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<span class="kw">let </span><font color="Maroon" title="c1">y</font>, <font color="Maroon" title="c2">z</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"> ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos2.html#K2" title="SIN_COS2:func.2">sinh</a> <font color="Maroon" title="c1">y</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="sin_cos2.html#K4" title="SIN_COS2:func.4">cosh</a> <font color="Maroon" title="c2">z</font></span>)</span></span>)</span> <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">y</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="sin_cos2.html#K4" title="SIN_COS2:func.4">cosh</a> <font color="Maroon" title="c2">z</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="sin_cos2.html#K6" title="SIN_COS2:func.6">tanh</a> <font color="Maroon" title="c1">y</font> &amp; <span class="p1">(<span class="default"><a href="sin_cos2.html#K2" title="SIN_COS2:func.2">sinh</a> <font color="Maroon" title="c1">y</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="sin_cos2.html#K4" title="SIN_COS2:func.4">cosh</a> <font color="Maroon" title="c2">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="sin_cos2.html#K6" title="SIN_COS2:func.6">tanh</a> <font color="Maroon" title="c1">y</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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">y</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="sin_cos2.html#K4" title="SIN_COS2:func.4">cosh</a> <font color="Maroon" title="c2">z</font></span>)</span></span>)</span> &amp;  <a href="sin_cos2.html#K2" title="SIN_COS2:func.2">sinh</a> <font color="Maroon" title="c1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="sin_cos2.html#K6" title="SIN_COS2:func.6">tanh</a> <font color="Maroon" title="c1">y</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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">y</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="sin_cos2.html#K2" title="SIN_COS2:func.2">sinh</a> <font color="Maroon" title="c1">y</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="sin_cos2.html#K2" title="SIN_COS2:func.2">sinh</a> <font color="Maroon" title="c2">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos2.html#K6" title="SIN_COS2:func.6">tanh</a> <font color="Maroon" title="c1">y</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="sin_cos2.html#K6" title="SIN_COS2:func.6">tanh</a> <font color="Maroon" title="c2">z</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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">y</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="sin_cos2.html#K4" title="SIN_COS2:func.4">cosh</a> <font color="Maroon" title="c2">z</font></span>)</span></span>)</span> )</span><br/>



<a NAME="E1:34"/><span class="lab"><font color="Green" title="E20">A1</font></span>: 
 <a href="sin_cos2.html#K4" title="SIN_COS2:func.4">cosh</a> <font color="Maroon" title="c1">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="sin_cos8.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E21">A2</font></span>:  <a href="sin_cos2.html#K2" title="SIN_COS2:func.2">sinh</a> <font color="Maroon" title="c1">y</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos2.html#K2" title="SIN_COS2:func.2">sinh</a> <font color="Maroon" title="c1">y</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#K4" title="SIN_COS2:func.4">cosh</a> <font color="Maroon" title="c1">y</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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">y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T87" onmouseover="rs('xcmplx_1/T87')" onmouseout="rh()">XCMPLX_1:87</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="sin_cos2.html#K6" title="SIN_COS2:func.6">tanh</a> <font color="Maroon" title="c1">y</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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">y</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/>
<a NAME="E3:34"/><span class="lab"><font color="Green" title="E22">A3</font></span>: 
 <a href="sin_cos2.html#K4" title="SIN_COS2:func.4">cosh</a> <font color="Maroon" title="c2">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="sin_cos8.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E23">A4</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos2.html#K2" title="SIN_COS2:func.2">sinh</a> <font color="Maroon" title="c1">y</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="sin_cos2.html#K4" title="SIN_COS2:func.4">cosh</a> <font color="Maroon" title="c2">z</font></span>)</span></span>)</span> <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">y</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="sin_cos2.html#K4" title="SIN_COS2:func.4">cosh</a> <font color="Maroon" title="c2">z</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="sin_cos2.html#K2" title="SIN_COS2:func.2">sinh</a> <font color="Maroon" title="c1">y</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">y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T91" onmouseover="rs('xcmplx_1/T91')" onmouseout="rh()">XCMPLX_1:91</a></span>
<br/>.= 
 <a href="sin_cos2.html#K6" title="SIN_COS2:func.6">tanh</a> <font color="Maroon" title="c1">y</font>
<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/>
<span class="p1">(<span class="default"><a href="sin_cos2.html#K2" title="SIN_COS2:func.2">sinh</a> <font color="Maroon" title="c1">y</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="sin_cos2.html#K2" title="SIN_COS2:func.2">sinh</a> <font color="Maroon" title="c2">z</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos2.html#K2" title="SIN_COS2:func.2">sinh</a> <font color="Maroon" title="c1">y</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><a href="sin_cos2.html#K2" title="SIN_COS2:func.2">sinh</a> <font color="Maroon" title="c2">z</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos2.html#K4" title="SIN_COS2:func.4">cosh</a> <font color="Maroon" title="c1">y</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><a href="sin_cos2.html#K4" title="SIN_COS2:func.4">cosh</a> <font color="Maroon" title="c2">z</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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">y</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="sin_cos2.html#K4" title="SIN_COS2:func.4">cosh</a> <font color="Maroon" title="c2">z</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E3:34"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="ref" href="xcmplx_1.html#T6" onmouseover="rs('xcmplx_1/T6')" onmouseout="rh()">XCMPLX_1:6</a>, <a class="ref" href="xcmplx_1.html#T87" onmouseover="rs('xcmplx_1/T87')" onmouseout="rh()">XCMPLX_1:87</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos2.html#K2" title="SIN_COS2:func.2">sinh</a> <font color="Maroon" title="c1">y</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default"><a href="sin_cos2.html#K4" title="SIN_COS2:func.4">cosh</a> <font color="Maroon" title="c1">y</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos2.html#K2" title="SIN_COS2:func.2">sinh</a> <font color="Maroon" title="c2">z</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default"><a href="sin_cos2.html#K4" title="SIN_COS2:func.4">cosh</a> <font color="Maroon" title="c2">z</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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">y</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="sin_cos2.html#K4" title="SIN_COS2:func.4">cosh</a> <font color="Maroon" title="c2">z</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T76" onmouseover="rs('xcmplx_1/T76')" onmouseout="rh()">XCMPLX_1:76</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos2.html#K6" title="SIN_COS2:func.6">tanh</a> <font color="Maroon" title="c1">y</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos2.html#K2" title="SIN_COS2:func.2">sinh</a> <font color="Maroon" title="c2">z</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default"><a href="sin_cos2.html#K4" title="SIN_COS2:func.4">cosh</a> <font color="Maroon" title="c2">z</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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">y</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="sin_cos2.html#K4" title="SIN_COS2:func.4">cosh</a> <font color="Maroon" title="c2">z</font></span>)</span></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/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos2.html#K6" title="SIN_COS2:func.6">tanh</a> <font color="Maroon" title="c1">y</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="sin_cos2.html#K6" title="SIN_COS2:func.6">tanh</a> <font color="Maroon" title="c2">z</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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">y</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="sin_cos2.html#K4" title="SIN_COS2:func.4">cosh</a> <font color="Maroon" title="c2">z</font></span>)</span></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/>
<span class="kw">hence </span><a NAME="E6:34"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos2.html#K2" title="SIN_COS2:func.2">sinh</a> <font color="Maroon" title="c1">y</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="sin_cos2.html#K4" title="SIN_COS2:func.4">cosh</a> <font color="Maroon" title="c2">z</font></span>)</span></span>)</span> <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">y</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="sin_cos2.html#K4" title="SIN_COS2:func.4">cosh</a> <font color="Maroon" title="c2">z</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="sin_cos2.html#K6" title="SIN_COS2:func.6">tanh</a> <font color="Maroon" title="c1">y</font> &amp; <span class="p1">(<span class="default"><a href="sin_cos2.html#K2" title="SIN_COS2:func.2">sinh</a> <font color="Maroon" title="c1">y</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="sin_cos2.html#K4" title="SIN_COS2:func.4">cosh</a> <font color="Maroon" title="c2">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="sin_cos2.html#K6" title="SIN_COS2:func.6">tanh</a> <font color="Maroon" title="c1">y</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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">y</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="sin_cos2.html#K4" title="SIN_COS2:func.4">cosh</a> <font color="Maroon" title="c2">z</font></span>)</span></span>)</span> &amp;  <a href="sin_cos2.html#K2" title="SIN_COS2:func.2">sinh</a> <font color="Maroon" title="c1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="sin_cos2.html#K6" title="SIN_COS2:func.6">tanh</a> <font color="Maroon" title="c1">y</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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">y</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="sin_cos2.html#K2" title="SIN_COS2:func.2">sinh</a> <font color="Maroon" title="c1">y</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="sin_cos2.html#K2" title="SIN_COS2:func.2">sinh</a> <font color="Maroon" title="c2">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos2.html#K6" title="SIN_COS2:func.6">tanh</a> <font color="Maroon" title="c1">y</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="sin_cos2.html#K6" title="SIN_COS2:func.6">tanh</a> <font color="Maroon" title="c2">z</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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">y</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="sin_cos2.html#K4" title="SIN_COS2:func.4">cosh</a> <font color="Maroon" title="c2">z</font></span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="txt" href="#E2:34"><span class="lab"><font color="Green" title="E21">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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