<?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="E1:70"/>
<font color="Maroon">x</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 ;<br/>

<a NAME="E2:70"/><b>then </b><i><font color="Green">E21</font></i>: 
(  <a href="square_1.html#K6">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K7">/</a> 2 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K7">/</a> 2 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <span class="p1">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K7">/</a> 4 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 
<b>by </b><i>, , , , </i>;<br/>
2 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="sin_cos7.html#K3" target="_self">cosh2"</a> <span class="p2">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span></span>)</span> = 
2 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p2">(<span class="default"><a href="power.html#K5">log</a> <a href="power.html#K8">number_e</a> ,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos7.html#T30" target="_self">Th30</a>, <a class="ref" href="square_1.html#D4">SQUARE_1:def 4</a></i>
<br/>.= 
 <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="power.html#K5">log</a> <a href="power.html#K8">number_e</a> ,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>

<br/>.= 
 <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><a href="power.html#K5">log</a> <a href="power.html#K8">number_e</a> ,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="power.html#K3">to_power</a> 2</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos7.html#T30" target="_self">Th30</a>, , <a class="ref" href="power.html#T63">POWER:63</a></i>
<br/>.= 
 <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><a href="power.html#K5">log</a> <a href="power.html#K8">number_e</a> ,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="power.html#T53">POWER:53</a></i>
<br/>.= 
 <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><a href="power.html#K5">log</a> <a href="power.html#K8">number_e</a> ,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p0">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span>

<br/>.= 
 <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><a href="power.html#K5">log</a> <a href="power.html#K8">number_e</a> ,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p0">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos7.html#T30" target="_self">Th30</a>, <a class="ref" href="square_1.html#D4">SQUARE_1:def 4</a></i>
<br/>.= 
 <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><a href="power.html#K5">log</a> <a href="power.html#K8">number_e</a> ,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p0">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos7.html#T30" target="_self">Th30</a>, <a class="ref" href="square_1.html#D4">SQUARE_1:def 4</a></i>
<br/>.= 
 <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><a href="power.html#K5">log</a> <a href="power.html#K8">number_e</a> ,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>

<br/>.= 
 <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><a href="power.html#K5">log</a> <a href="power.html#K8">number_e</a> ,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos7.html#T30" target="_self">Th30</a>, <a class="ref" href="square_1.html#T97">SQUARE_1:97</a></i>
<br/>.= 
 <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><a href="power.html#K5">log</a> <a href="power.html#K8">number_e</a> ,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p5">(<span class="default">2 <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K7">/</a> 4</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="square_1.html#T89">SQUARE_1:89</a></i>
<br/>.= 
 <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><a href="power.html#K5">log</a> <a href="power.html#K8">number_e</a> ,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p4">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K7">/</a> 4</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos7.html#T30" target="_self">Th30</a>, <a class="ref" href="square_1.html#T97">SQUARE_1:97</a></i>
<br/>.= 
 <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><a href="power.html#K5">log</a> <a href="power.html#K8">number_e</a> ,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">x</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>

;<br/>
<b>hence </b><a NAME="E4:70"/>
 <a href="sin_cos7.html#K3" target="_self">cosh2"</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 2 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="sin_cos7.html#K3" target="_self">cosh2"</a> <span class="p2">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span></span>)</span>
 ;<br/>


</div>
