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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>



<i><font color="Green">E14</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos2.html#K6">cosh</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="sin_cos2.html#K3">sinh</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="prepower.html#K3">|^</a> <font color="Maroon">c<sub>2</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos2.html#K4">cosh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="sin_cos2.html#K3">sinh</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="prepower.html#K3">|^</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="sin_cos2.html#D4">SIN_COS2:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos2.html#K4">cosh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="sin_cos2.html#K1">sinh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="prepower.html#K3">|^</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="sin_cos2.html#D2">SIN_COS2:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="sin_cos2.html#K4">cosh</a>  <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="sin_cos2.html#K1">sinh</a>  <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos2.html#T29">SIN_COS2:29</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="sin_cos2.html#K6">cosh</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="sin_cos2.html#K1">sinh</a>  <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos2.html#D4">SIN_COS2:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="sin_cos2.html#K6">cosh</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="sin_cos2.html#K3">sinh</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos2.html#D2">SIN_COS2:def 2</a></i>
;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos2.html#K6">cosh</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="sin_cos2.html#K3">sinh</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="prepower.html#K3">|^</a> <font color="Maroon">c<sub>2</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos2.html#K6">cosh</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><a href="sin_cos2.html#K3">sinh</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="prepower.html#K3">|^</a> <font color="Maroon">c<sub>2</sub></font>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos2.html#K6">cosh</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="sin_cos2.html#K3">sinh</a> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="prepower.html#K3">|^</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="sin_cos8.html#E10"><i><font color="Green">Lemma10</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos2.html#K4">cosh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="sin_cos2.html#K3">sinh</a> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="prepower.html#K3">|^</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="sin_cos2.html#D4">SIN_COS2:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos2.html#K4">cosh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="sin_cos2.html#K1">sinh</a>  <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="prepower.html#K3">|^</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="sin_cos2.html#D2">SIN_COS2:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos2.html#K4">cosh</a>  <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="sin_cos2.html#K1">sinh</a>  <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="prepower.html#K3">|^</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="sin_cos2.html#T19">SIN_COS2:19</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="sin_cos2.html#K4">cosh</a>  <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="sin_cos2.html#K1">sinh</a>  <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos2.html#T29">SIN_COS2:29</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="sin_cos2.html#K4">cosh</a>  <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="sin_cos2.html#K1">sinh</a>  <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos2.html#T19">SIN_COS2:19</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="sin_cos2.html#K4">cosh</a>  <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</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#K1">sinh</a>  <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos2.html#T19">SIN_COS2:19</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="sin_cos2.html#K4">cosh</a>  <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="sin_cos2.html#K1">sinh</a>  <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="sin_cos2.html#K6">cosh</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="sin_cos2.html#K1">sinh</a>  <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos2.html#D4">SIN_COS2:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="sin_cos2.html#K6">cosh</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="sin_cos2.html#K3">sinh</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos2.html#D2">SIN_COS2:def 2</a></i>
;<br/>
<b>hence </b><a NAME="E3:15"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos2.html#K6">cosh</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="sin_cos2.html#K3">sinh</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="prepower.html#K3">|^</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="sin_cos2.html#K6">cosh</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="sin_cos2.html#K3">sinh</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos2.html#K6">cosh</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="sin_cos2.html#K3">sinh</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="prepower.html#K3">|^</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="sin_cos2.html#K6">cosh</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="sin_cos2.html#K3">sinh</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="sin_cos8.html#E1:15"><i><font color="Green">E14</font></i></a></i>;<br/>


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