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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>2</sub></font> =  <a href="binop_2.html#K31">compreal</a>  as   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a>  ;<br/>
<a NAME="E2:61"/><i><font color="Green">E40</font></i>: 
 <a href="sin_cos2.html#K4" target="_self">cosh</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K6">+</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="sin_cos2.html#E59"><i><font color="Green">Lemma38</font></i></a></i>;<br/>
 <a href="fdiff_1.html#K1">diff</a> <a href="sin_cos2.html#K4" target="_self">cosh</a> ,<font color="Maroon">c<sub>1</sub></font> = 
 <a href="fdiff_1.html#K1">diff</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K6">+</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="sin_cos2.html#E59"><i><font color="Green">Lemma38</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K6">+</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="sin_cos2.html#E57"><i><font color="Green">Lemma36</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="sin_cos2.html#E52"><i><font color="Green">Lemma31</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> 2

<br/>.= 
<a href="sin_cos2.html#K1" target="_self">sinh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="sin_cos2.html#D1" target="_self">Def1</a></i>
;<br/>
<b>hence </b><a NAME="E4:61"/>
(  <a href="sin_cos2.html#K4" target="_self">cosh</a>  <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="fdiff_1.html#K1">diff</a> <a href="sin_cos2.html#K4" target="_self">cosh</a> ,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <a href="sin_cos2.html#K1" target="_self">sinh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="sin_cos2.html#E2:61"><i><font color="Green">E40</font></i></a>, <a class="txt" href="sin_cos2.html#E57"><i><font color="Green">Lemma36</font></i></a></i>;<br/>


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