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

<b>let </b><font color="Maroon" title="c1">p</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</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_cos2.html#K1" title="SIN_COS2:func.1">sinh</a>  <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c1">p</font> &amp;  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <a href="sin_cos2.html#K1" title="SIN_COS2:func.1">sinh</a> ,<font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="sin_cos2.html#K4" title="SIN_COS2:func.4">cosh</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">p</font> )</span><br/>

<b>set </b><font color="Maroon" title="c2">ff</font> =  <a href="binop_2.html#K31" title="BINOP_2:func.31">compreal</a> ;<br/>
<a NAME="E1:56"/><i><font color="Green" title="E36">A1</font></i>: 
 <a href="sin_cos2.html#K1" title="SIN_COS2:func.1">sinh</a>  <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="binop_2.html#K31" title="BINOP_2:func.31">compreal</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="sin_cos2.html#E55"><i><font color="Green" title="E34">Lm19</font></i></a></i>;<br/>
 <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <a href="sin_cos2.html#K1" title="SIN_COS2:func.1">sinh</a> ,<font color="Maroon" title="c1">p</font> = 
 <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="binop_2.html#K31" title="BINOP_2:func.31">compreal</a> </span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c1">p</font>
<b>by </b><i><a class="txt" href="sin_cos2.html#E55"><i><font color="Green" title="E34">Lm19</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="binop_2.html#K31" title="BINOP_2:func.31">compreal</a> </span>)</span></span>)</span>,<font color="Maroon" title="c1">p</font></span>)</span>
<b>by </b><i><a class="txt" href="sin_cos2.html#E53"><i><font color="Green" title="E32">Lm17</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="sin_cos2.html#E52"><i><font color="Green" title="E31">Lm16</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2

<br/>.= 
<a href="sin_cos2.html#K4" title="SIN_COS2:func.4">cosh</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">p</font>
<b>by </b><i><a class="ref" href="sin_cos2.html#D3" target="_self" title="SIN_COS2:def.3">Def3</a></i>
;<br/>
<b>hence </b><a NAME="E3:56"/>
(  <a href="sin_cos2.html#K1" title="SIN_COS2:func.1">sinh</a>  <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c1">p</font> &amp;  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <a href="sin_cos2.html#K1" title="SIN_COS2:func.1">sinh</a> ,<font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="sin_cos2.html#K4" title="SIN_COS2:func.4">cosh</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">p</font> )
 <b>by </b><i><a class="txt" href="sin_cos2.html#E1:56"><i><font color="Green" title="E36">A1</font></i></a>, <a class="txt" href="sin_cos2.html#E53"><i><font color="Green" title="E32">Lm17</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
