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

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

<a NAME="E1:37"/><i><font color="Green">E25</font></i>: 
<a href="sin_cos2.html#K4" target="_self">cosh</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a> <a href="sin_cos2.html#K4" target="_self">cosh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1"><b>proof </b></a><div class="add">

<a href="sin_cos2.html#K4" target="_self">cosh</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">p</font></span>)</span> = 
<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> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</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> <span class="p4">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> 2
<b>by </b><i/>
<br/>.= 
<a href="sin_cos2.html#K4" target="_self">cosh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E2:37_1"/>
<a href="sin_cos2.html#K4" target="_self">cosh</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a> <a href="sin_cos2.html#K4" target="_self">cosh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:37"/><i><font color="Green">E26</font></i>: 
<a href="sin_cos2.html#K1" target="_self">sinh</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="binop_2.html#K7">-</a> <span class="p1">(<span class="default"><a href="sin_cos2.html#K1" target="_self">sinh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2"><b>proof </b></a><div class="add">

<a href="sin_cos2.html#K1" target="_self">sinh</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">p</font></span>)</span> = 
<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> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">p</font></span>)</span></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> <span class="p4">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> 2
<b>by </b><i/>
<br/>.= 
 <a href="binop_2.html#K7">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> 2</span>)</span>

<br/>.= 
 <a href="binop_2.html#K7">-</a> <span class="p1">(<span class="default"><a href="sin_cos2.html#K1" target="_self">sinh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E2:37_2"/>
<a href="sin_cos2.html#K1" target="_self">sinh</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="binop_2.html#K7">-</a> <span class="p1">(<span class="default"><a href="sin_cos2.html#K1" target="_self">sinh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:37"/>
<a href="sin_cos2.html#K7" target="_self">tanh</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="binop_2.html#K7">-</a> <span class="p1">(<span class="default"><a href="sin_cos2.html#K7" target="_self">tanh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_3"><b>proof </b></a><div class="add">

<a href="sin_cos2.html#K7" target="_self">tanh</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">p</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> <span class="p2">(<span class="default"><a href="sin_cos2.html#K1" target="_self">sinh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><a href="sin_cos2.html#K4" target="_self">cosh</a>  <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">p</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
 <a href="binop_2.html#K7">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos2.html#K1" target="_self">sinh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><a href="sin_cos2.html#K4" target="_self">cosh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T188">XCMPLX_1:188</a></i>
<br/>.= 
 <a href="binop_2.html#K7">-</a> <span class="p1">(<span class="default"><a href="sin_cos2.html#K7" target="_self">tanh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E2:37_3"/>
<a href="sin_cos2.html#K7" target="_self">tanh</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="binop_2.html#K7">-</a> <span class="p1">(<span class="default"><a href="sin_cos2.html#K7" target="_self">tanh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:37"/>
( <a href="sin_cos2.html#K4" target="_self">cosh</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a> <a href="sin_cos2.html#K4" target="_self">cosh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font> &amp; <a href="sin_cos2.html#K1" target="_self">sinh</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="binop_2.html#K7">-</a> <span class="p1">(<span class="default"><a href="sin_cos2.html#K1" target="_self">sinh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span> &amp; <a href="sin_cos2.html#K7" target="_self">tanh</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="binop_2.html#K7">-</a> <span class="p1">(<span class="default"><a href="sin_cos2.html#K7" target="_self">tanh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span> )
 <b>by </b><i>, </i>;<br/>


</div>
