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<span class="kw">let </span><font color="Maroon" title="c1">z</font> be    <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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="sin_cos.html#K16" title="SIN_COS:func.16">exp</a> <font color="Maroon" title="c1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="sin_cos3.html#K4" title="SIN_COS3:func.4">cosh_C</a> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c1">z</font></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p1">(<span class="default"><a href="sin_cos3.html#K3" title="SIN_COS3:func.3">sinh_C</a> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c1">z</font></span>)</span></span><br/>

<span class="p1">(<span class="default"><a href="sin_cos3.html#K4" title="SIN_COS3:func.4">cosh_C</a> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c1">z</font></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p1">(<span class="default"><a href="sin_cos3.html#K3" title="SIN_COS3:func.3">sinh_C</a> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c1">z</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K16" title="SIN_COS:func.16">exp</a> <font color="Maroon" title="c1">z</font></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K16" title="SIN_COS:func.16">exp</a> <span class="p4">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <font color="Maroon" title="c1">z</font></span>)</span></span>)</span></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> 2</span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p1">(<span class="default"><a href="sin_cos3.html#K3" title="SIN_COS3:func.3">sinh_C</a> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c1">z</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos3.html#D4" target="_self" title="SIN_COS3:def.4">Def4</a></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#K16" title="SIN_COS:func.16">exp</a> <font color="Maroon" title="c1">z</font></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K16" title="SIN_COS:func.16">exp</a> <span class="p4">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <font color="Maroon" title="c1">z</font></span>)</span></span>)</span></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> 2</span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K16" title="SIN_COS:func.16">exp</a> <font color="Maroon" title="c1">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#K16" title="SIN_COS:func.16">exp</a> <span class="p4">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <font color="Maroon" title="c1">z</font></span>)</span></span>)</span></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> 2</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos3.html#D3" target="_self" title="SIN_COS3:def.3">Def3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K16" title="SIN_COS:func.16">exp</a> <font color="Maroon" title="c1">z</font></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K16" title="SIN_COS:func.16">exp</a> <span class="p5">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <font color="Maroon" title="c1">z</font></span>)</span></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K16" title="SIN_COS:func.16">exp</a> <font color="Maroon" title="c1">z</font></span>)</span></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K16" title="SIN_COS:func.16">exp</a> <span class="p4">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <font color="Maroon" title="c1">z</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> 2

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K16" title="SIN_COS:func.16">exp</a> <font color="Maroon" title="c1">z</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K16" title="SIN_COS:func.16">exp</a> <font color="Maroon" title="c1">z</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p5">(<span class="default"><a href="sin_cos.html#K16" title="SIN_COS:func.16">exp</a> <font color="Maroon" title="c1">z</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p5">(<span class="default"><a href="sin_cos.html#K16" title="SIN_COS:func.16">exp</a> <font color="Maroon" title="c1">z</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a></span>)</span></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> 2
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#D5" title="COMPLEX1:def.5">COMPLEX1:def 5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K16" title="SIN_COS:func.16">exp</a> <font color="Maroon" title="c1">z</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p5">(<span class="default"><a href="sin_cos.html#K16" title="SIN_COS:func.16">exp</a> <font color="Maroon" title="c1">z</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a></span>)</span></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> 2

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K16" title="SIN_COS:func.16">exp</a> <font color="Maroon" title="c1">z</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p5">(<span class="default"><a href="sin_cos.html#K16" title="SIN_COS:func.16">exp</a> <font color="Maroon" title="c1">z</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a></span>)</span></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> 2
<span class="kw">by</span> <span class="lab"><a class="ref" href="comseq_3.html#T17" title="COMSEQ_3:th.17">COMSEQ_3:17</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K16" title="SIN_COS:func.16">exp</a> <font color="Maroon" title="c1">z</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p5">(<span class="default"><a href="sin_cos.html#K16" title="SIN_COS:func.16">exp</a> <font color="Maroon" title="c1">z</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a></span>)</span></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> 2
<span class="kw">by</span> <span class="lab"><a class="ref" href="comseq_3.html#T17" title="COMSEQ_3:th.17">COMSEQ_3:17</a></span>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K16" title="SIN_COS:func.16">exp</a> <font color="Maroon" title="c1">z</font></span>)</span></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> 2
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T13" title="COMPLEX1:th.13">COMPLEX1:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="sin_cos.html#K16" title="SIN_COS:func.16">exp</a> <font color="Maroon" title="c1">z</font></span>)</span> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> 1

;<br/>
<span class="kw">hence </span><a NAME="E2:32"/>
 <a href="sin_cos.html#K16" title="SIN_COS:func.16">exp</a> <font color="Maroon" title="c1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="sin_cos3.html#K4" title="SIN_COS3:func.4">cosh_C</a> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c1">z</font></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p1">(<span class="default"><a href="sin_cos3.html#K3" title="SIN_COS3:func.3">sinh_C</a> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c1">z</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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