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<div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E1:42"/>
 <a href="sin_cos.html#K18">exp</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default">0,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span></span>)</span></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"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="arytm_0.html#K5">*]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1"><b>proof </b></a><div class="add">

 <a href="sin_cos.html#K18">exp</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default">0,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span></span>)</span></span><a href="arytm_0.html#K5">*]</a></span> = 
<span class="p1">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span></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="sin_cos.html#K21">sin</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span></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="sin_cos.html#T28">SIN_COS:28</a></i>
<br/>.= 
<span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span></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/>.= 
<span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span></span>)</span></span>)</span></span><a href="arytm_0.html#K5">*]</a></span>
<b>by </b><i><a class="ref" href="sin_cos.html#T34">SIN_COS:34</a></i>
<br/>.= 
<span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K4">*</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K3">+</a> <a href="sin_cos.html#K35">PI</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K4">*</a> <span class="p0">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">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_cos.html#T34">SIN_COS:34</a></i>
<br/>.= 
<span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K4">*</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K3">+</a> <a href="sin_cos.html#K35">PI</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K4">*</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K3">+</a> <a href="sin_cos.html#K35">PI</a> </span>)</span></span>)</span></span>)</span></span><a href="arytm_0.html#K5">*]</a></span>
<b>by </b><i><a class="ref" href="sin_cos.html#D23">SIN_COS:def 23</a></i>
<br/>.= 
<span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K4">*</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K3">+</a> <a href="sin_cos.html#K35">PI</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</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"><span class="p0">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K4">*</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K3">+</a> <a href="sin_cos.html#K35">PI</a> </span>)</span></span>)</span></span>)</span></span><a href="arytm_0.html#K5">*]</a></span>
<b>by </b><i><a class="ref" href="sin_cos.html#D21">SIN_COS:def 21</a></i>
<br/>.= 
<span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="seq_1.html#K2">.</a> <a href="sin_cos.html#K35">PI</a> </span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</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"><span class="p0">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K4">*</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K3">+</a> <a href="sin_cos.html#K35">PI</a> </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"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <a href="sin_cos.html#K35">PI</a> </span>)</span></span>)</span></span><a href="arytm_0.html#K5">*]</a></span>
<b>by </b><i><a class="ref" href="sin_cos.html#T81">SIN_COS:81</a>, <a class="ref" href="sin_cos2.html#T10">SIN_COS2:10</a></i>
;<br/>
<b>hence </b><a NAME="E2:42_1"/>
 <a href="sin_cos.html#K18">exp</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default">0,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span></span>)</span></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"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="arytm_0.html#K5">*]</a></span>
 <b>by </b><i><a class="ref" href="sin_cos.html#T81">SIN_COS:81</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:42"/>
(  <a href="sin_cos.html#K18">exp</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default">0,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span></span>)</span></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"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="arytm_0.html#K5">*]</a></span> &amp;  <a href="sin_cos.html#K18">exp</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"><a href="real_1.html#K1">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span></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> <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="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="arytm_0.html#K5">*]</a></span> )
 <b>by </b><i><a class="txt" href="sin_cos3.html#E6"><i><font color="Green">Lemma33</font></i></a></i>;<br/>


</div>
