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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="E1:5"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>

<a NAME="E2:5"/><i><font color="Green">E26</font></i>: 
 <a href="sin_cos2.html#K6">cosh</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E3:5"/><i><font color="Green">E30</font></i>: 
 <a href="sin_cos.html#K28">exp_R</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 
<b>by </b><i>, <a class="ref" href="sin_cos7.html#T29">SIN_COS7:29</a></i>;<br/>
<a NAME="E4:5"/>
 <a href="sin_cos.html#K28">exp_R</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="sin_cos.html#T60">SIN_COS:60</a></i>;<br/>
<a NAME="E5:5"/><b>then </b>
1 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K28">exp_R</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="sin_cos.html#K28">exp_R</a> <font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T200">XCMPLX_1:200</a></i>;<br/>
<a NAME="E6:5"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K28">exp_R</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K28">exp_R</a> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="taylor_1.html#T4">TAYLOR_1:4</a></i>;<br/>
<a NAME="E7:5"/><b>then </b>
 <a href="sin_cos2.html#K3">sinh</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E8:5"/><b>then </b>
<span class="p1">(<span class="default"><a href="sin_cos2.html#K3">sinh</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><a href="sin_cos2.html#K6">cosh</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T50">XCMPLX_1:50</a></i>;<br/>
<b>hence </b><a NAME="E9:5"/>
(  <a href="sin_cos2.html#K3">sinh</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp;  <a href="sin_cos2.html#K9">tanh</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 <b>by </b><i/>;<br/>


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