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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:16"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/>

<a NAME="E2:16"/>
<span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <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="E3:16"/><b>then </b>
(  <a href="complex1.html#K3">Re</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="complex1.html#K4">Im</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i><a class="ref" href="complex1.html#T28">COMPLEX1:28</a></i>;<br/>
<b>then </b><i><font color="Green">E21</font></i>: <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,0</span><a href="arytm_0.html#K5">*]</a></span></span><a href="complex1.html#K17">.|</a></span> = 
 <a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="complex1.html#T136">COMPLEX1:136</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="complex2.html#E1:16"><i><font color="Green">E20</font></i></a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
;<br/>
<a NAME="E5:16"/>
<span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,0</span><a href="arytm_0.html#K5">*]</a></span> <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"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,0</span><a href="arytm_0.html#K5">*]</a></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"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,0</span><a href="arytm_0.html#K5">*]</a></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"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,0</span><a href="arytm_0.html#K5">*]</a></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"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,0</span><a href="arytm_0.html#K5">*]</a></span></span>)</span></span>)</span></span>)</span></span><a href="arytm_0.html#K5">*]</a></span>
 
<b>by </b><i><a class="ref" href="complex2.html#T19" target="_self">Th19</a></i>;<br/>
<a NAME="E6:16"/><b>then </b><i><font color="Green">E22</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,0</span><a href="arytm_0.html#K5">*]</a></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"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,0</span><a href="arytm_0.html#K5">*]</a></span></span>)</span></span>)</span> &amp; 0 <a href="hidden.html#R1">=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,0</span><a href="arytm_0.html#K5">*]</a></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"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,0</span><a href="arytm_0.html#K5">*]</a></span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="arytm_0.html#T12">ARYTM_0:12</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_2"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 or <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#E1:16"><i><font color="Green">E20</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2_1"><b>suppose </b></a><a NAME="E1:16_2_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_2_1"/>
 <a href="comptrig.html#K1">Arg</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="complex2.html#T20" target="_self">Th20</a>, <a class="ref" href="arytm_0.html#D7">ARYTM_0:def 7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2_2"><b>suppose </b></a><a NAME="E1:16_2_2"/><i><font color="Green">E23</font></i>: 
<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="E2:16_2_2"/><b>then </b><i><font color="Green">E24</font></i>: 
 <a href="sin_cos.html#K24">cos</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,0</span><a href="arytm_0.html#K5">*]</a></span></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="complex2.html#E4:16"><i><font color="Green">E21</font></i></a>, <a class="txt" href="complex2.html#E6:16"><i><font color="Green">E22</font></i></a>, <a class="ref" href="xcmplx_1.html#T7">XCMPLX_1:7</a></i>;<br/><a NAME="E3:16_2_2"/><i><font color="Green">E25</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"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,0</span><a href="arytm_0.html#K5">*]</a></span></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="complex2.html#E4:16"><i><font color="Green">E21</font></i></a>, <a class="txt" href="complex2.html#E6:16"><i><font color="Green">E22</font></i></a>, <a class="txt" href="complex2.html#E1:16_2_2"><i><font color="Green">E23</font></i></a>, <a class="ref" href="xcmplx_1.html#T6">XCMPLX_1:6</a></i>;<br/><a NAME="E4:16_2_2"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="comptrig.html#K1">Arg</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,0</span><a href="arytm_0.html#K5">*]</a></span> &amp;  <a href="comptrig.html#K1">Arg</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="xxreal_0.html#R1">&lt;</a> 2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a>  )
 <b>by </b><i><a class="txt" href="complex2.html#E18"><i><font color="Green">Lemma12</font></i></a></i>;<br/><b>hence </b><a NAME="E5:16_2_2"/>
 <a href="comptrig.html#K1">Arg</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="complex2.html#E2:16_2_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="complex2.html#E3:16_2_2"><i><font color="Green">E25</font></i></a>, <a class="ref" href="complex2.html#T12" target="_self">Th12</a>, <a class="ref" href="sin_cos.html#T34">SIN_COS:34</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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