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

<b>let </b><font color="Maroon" title="c1">z</font> be  <a href="xcmplx_0.html#V1" title="XCMPLX_0:attr.1">complex</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <a href="sin_cos3.html#K1" title="SIN_COS3:func.1">sin_C</a>  <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a>  <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <font color="Maroon" title="c1">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a>  <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p1">(<span class="default"><a href="sin_cos3.html#K3" title="SIN_COS3:func.3">sinh_C</a>  <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">z</font></span>)</span></span><br/>

<b>reconsider </b><font color="Maroon" title="c2">z</font> = <font color="Maroon" title="c1">z</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  <b>by </b><i><a class="ref" href="xcmplx_0.html#D2" title="XCMPLX_0:def.2">XCMPLX_0:def 2</a></i>;<br/>
<a href="sin_cos3.html#K1" title="SIN_COS3:func.1">sin_C</a>  <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a>  <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <font color="Maroon" title="c2">z</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K19" title="SIN_COS:func.19">exp</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a>  <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> </span>)</span> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <font color="Maroon" title="c2">z</font></span>)</span></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K19" title="SIN_COS:func.19">exp</a> <span class="p3">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <span class="p4">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a>  <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p5">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a>  <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <font color="Maroon" title="c2">z</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a>  <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> 2</span>)</span>
<b>by </b><i><a class="ref" href="sin_cos3.html#D1" target="_self" title="SIN_COS3:def.1">Def1</a></i>
<br/>.= 
<a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a>  <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K19" title="SIN_COS:func.19">exp</a> <font color="Maroon" title="c2">z</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K19" title="SIN_COS:func.19">exp</a> <span class="p4">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <font color="Maroon" title="c2">z</font></span>)</span></span>)</span></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> 2</span>)</span>

;<br/>
<a NAME="E3:22"/><b>then </b>
<a href="sin_cos3.html#K1" title="SIN_COS3:func.1">sin_C</a>  <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a>  <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <font color="Maroon" title="c2">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a>  <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p1">(<span class="default"><a href="sin_cos3.html#K3" title="SIN_COS3:func.3">sinh_C</a>  <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c2">z</font></span>)</span>
 
<b>by </b><i><a class="ref" href="sin_cos3.html#D3" target="_self" title="SIN_COS3:def.3">Def3</a></i>;<br/>
<b>hence </b><a NAME="E4:22"/>
<a href="sin_cos3.html#K1" title="SIN_COS3:func.1">sin_C</a>  <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a>  <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <font color="Maroon" title="c1">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a>  <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p1">(<span class="default"><a href="sin_cos3.html#K3" title="SIN_COS3:func.3">sinh_C</a>  <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">z</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
