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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>



<a NAME="E1:34"/><i><font color="Green">E17</font></i>: 
( <a href="sin_cos2.html#K4" target="_self">cosh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <a href="sin_cos2.html#K4" target="_self">cosh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <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">c<sub>1</sub></font></span>)</span> <a href="binop_2.html#K11">*</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">c<sub>2</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</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">c<sub>1</sub></font></span>)</span> <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">c<sub>2</sub></font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><b>proof </b></a><div class="add">

<a NAME="E1:34_1"/>
<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">c<sub>1</sub></font></span>)</span> <a href="binop_2.html#K11">*</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">c<sub>2</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</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">c<sub>1</sub></font></span>)</span> <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">c<sub>2</sub></font></span>)</span></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> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="sin_cos2.html#E23"><i><font color="Green">Lemma10</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:34_1"/>
( <a href="sin_cos2.html#K4" target="_self">cosh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <a href="sin_cos2.html#K4" target="_self">cosh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <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">c<sub>1</sub></font></span>)</span> <a href="binop_2.html#K11">*</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">c<sub>2</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</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">c<sub>1</sub></font></span>)</span> <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">c<sub>2</sub></font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 <b>by </b><i><a class="ref" href="sin_cos2.html#T15" target="_self">Th15</a></i>;<br/>


</div><b>end;</b></div>
<a href="sin_cos2.html#K7" target="_self">tanh</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="sin_cos2.html#K1" target="_self">sinh</a>  <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>2</sub></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"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos2.html#T17" target="_self">Th17</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><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></span>)</span> <a href="binop_2.html#K11">*</a> <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">c<sub>2</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</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">c<sub>1</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><a href="sin_cos2.html#K1" target="_self">sinh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></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"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="sin_cos2.html#E24"><i><font color="Green">Lemma11</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><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></span>)</span> <a href="binop_2.html#K11">*</a> <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">c<sub>2</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</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">c<sub>1</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><a href="sin_cos2.html#K1" target="_self">sinh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><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">c<sub>1</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <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">c<sub>2</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><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></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><a href="sin_cos2.html#K1" target="_self">sinh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="sin_cos2.html#E23"><i><font color="Green">Lemma10</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><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></span>)</span> <a href="binop_2.html#K12">/</a> <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">c<sub>1</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos2.html#K1" target="_self">sinh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_2.html#K12">/</a> <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">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><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></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p4">(<span class="default"><a href="sin_cos2.html#K4" target="_self">cosh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos2.html#K1" target="_self">sinh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p4">(<span class="default"><a href="sin_cos2.html#K4" target="_self">cosh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="sin_cos2.html#E1:34"><i><font color="Green">E17</font></i></a>, <a class="txt" href="sin_cos2.html#E28"><i><font color="Green">Lemma15</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos2.html#K7" target="_self">tanh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos2.html#K1" target="_self">sinh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_2.html#K12">/</a> <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">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><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></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p4">(<span class="default"><a href="sin_cos2.html#K4" target="_self">cosh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos2.html#K1" target="_self">sinh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p4">(<span class="default"><a href="sin_cos2.html#K4" target="_self">cosh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos2.html#T17" target="_self">Th17</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos2.html#K7" target="_self">tanh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><a href="sin_cos2.html#K7" target="_self">tanh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><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></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p4">(<span class="default"><a href="sin_cos2.html#K4" target="_self">cosh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos2.html#K1" target="_self">sinh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p4">(<span class="default"><a href="sin_cos2.html#K4" target="_self">cosh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos2.html#T17" target="_self">Th17</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos2.html#K7" target="_self">tanh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><a href="sin_cos2.html#K7" target="_self">tanh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos2.html#K7" target="_self">tanh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos2.html#K1" target="_self">sinh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p4">(<span class="default"><a href="sin_cos2.html#K4" target="_self">cosh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos2.html#T17" target="_self">Th17</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos2.html#K7" target="_self">tanh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><a href="sin_cos2.html#K7" target="_self">tanh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos2.html#K7" target="_self">tanh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><a href="sin_cos2.html#K7" target="_self">tanh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos2.html#T17" target="_self">Th17</a></i>
;<br/>
<b>hence </b><a NAME="E3:34"/>
<a href="sin_cos2.html#K7" target="_self">tanh</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos2.html#K7" target="_self">tanh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><a href="sin_cos2.html#K7" target="_self">tanh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos2.html#K7" target="_self">tanh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><a href="sin_cos2.html#K7" target="_self">tanh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span>
 ;<br/>


</div>
