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<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="E1:79"/><i><font color="Green">E21</font></i>: 
( <font color="Maroon">x</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/>

<a NAME="E2:79"/><b>then </b><i><font color="Green">E25</font></i>: 
<font color="Maroon">x</font> <a href="square_1.html#K3">^2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="square_1.html#T74">SQUARE_1:74</a></i>;<br/>
<a NAME="E3:79"/><i><font color="Green">E27</font></i>: 
( 1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; 1 <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 
<b>by </b><i><a class="ref" href="sin_cos7.html#T30" target="_self">Th30</a>, <a class="ref" href="sin_cos7.html#T5" target="_self">Th5</a>, <a class="txt" href="sin_cos7.html#E10"><i><font color="Green">Lemma31</font></i></a></i>;<br/>
 <a href="sin_cos7.html#K3" target="_self">cosh2"</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">x</font></span>)</span> = 
 <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">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">x</font></span>)</span> <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">1 <a href="xcmplx_0.html#K7">/</a> <span class="p0">(<span class="default"><font color="Maroon">x</font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="square_1.html#T59">SQUARE_1:59</a>, <a class="ref" href="xcmplx_1.html#T77">XCMPLX_1:77</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">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">x</font></span>)</span> <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">1 <a href="xcmplx_0.html#K6">-</a> <span class="p0">(<span class="default">1 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p5">(<span class="default"><font color="Maroon">x</font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="sin_cos7.html#E42"><i><font color="Green">Lemma64</font></i></a>, <a class="ref" href="xcmplx_1.html#T127">XCMPLX_1:127</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">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">x</font></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">1 <a href="xcmplx_0.html#K6">-</a> <span class="p0">(<span class="default"><font color="Maroon">x</font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p5">(<span class="default"><font color="Maroon">x</font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos7.html#T31" target="_self">Th31</a>, <a class="txt" href="sin_cos7.html#E42"><i><font color="Green">Lemma64</font></i></a>, <a class="ref" href="square_1.html#T99">SQUARE_1:99</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">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">x</font></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">1 <a href="xcmplx_0.html#K6">-</a> <span class="p0">(<span class="default"><font color="Maroon">x</font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">x</font></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#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"><span class="p3">(<span class="default">1 <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">1 <a href="xcmplx_0.html#K6">-</a> <span class="p0">(<span class="default"><font color="Maroon">x</font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T63">XCMPLX_1:63</a></i>
;<br/>
<b>hence </b><a NAME="E5:79"/>
 <a href="sin_cos7.html#K7" target="_self">sech2"</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="sin_cos7.html#K3" target="_self">cosh2"</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">x</font></span>)</span>
 ;<br/>


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