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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/>

<i><font color="Green">E25</font></i>:  <a href="sin_cos2.html#K3">sinh</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">x</font></span>)</span> = 
<a href="sin_cos2.html#K1">sinh</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">x</font></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="sin_cos2.html#K1">sinh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos2.html#T19">SIN_COS2:19</a></i>
<br/>.= 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="sin_cos2.html#K3">sinh</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos2.html#D2">SIN_COS2:def 2</a></i>
;<br/>
<i><font color="Green">E26</font></i>:  <a href="sin_cos2.html#K6">cosh</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">x</font></span>)</span> = 
<a href="sin_cos2.html#K4">cosh</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos2.html#D4">SIN_COS2:def 4</a></i>
<br/>.= 
<a href="sin_cos2.html#K4">cosh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="sin_cos2.html#T19">SIN_COS2:19</a></i>
<br/>.= 
 <a href="sin_cos2.html#K6">cosh</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="sin_cos2.html#D4">SIN_COS2:def 4</a></i>
;<br/>
<i><font color="Green">E30</font></i>:  <a href="sin_cos2.html#K9">tanh</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">x</font></span>)</span> = 
<a href="sin_cos2.html#K7">tanh</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos2.html#D6">SIN_COS2:def 6</a></i>
<br/>.= 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="sin_cos2.html#K7">tanh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos2.html#T19">SIN_COS2:19</a></i>
<br/>.= 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="sin_cos2.html#K9">tanh</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos2.html#D6">SIN_COS2:def 6</a></i>
;<br/>
<i><font color="Green">E35</font></i>:  <a href="sin_cos5.html#K1">coth</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"><a href="sin_cos2.html#K6">cosh</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K6">/</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> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="sin_cos5.html#D1">SIN_COS5:def 1</a></i>
<br/>.= 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos2.html#K6">cosh</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><a href="sin_cos2.html#K3">sinh</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T189">XCMPLX_1:189</a></i>
<br/>.= 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="sin_cos5.html#K1">coth</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos5.html#D1">SIN_COS5:def 1</a></i>
;<br/>
<i><font color="Green">E36</font></i>:  <a href="sin_cos5.html#K2">sech</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">x</font></span>)</span> = 
1 <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>
<b>by </b><i>, <a class="ref" href="sin_cos5.html#D2">SIN_COS5:def 2</a></i>
<br/>.= 
 <a href="sin_cos5.html#K2">sech</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="sin_cos5.html#D2">SIN_COS5:def 2</a></i>
;<br/>
 <a href="sin_cos5.html#K3">cosech</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">x</font></span>)</span> = 
1 <a href="real_1.html#K6">/</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> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="sin_cos5.html#D3">SIN_COS5:def 3</a></i>
<br/>.= 
 <a href="real_1.html#K1">-</a> <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#K3">sinh</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T189">XCMPLX_1:189</a></i>
<br/>.= 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="sin_cos5.html#K3">cosech</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos5.html#D3">SIN_COS5:def 3</a></i>
;<br/>
<b>hence </b><a NAME="E7:9"/>
(  <a href="sin_cos2.html#K3">sinh</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>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="sin_cos2.html#K3">sinh</a> <font color="Maroon">x</font></span>)</span> &amp;  <a href="sin_cos2.html#K6">cosh</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>  <a href="sin_cos2.html#K6">cosh</a> <font color="Maroon">x</font> &amp;  <a href="sin_cos2.html#K9">tanh</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>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="sin_cos2.html#K9">tanh</a> <font color="Maroon">x</font></span>)</span> &amp;  <a href="sin_cos5.html#K1">coth</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>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="sin_cos5.html#K1">coth</a> <font color="Maroon">x</font></span>)</span> &amp;  <a href="sin_cos5.html#K2">sech</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>  <a href="sin_cos5.html#K2">sech</a> <font color="Maroon">x</font> &amp;  <a href="sin_cos5.html#K3">cosech</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>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="sin_cos5.html#K3">cosech</a> <font color="Maroon">x</font></span>)</span> )
 <b>by </b><i>, , , , </i>;<br/>


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