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<b>let </b><font color="Maroon" title="c1">x</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="sin_cos2.html#K9" title="SIN_COS2:func.9">tanh</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="sin_cos2.html#K3" title="SIN_COS2:func.3">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#K6" title="SIN_COS2:func.6">cosh</a> <font color="Maroon" title="c1">x</font></span>)</span> &amp;  <a href="sin_cos2.html#K9" title="SIN_COS2:func.9">tanh</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/>

<i><font color="Green" title="E5">A1</font></i>:  <a href="sin_cos2.html#K9" title="SIN_COS2:func.9">tanh</a> <font color="Maroon" title="c1">x</font> = 
<a href="sin_cos2.html#K7" title="SIN_COS2:func.7">tanh</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">x</font>
<b>by </b><i><a class="ref" href="sin_cos2.html#D6" title="SIN_COS2:def.6">SIN_COS2:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="sin_cos2.html#K1" title="SIN_COS2:func.1">sinh</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</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>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">x</font></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos2.html#T17" title="SIN_COS2:th.17">SIN_COS2:17</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="sin_cos2.html#K3" title="SIN_COS2:func.3">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>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">x</font></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos2.html#D2" title="SIN_COS2:def.2">SIN_COS2:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="sin_cos2.html#K3" title="SIN_COS2:func.3">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#K6" title="SIN_COS2:func.6">cosh</a> <font color="Maroon" title="c1">x</font></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos2.html#D4" title="SIN_COS2:def.4">SIN_COS2:def 4</a></i>
;<br/>
 <a href="sin_cos2.html#K9" title="SIN_COS2:func.9">tanh</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  = 
<a href="sin_cos2.html#K7" title="SIN_COS2:func.7">tanh</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="ref" href="sin_cos2.html#D6" title="SIN_COS2:def.6">SIN_COS2:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="sin_cos2.html#K1" title="SIN_COS2:func.1">sinh</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
<b>by </b><i><a class="ref" href="sin_cos2.html#T17" title="SIN_COS2:th.17">SIN_COS2:17</a></i>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="ref" href="sin_cos2.html#T16" title="SIN_COS2:th.16">SIN_COS2:16</a></i>
;<br/>
<b>hence </b><a NAME="E3:4"/>
(  <a href="sin_cos2.html#K9" title="SIN_COS2:func.9">tanh</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="sin_cos2.html#K3" title="SIN_COS2:func.3">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#K6" title="SIN_COS2:func.6">cosh</a> <font color="Maroon" title="c1">x</font></span>)</span> &amp;  <a href="sin_cos2.html#K9" title="SIN_COS2:func.9">tanh</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="sin_cos8.html#E1:4"><i><font color="Green" title="E5">A1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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