span>
 <b>by </b><i><a class="txt" href="complex2.html#E5:22_1_1_1_1_1_1_1_1_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><b>hence </b><a NAME="E8:22_1_1_1_1_1_1_1_1_2"/>
 <a href="comptrig.html#K1">Arg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>2</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#E4:22_1_1_1_1_1_1_1_1_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="comptrig.html#T21">COMPTRIG:21</a>, <a class="ref" href="comptrig.html#T53">COMPTRIG:53</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E7:22_1_1_1_1_1_1"/>
 <a href="comptrig.html#K1">Arg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a>  <a href="sin_cos.html#K33">PI</a> 
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:22_1_1_1_1_1_2"/>
 <a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:22_1_1_1_1_1_2"/><b>then </b>
 <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#E1:22"><i><font color="Green">E24</font></i></a>, <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/><a NAME="E3:22_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E29</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/><a NAME="E4:22_1_1_1_1_1_2"/>
 <a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <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#K3">+</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#T19">COMPLEX1:19</a></i>;<br/><a NAME="E5:22_1_1_1_1_1_2"/><b>then </b>
 <a href="comptrig.html#K1">Arg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>2</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,<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#E3:22_1_1_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="complex2.html#E3:22_1_1_1_1_1_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="complex2.html#T27" target="_self">Th27</a></i>;<br/><b>hence </b><a NAME="E6:22_1_1_1_1_1_2"/>
 <a href="comptrig.html#K1">Arg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>2</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="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:22_1_1_1"/>
 <a href="comptrig.html#K1">Arg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a>  <a href="sin_cos.html#K33">PI</a> 
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_1_2"><b>case </b></a><a NAME="E1:22_1_1_2"/>
 <a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:22_1_1_2"/><b>then </b>
 <a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>1</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#E1:22"><i><font color="Green">E24</font></i></a>, <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/><a NAME="E3:22_1_1_2"/><b>then </b><i><font color="Green">E27</font></i>: 
 <a href="complex1.html#K4">Im</a> <font color="Maroon">c<sub>1</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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:22_1_1_2_1_1"/>
(  <a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 or  <a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span><b>by </b><i><a class="txt" href="complex2.html#E18"><i><font color="Green">Lemma12</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_1_2_1_1_1"><b>case </b></a><a NAME="E1:22_1_1_2_1_1_1"/>
 <a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:22_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="complex1.html#K17">.|</a></span>,0</span><a href="arytm_0.html#K5">*]</a></span>
 <b>by </b><i><a class="ref" href="complex2.html#T24" target="_self">Th24</a></i>;<br/><a NAME="E3:22_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E28</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="complex1.html#K17">.|</a></span>,0</span><a href="arytm_0.html#K5">*]</a></span> &amp;  <a href="complex1.html#K4">Im</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="ref" href="complex1.html#T28">COMPLEX1:28</a>, <a class="txt" href="complex2.html#E2:22"><i><font color="Green">E25</font></i></a></i>;<br/><a NAME="E4:22_1_1_2_1_1_1"/>
 <a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <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#K3">+</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#T19">COMPLEX1:19</a></i>;<br/><a NAME="E5:22_1_1_2_1_1_1"/><b>then </b>
 <a href="comptrig.html#K1">Arg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>2</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,<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#E3:22_1_1_2"><i><font color="Green">E27</font></i></a>, <a class="txt" href="complex2.html#E3:22_1_1_2_1_1_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="complex2.html#T27" target="_self">Th27</a></i>;<br/><b>hence </b><a NAME="E6:22_1_1_2_1_1_1"/>
 <a href="comptrig.html#K1">Arg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>2</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="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_1_2_1_1_2"><b>case </b></a><a NAME="E1:22_1_1_2_1_1_2"/>
 <a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:22_1_1_2_1_1_2"/><b>then </b>
 <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#E1:22"><i><font color="Green">E24</font></i></a>, <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/><a NAME="E3:22_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E28</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/><a NAME="E4:22_1_1_2_1_1_2"/>
 <a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <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#K3">+</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#T19">COMPLEX1:19</a></i>;<br/><a NAME="E5:22_1_1_2_1_1_2"/><b>then </b>
 <a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>2</sub></font></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="txt" href="complex2.html#E3:22_1_1_2"><i><font color="Green">E27</font></i></a>, <a class="txt" href="complex2.html#E3:22_1_1_2_1_1_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E6:22_1_1_2_1_1_2"/><b>then </b>
 <a href="comptrig.html#K1">Arg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>2</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,<a href="sin_cos.html#K33">PI</a> </span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="ref" href="complex2.html#T27" target="_self">Th27</a></i>;<br/><b>hence </b><a NAME="E7:22_1_1_2_1_1_2"/>
 <a href="comptrig.html#K1">Arg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>2</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="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:22_1_1_2"/>
 <a href="comptrig.html#K1">Arg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a>  <a href="sin_cos.html#K33">PI</a> 
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:22"/>
 <a href="comptrig.html#K1">Arg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a>  <a href="sin_cos.html#K33">PI</a> 
 ;<br/>


</div>
