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

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

<a NAME="E1:38"/>
 <a href="sin_cos.html#K18">exp</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complex1.html#K8">+</a> <span class="p2">(<span class="default"><span class="p3"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="complex1.html#K9">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="sin_cos.html#K18">exp</a> <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1"><b>proof </b></a><div class="add">

<i><font color="Green">E34</font></i>: <font color="Maroon">c<sub>1</sub></font> <a href="complex1.html#K8">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="complex1.html#K9">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span> = 
<font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default">0 <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </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#T27">COMPLEX1:27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default">0 <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </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">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K3">+</a> 0</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span>

<br/>.= 
<span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K3">+</a> 0</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<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="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </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="E2:38_1"/><i><font color="Green">E35</font></i>: 
<span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</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"><a href="complex1.html#K3">Re</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="arytm_0.html#K5">*]</a></span>
 
<b>by </b><i><a class="ref" href="complex1.html#T27">COMPLEX1:27</a></i>;<br/>
 <a href="sin_cos.html#K18">exp</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complex1.html#K8">+</a> <span class="p2">(<span class="default"><span class="p3"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="complex1.html#K9">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span></span>)</span> = 
 <a href="sin_cos.html#K18">exp</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K3">+</a> 0</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<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="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span></span>)</span></span><a href="arytm_0.html#K5">*]</a></span>
<b>by </b><i><a class="txt" href="sin_cos3.html#E1:38_1"><i><font color="Green">E34</font></i></a></i>
<br/>.= 
<span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><span class="p5">(<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="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><span class="p5">(<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="p5">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span></span>)</span></span>)</span></span>)</span></span><a href="arytm_0.html#K5">*]</a></span>
<b>by </b><i><a class="ref" href="sin_cos3.html#T19" target="_self">Th19</a></i>
<br/>.= 
<span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><span class="p5">(<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="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="arytm_0.html#K5">*]</a></span>
<b>by </b><i><a class="ref" href="sin_cos2.html#T11">SIN_COS2:11</a></i>
<br/>.= 
<span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span><a href="arytm_0.html#K5">*]</a></span>
<b>by </b><i><a class="ref" href="sin_cos2.html#T10">SIN_COS2:10</a></i>
<br/>.= 
 <a href="sin_cos.html#K18">exp</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="arytm_0.html#K5">*]</a></span>
<b>by </b><i><a class="ref" href="sin_cos3.html#T19" target="_self">Th19</a></i>
;<br/>
<b>hence </b><a NAME="E4:38_1"/>
 <a href="sin_cos.html#K18">exp</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complex1.html#K8">+</a> <span class="p2">(<span class="default"><span class="p3"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="complex1.html#K9">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="sin_cos.html#K18">exp</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="complex1.html#T29">COMPLEX1:29</a>, <a class="txt" href="sin_cos3.html#E2:38_1"><i><font color="Green">E35</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:38"/>
(  <a href="sin_cos.html#K18">exp</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complex1.html#K8">+</a> <span class="p2">(<span class="default"><span class="p3"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="complex1.html#K9">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="sin_cos.html#K18">exp</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="sin_cos.html#K18">exp</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complex1.html#K8">+</a> <span class="p2"><a href="arytm_0.html#K5">[*</a><span class="default">0,<span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span></span><a href="arytm_0.html#K5">*]</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="sin_cos.html#K18">exp</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="sin_cos3.html#E40"><i><font color="Green">Lemma33</font></i></a></i>;<br/>


</div>
