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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/>

<a href="sin_cos3.html#K3" target="_self">sinh_C</a>  <a href="finseq_4.html#K4">/.</a> <span class="p1">(<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 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> 2
<b>by </b><i><a class="txt" href="sin_cos3.html#E8"><i><font color="Green">Lemma8</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="complex1.html#K9">*</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#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="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></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="xcmplx_0.html#K3">*</a> 2</span>)</span>

<br/>.= 
<a href="complex1.html#K7">&lt;i&gt;</a>  <a href="xcmplx_0.html#K3">*</a> <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#K11">-</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> <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="p2">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="xcmplx_0.html#K3">*</a> 2</span>)</span></span>)</span>

;<br/>
<b>hence </b><a NAME="E2:26"/>
<a href="sin_cos3.html#K3" target="_self">sinh_C</a>  <a href="finseq_4.html#K4">/.</a> <span class="p1">(<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="hidden.html#R1">=</a> <a href="complex1.html#K7">&lt;i&gt;</a>  <a href="complex1.html#K9">*</a> <span class="p1">(<span class="default"><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></span>)</span>
 <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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