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<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/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>



<a href="sin_cos3.html#K1" target="_self">sin_C</a>  <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span></span>)</span> = 
<a href="sin_cos3.html#K1" target="_self">sin_C</a>  <a href="finseq_4.html#K4">/.</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"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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></span>)</span>
<b>by </b><i><a class="ref" href="arytm_0.html#D7">ARYTM_0:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K18">exp</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="complex1.html#K9">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p4">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="complex1.html#K9">*</a> <span class="p5"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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></span>)</span></span>)</span></span>)</span> <a href="complex1.html#K11">-</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K18">exp</a> <span class="p3">(<span class="default"><a href="complex1.html#K10">-</a> <span class="p4">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="complex1.html#K9">*</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complex1.html#K8">+</a> <span class="p0"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <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="sin_cos3.html#E6"><i><font color="Green">Lemma6</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K18">exp</a> <span class="p4">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="complex1.html#K9">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="complex1.html#K9">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K18">exp</a> <span class="p4">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="complex1.html#K9">*</a> <span class="p5"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,0</span><a href="arytm_0.html#K5">*]</a></span></span>)</span></span>)</span></span>)</span> <a href="complex1.html#K11">-</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K18">exp</a> <span class="p3">(<span class="default"><a href="complex1.html#K10">-</a> <span class="p4">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="complex1.html#K9">*</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complex1.html#K8">+</a> <span class="p0"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <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="sin_cos.html#T24">SIN_COS:24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K18">exp</a> <span class="p4">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="complex1.html#K9">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K18">exp</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="complex1.html#K10">-</a> <span class="p5">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="complex1.html#K9">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="complex1.html#K8">+</a> <span class="p4">(<span class="default"><a href="complex1.html#K10">-</a> <span class="p5">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="complex1.html#K9">*</a> <span class="p0"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <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="sin_cos3.html#T28" target="_self">Th28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K18">exp</a> <span class="p4">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="complex1.html#K9">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K18">exp</a> <span class="p4">(<span class="default"><a href="complex1.html#K10">-</a> <span class="p5">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="complex1.html#K9">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="complex1.html#K9">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K18">exp</a> <span class="p4">(<span class="default"><a href="complex1.html#K10">-</a> <span class="p5">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="complex1.html#K9">*</a> <span class="p0"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,0</span><a href="arytm_0.html#K5">*]</a></span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <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="sin_cos.html#T24">SIN_COS:24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K18">exp</a> <span class="p4">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="complex1.html#K9">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K18">exp</a> <span class="p4">(<span class="default"><a href="complex1.html#K10">-</a> <span class="p5">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="complex1.html#K9">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="complex1.html#K9">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K18">exp</a> <span class="p4">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="complex1.html#K9">*</a> <span class="p5">(<span class="default"><a href="complex1.html#K10">-</a> <span class="p0"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,0</span><a href="arytm_0.html#K5">*]</a></span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K18">exp</a> <span class="p4">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="complex1.html#K9">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K18">exp</a> <span class="p4">(<span class="default"><a href="complex1.html#K10">-</a> <span class="p5">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="complex1.html#K9">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="complex1.html#K9">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K18">exp</a> <span class="p4">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="complex1.html#K9">*</a> <span class="p5"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> 0</span>)</span></span><a href="arytm_0.html#K5">*]</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <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="sin_cos3.html#E48"><i><font color="Green">Lemma36</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K18">exp</a> <span class="p4">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="complex1.html#K9">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K18">exp</a> <span class="p4">(<span class="default"><a href="complex1.html#K10">-</a> <span class="p5">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="complex1.html#K9">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="complex1.html#K9">*</a> <a href="complex1.html#K6">1r</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <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="sin_cos3.html#E11"><i><font color="Green">Lemma11</font></i></a>, <a class="ref" href="sin_cos3.html#T29" target="_self">Th29</a>, <a class="txt" href="sin_cos3.html#E49"><i><font color="Green">Lemma37</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K18">exp</a> <span class="p3">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="complex1.html#K9">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="complex1.html#K11">-</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K18">exp</a> <span class="p3">(<span class="default"><a href="complex1.html#K10">-</a> <span class="p4">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="complex1.html#K9">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <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#D7">COMPLEX1:def 7</a></i>
;<br/>
<b>hence </b><a NAME="E2:46"/>
<a href="sin_cos3.html#K1" target="_self">sin_C</a>  <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <a href="sin_cos3.html#K1" target="_self">sin_C</a>  <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="sin_cos3.html#E6"><i><font color="Green">Lemma6</font></i></a></i>;<br/>


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