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

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> ;<br/>

<a NAME="E1:54"/><i><font color="Green">E61</font></i>: 
<span class="p1"><a href="arytm_0.html#K5">[*</a><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">x</font></span>)</span> <a href="real_1.html#K3">+</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="real_1.html#K1">-</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos2.html#K4">cosh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="xcmplx_0.html#K3">*</a> 2
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1"><b>proof </b></a><div class="add">

<span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos2.html#K4">cosh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="xcmplx_0.html#K3">*</a> 2 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos2.html#K4">cosh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default">0 <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> 2
<b>by </b><i><a class="ref" href="complex1.html#T27">COMPLEX1:27</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#K4">cosh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</a> 2</span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K2">*</a> 0</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos2.html#K4">cosh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</a> 0</span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">0 <a href="nat_1.html#K2">*</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span>

<br/>.= 
<span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos2.html#K4">cosh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</a> 2</span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">0 <a href="nat_1.html#K2">*</a> 0</span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos2.html#K4">cosh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</a> 0</span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">0 <a href="nat_1.html#K2">*</a> 2</span>)</span></span>)</span></span><a href="arytm_0.html#K5">*]</a></span>
<b>by </b><i><a class="ref" href="complex1.html#T27">COMPLEX1:27</a></i>
<br/>.= 
<span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>,0</span><a href="arytm_0.html#K5">*]</a></span>
<b>by </b><i><a class="ref" href="sin_cos2.html#D3">SIN_COS2:def 3</a></i>
<br/>.= 
<span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><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">x</font></span>)</span> <a href="real_1.html#K3">+</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="real_1.html#K1">-</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>,0</span><a href="arytm_0.html#K5">*]</a></span>

;<br/>
<b>hence </b><a NAME="E2:54_1"/>
<span class="p1"><a href="arytm_0.html#K5">[*</a><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">x</font></span>)</span> <a href="real_1.html#K3">+</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="real_1.html#K1">-</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos2.html#K4">cosh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="xcmplx_0.html#K3">*</a> 2
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:54"/><i><font color="Green">E66</font></i>: 
( <span class="p1"><a href="arytm_0.html#K5">[*</a><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">x</font></span>)</span>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">0 <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span> &amp; <span class="p1"><a href="arytm_0.html#K5">[*</a><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="real_1.html#K1">-</a> <font color="Maroon">x</font></span>)</span></span>)</span>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">0 <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span> )
 
<b>by </b><i><a class="ref" href="complex1.html#T27">COMPLEX1:27</a></i>;<br/>
<a href="sin_cos3.html#K4" target="_self">cosh_C</a>  <a href="finseq_4.html#K4">/.</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">x</font>,0</span><a href="arytm_0.html#K5">*]</a></span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K18">exp</a> <span class="p3"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">x</font>,0</span><a href="arytm_0.html#K5">*]</a></span></span>)</span> <a href="complex1.html#K8">+</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K18">exp</a> <span class="p3">(<span class="default"><a href="complex1.html#K10">-</a> <span class="p4"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">x</font>,0</span><a href="arytm_0.html#K5">*]</a></span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="arytm_0.html#K5">[*</a><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">x</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="seq_1.html#K2">.</a> 0</span>)</span></span>)</span>,<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">x</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> 0</span>)</span></span>)</span></span><a href="arytm_0.html#K5">*]</a></span> <a href="complex1.html#K8">+</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K18">exp</a> <span class="p3">(<span class="default"><a href="complex1.html#K10">-</a> <span class="p4"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">x</font>,0</span><a href="arytm_0.html#K5">*]</a></span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="arytm_0.html#K5">[*</a><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">x</font></span>)</span>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="complex1.html#K8">+</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K18">exp</a> <span class="p3"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">x</font></span>)</span>,<span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> 0</span>)</span></span><a href="arytm_0.html#K5">*]</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2
<b>by </b><i><a class="txt" href="sin_cos3.html#E9"><i><font color="Green">Lemma36</font></i></a>, <a class="ref" href="sin_cos.html#T33">SIN_COS:33</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="arytm_0.html#K5">[*</a><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">x</font></span>)</span>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="complex1.html#K8">+</a> <span class="p2"><a href="arytm_0.html#K5">[*</a><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> <span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> 1</span>)</span>,<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> <span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> 0</span>)</span></span><a href="arytm_0.html#K5">*]</a></span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2
<b>by </b><i>, <a class="ref" href="sin_cos.html#T33">SIN_COS:33</a></i>
<br/>.= 
<span class="p1">(<span class="default"><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">x</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default">0 <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</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="real_1.html#K1">-</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default">0 <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p3">(<span class="default"><a href="sin_cos2.html#K4">cosh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="xcmplx_0.html#K3">*</a> 2</span>)</span> <a href="xcmplx_0.html#K7">/</a> 2
<b>by </b><i><a class="ref" href="complex1.html#T27">COMPLEX1:27</a>, </i>
;<br/>
<b>hence </b><a NAME="E4:54"/>
<a href="sin_cos3.html#K4" target="_self">cosh_C</a>  <a href="finseq_4.html#K4">/.</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">x</font>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="hidden.html#R1">=</a> <a href="sin_cos2.html#K4">cosh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="arytm_0.html#D7">ARYTM_0:def 7</a></i>;<br/>


</div>
