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

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

<b>assume </b><a NAME="E1:17"/>
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>

<a NAME="E2:17"/><b>then </b><i><font color="Green">E25</font></i>: 
<font color="Maroon">x</font> <a href="xcmplx_0.html#K7">/</a> 2 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
;<br/>
<a NAME="E3:17"/><i><font color="Green">E26</font></i>: 
 <a href="sin_cos2.html#K9">tanh</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i>, </i>;<br/>
<i><font color="Green">E30</font></i>: <span class="p1">(<span class="default"><a href="sin_cos5.html#K1">coth</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1 = 
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><a href="sin_cos5.html#K1">coth</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1
<b>by </b><i><a class="ref" href="xcmplx_1.html#T56">XCMPLX_1:56</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><a href="sin_cos2.html#K9">tanh</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><a href="sin_cos2.html#K9">tanh</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos2.html#K9">tanh</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><a href="sin_cos2.html#K9">tanh</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="sin_cos2.html#K9">tanh</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><a href="sin_cos2.html#K9">tanh</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T63">XCMPLX_1:63</a></i>
;<br/>
<i><font color="Green">E35</font></i>: <span class="p1">(<span class="default"><a href="sin_cos5.html#K1">coth</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1 = 
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><a href="sin_cos5.html#K1">coth</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1
<b>by </b><i><a class="ref" href="xcmplx_1.html#T56">XCMPLX_1:56</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><a href="sin_cos2.html#K9">tanh</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><a href="sin_cos2.html#K9">tanh</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos2.html#K9">tanh</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><a href="sin_cos2.html#K9">tanh</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="sin_cos2.html#K9">tanh</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><a href="sin_cos2.html#K9">tanh</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T121">XCMPLX_1:121</a></i>
;<br/>
<i><font color="Green">E36</font></i>:  <a href="sin_cos.html#K28">exp_R</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="sin_cos2.html#K9">tanh</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="sin_cos2.html#K9">tanh</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos5.html#K1">coth</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos5.html#K1">coth</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>
<b>by </b><i>, , , <a class="ref" href="xcmplx_1.html#T55">XCMPLX_1:55</a></i>
;<br/>
 <a href="sin_cos.html#K28">exp_R</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">x</font></span>)</span> = 
<span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="sin_cos2.html#K9">tanh</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="sin_cos2.html#K9">tanh</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos5.html#K1">coth</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos5.html#K1">coth</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span>
<b>by </b><i>, , , <a class="ref" href="xcmplx_1.html#T55">XCMPLX_1:55</a></i>
;<br/>
<b>hence </b><a NAME="E8:17"/>
(  <a href="sin_cos.html#K28">exp_R</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos5.html#K1">coth</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos5.html#K1">coth</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> &amp;  <a href="sin_cos.html#K28">exp_R</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos5.html#K1">coth</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos5.html#K1">coth</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span> )
 <b>by </b><i/>;<br/>


</div>
