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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="E1:39"/>
<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;</a> 0
 ;<br/>

<a NAME="E2:39"/><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="xxreal_0.html#R1">&lt;</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T143">XREAL_1:143</a></i>;<br/>
<a NAME="E3:39"/><i><font color="Green">E26</font></i>: 
 <a href="sin_cos2.html#K3">sinh</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0
 
<b>by </b><i><a class="txt" href="sin_cos8.html#E3"><i><font color="Green">Lemma24</font></i></a>, <a class="ref" href="sin_cos5.html#T47">SIN_COS5:47</a></i>;<br/>
 <a href="sin_cos2.html#K3">sinh</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> = 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><a href="sin_cos2.html#K3">sinh</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>

<br/>.= 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos2.html#K3">sinh</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="sin_cos8.html#E2:39"><i><font color="Green">E25</font></i></a>, <a class="ref" href="square_1.html#T90">SQUARE_1:90</a></i>
<br/>.= 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos2.html#K1">sinh</a>  <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos2.html#D2">SIN_COS2:def 2</a></i>
<br/>.= 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <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> <span class="p5">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p0">(<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</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos2.html#T18">SIN_COS2:18</a></i>
<br/>.= 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <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#K5">-</a> 1</span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>

<br/>.= 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos2.html#K6">cosh</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos2.html#D4">SIN_COS2:def 4</a></i>
;<br/>
<b>hence </b><a NAME="E5:39"/>
 <a href="sin_cos2.html#K3">sinh</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#R1">=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos2.html#K6">cosh</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>
 ;<br/>


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