<?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="E1:36"/><i><font color="Green">E25</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">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K11">*</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#K11">*</a> <span class="p1">(<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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1"><b>proof </b></a><div class="add">

<span class="p1">(<span class="default">2 <a href="binop_2.html#K11">*</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#K11">*</a> <span class="p1">(<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 class="p1">(<span class="default">2 <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<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="p4">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <span class="p5">(<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></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<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>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<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="p4">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <span class="p5">(<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></span>)</span> <a href="binop_2.html#K11">*</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#K9">+</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>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="binop_2.html#K12">/</a> 4</span>)</span> <a href="binop_2.html#K11">*</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="square_1.html#K5">^2</a> </span>)</span> <a href="binop_2.html#K10">-</a> <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> <span class="p4">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default">2 <a href="binop_2.html#K12">/</a> 4</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> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <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> <span class="p4">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</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></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="binop_2.html#K12">/</a> 4</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> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3">*</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"><span class="p4">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.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></span>)</span>
<b>by </b><i/>
<br/>.= 
<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> <span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">p</font></span>)</span></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> <span class="p5">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="binop_2.html#K11">*</a> 1

<br/>.= 
<a href="sin_cos2.html#K1" target="_self">sinh</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">p</font></span>)</span>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E2:36_1"/>
<a href="sin_cos2.html#K1" target="_self">sinh</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K11">*</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#K11">*</a> <span class="p1">(<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>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:36"/>
<a href="sin_cos2.html#K4" target="_self">cosh</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><span class="p3">(<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> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_2"><b>proof </b></a><div class="add">

<a NAME="E1:36_2"/>
2 <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><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> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="hidden.html#R1">=</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">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_2_1"><b>proof </b></a><div class="add">

2 <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><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> <a href="square_1.html#K5">^2</a> </span>)</span> = 
2 <a href="binop_2.html#K11">*</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="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos2.html#K4" target="_self">cosh</a>  <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<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">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1

;<br/>
<b>hence </b><a NAME="E2:36_2_1"/>
2 <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><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> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="hidden.html#R1">=</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">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:36_2"/>
<a href="sin_cos2.html#K4" target="_self">cosh</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><span class="p3">(<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> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> 1
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:36"/>
( <a href="sin_cos2.html#K1" target="_self">sinh</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K11">*</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#K11">*</a> <span class="p1">(<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> &amp; <a href="sin_cos2.html#K4" target="_self">cosh</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><span class="p3">(<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> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> 1 )
 <b>by </b><i/>;<br/>


</div>
