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<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<a NAME="E1:85"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K7" target="_self">ExpSeq</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K3">Re</a> <span class="p2">(<span class="default"><span class="p3"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="sin_cos.html#K6" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<span class="p1">(<span class="default"><a href="comseq_3.html#K3">Re</a> <span class="p2">(<span class="default"><span class="p3"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="sin_cos.html#K6" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> = 
 <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="sin_cos.html#K6" target="_self">ExpSeq</a> </span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="comseq_3.html#D3">COMSEQ_3:def 3</a></i>
<br/>.= 
 <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="newton.html#K2">|^</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="sin_cos.html#K4" target="_self">!c</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos.html#D8" target="_self">Def8</a></i>
<br/>.= 
 <a href="complex1.html#K3">Re</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"><font color="Maroon">c<sub>3</sub></font> <a href="prepower.html#K3">|^</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="sin_cos.html#K5" target="_self">!</a> </span>)</span></span>)</span>,0</span><a href="arytm_0.html#K5">*]</a></span>
<b>by </b><i><a class="ref" href="sin_cos.html#T45" target="_self">Th45</a></i>
<br/>.= 
 <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prepower.html#K3">|^</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="sin_cos.html#K5" target="_self">!</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default">0 <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T27">COMPLEX1:27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prepower.html#K3">|^</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="sin_cos.html#K5" target="_self">!</a> </span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T28">COMPLEX1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K7" target="_self">ExpSeq</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="sin_cos.html#D9" target="_self">Def9</a></i>
;<br/>
<b>hence </b><a NAME="E2:85_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K7" target="_self">ExpSeq</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K3">Re</a> <span class="p2">(<span class="default"><span class="p3"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="sin_cos.html#K6" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:85"/>
<font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K7" target="_self">ExpSeq</a>  <a href="funct_2.html#R2">=</a>  <a href="comseq_3.html#K3">Re</a> <span class="p1">(<span class="default"><span class="p2"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="sin_cos.html#K6" target="_self">ExpSeq</a> </span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>


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