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<span class="kw">let </span><font color="Maroon" title="c1">x</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="sin_cos2.html#K2" title="SIN_COS2:func.2">sinh</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><a href="sin_cos5.html#K3" title="SIN_COS5:func.3">cosech</a> <font color="Maroon" title="c1">x</font></span>)</span> &amp;  <a href="sin_cos2.html#K4" title="SIN_COS2:func.4">cosh</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><a href="sin_cos5.html#K2" title="SIN_COS5:func.2">sech</a> <font color="Maroon" title="c1">x</font></span>)</span> &amp;  <a href="sin_cos2.html#K6" title="SIN_COS2:func.6">tanh</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><a href="sin_cos5.html#K1" title="SIN_COS5:func.1">coth</a> <font color="Maroon" title="c1">x</font></span>)</span> )</span><br/>

<span class="lab"><font color="Green" title="E11">A1</font></span>:  <a href="sin_cos2.html#K2" title="SIN_COS2:func.2">sinh</a> <font color="Maroon" title="c1">x</font> = 
1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><a href="sin_cos2.html#K2" title="SIN_COS2:func.2">sinh</a> <font color="Maroon" title="c1">x</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T56" onmouseover="rs('xcmplx_1/T56')" onmouseout="rh()">XCMPLX_1:56</a></span>
<br/>.= 
1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><a href="sin_cos5.html#K3" title="SIN_COS5:func.3">cosech</a> <font color="Maroon" title="c1">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos5.html#D3" onmouseover="rs('sin_cos5/D3')" onmouseout="rh()">SIN_COS5:def 3</a></span>
;<br/>
<span class="lab"><font color="Green" title="E12">A2</font></span>:  <a href="sin_cos2.html#K4" title="SIN_COS2:func.4">cosh</a> <font color="Maroon" title="c1">x</font> = 
1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><a href="sin_cos2.html#K4" title="SIN_COS2:func.4">cosh</a> <font color="Maroon" title="c1">x</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T56" onmouseover="rs('xcmplx_1/T56')" onmouseout="rh()">XCMPLX_1:56</a></span>
<br/>.= 
1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><a href="sin_cos5.html#K2" title="SIN_COS5:func.2">sech</a> <font color="Maroon" title="c1">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos5.html#D2" onmouseover="rs('sin_cos5/D2')" onmouseout="rh()">SIN_COS5:def 2</a></span>
;<br/>
 <a href="sin_cos2.html#K6" title="SIN_COS2:func.6">tanh</a> <font color="Maroon" title="c1">x</font> = 
<span class="p1">(<span class="default"><a href="sin_cos2.html#K2" title="SIN_COS2:func.2">sinh</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><a href="sin_cos2.html#K4" title="SIN_COS2:func.4">cosh</a> <font color="Maroon" title="c1">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos8.html#T1" target="_self" onmouseover="rs('sin_cos8/T1')" onmouseout="rh()">Th1</a></span>
<br/>.= 
1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos2.html#K4" title="SIN_COS2:func.4">cosh</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><a href="sin_cos2.html#K2" title="SIN_COS2:func.2">sinh</a> <font color="Maroon" title="c1">x</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T57" onmouseover="rs('xcmplx_1/T57')" onmouseout="rh()">XCMPLX_1:57</a></span>
<br/>.= 
1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><a href="sin_cos5.html#K1" title="SIN_COS5:func.1">coth</a> <font color="Maroon" title="c1">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos5.html#D1" onmouseover="rs('sin_cos5/D1')" onmouseout="rh()">SIN_COS5:def 1</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E4:10"/>
(  <a href="sin_cos2.html#K2" title="SIN_COS2:func.2">sinh</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><a href="sin_cos5.html#K3" title="SIN_COS5:func.3">cosech</a> <font color="Maroon" title="c1">x</font></span>)</span> &amp;  <a href="sin_cos2.html#K4" title="SIN_COS2:func.4">cosh</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><a href="sin_cos5.html#K2" title="SIN_COS5:func.2">sech</a> <font color="Maroon" title="c1">x</font></span>)</span> &amp;  <a href="sin_cos2.html#K6" title="SIN_COS2:func.6">tanh</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><a href="sin_cos5.html#K1" title="SIN_COS5:func.1">coth</a> <font color="Maroon" title="c1">x</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="txt" href="#E2:10"><span class="lab"><font color="Green" title="E12">A2</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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