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<a href="sin_cos2.html#K3" title="SIN_COS2:func.3">cosh</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 = 
<span class="p1">(<span class="default"><a href="power.html#K4" title="POWER:func.4">number_e</a> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">exp_R</a> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
<span class="kw">by</span> <span class="lab"><a class="ref" href="irrat_1.html#D7" onmouseover="rs('irrat_1/D7')" onmouseout="rh()">IRRAT_1:def 7</a>, <a class="ref" href="sin_cos2.html#D3" onmouseover="rs('sin_cos2/D3')" onmouseout="rh()">SIN_COS2:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="power.html#K4" title="POWER:func.4">number_e</a> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <a href="power.html#K4" title="POWER:func.4">number_e</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
<span class="kw">by</span> <span class="lab"><a class="ref" href="irrat_1.html#D7" onmouseover="rs('irrat_1/D7')" onmouseout="rh()">IRRAT_1:def 7</a>, <a class="ref" href="taylor_1.html#T4" onmouseover="rs('taylor_1/T4')" onmouseout="rh()">TAYLOR_1:4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="power.html#K4" title="POWER:func.4">number_e</a> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="power.html#K4" title="POWER:func.4">number_e</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="power.html#K4" title="POWER:func.4">number_e</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <a href="power.html#K4" title="POWER:func.4">number_e</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
<span class="kw">by</span> <span class="lab"><a class="ref" href="taylor_1.html#T11" onmouseover="rs('taylor_1/T11')" onmouseout="rh()">TAYLOR_1:11</a>, <a class="ref" href="xcmplx_1.html#T91" onmouseover="rs('xcmplx_1/T91')" onmouseout="rh()">XCMPLX_1:91</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="power.html#K4" title="POWER:func.4">number_e</a> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <a href="power.html#K4" title="POWER:func.4">number_e</a></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <a href="power.html#K4" title="POWER:func.4">number_e</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="power.html#K4" title="POWER:func.4">number_e</a> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <a href="power.html#K4" title="POWER:func.4">number_e</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T62" onmouseover="rs('xcmplx_1/T62')" onmouseout="rh()">XCMPLX_1:62</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="power.html#K4" title="POWER:func.4">number_e</a> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="power.html#K4" title="POWER:func.4">number_e</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T78" onmouseover="rs('xcmplx_1/T78')" onmouseout="rh()">XCMPLX_1:78</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E2:22"/>
<a href="sin_cos2.html#K3" title="SIN_COS2:func.3">cosh</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="power.html#K4" title="POWER:func.4">number_e</a> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="power.html#K4" title="POWER:func.4">number_e</a></span>)</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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