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<b>let </b><font color="Maroon">x1</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">x2</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>



<b>assume </b><a NAME="E1:43"/><i><font color="Green">E20</font></i>: 
(  <a href="sin_cos2.html#K3">sinh</a> <font color="Maroon">x1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp;  <a href="sin_cos2.html#K3">sinh</a> <font color="Maroon">x2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 ;<br/>

<a NAME="E2:43"/><b>then </b>
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="sin_cos2.html#K3">sinh</a> <font color="Maroon">x2</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
;<br/>
<a NAME="E3:43"/><b>then </b>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="sin_cos2.html#K3">sinh</a> <font color="Maroon">x2</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
;<br/>
<a NAME="E4:43"/><b>then </b>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="sin_cos2.html#K1">sinh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="sin_cos2.html#D2">SIN_COS2:def 2</a></i>;<br/>
<a NAME="E5:43"/><b>then </b>
<a href="sin_cos2.html#K1">sinh</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">x2</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="sin_cos2.html#T19">SIN_COS2:19</a></i>;<br/>
<a NAME="E6:43"/><b>then </b><i><font color="Green">E24</font></i>: 
 <a href="sin_cos2.html#K3">sinh</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">x2</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="sin_cos2.html#D2">SIN_COS2:def 2</a></i>;<br/>
 <a href="sin_cos5.html#K1" target="_self">coth</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x2</font></span>)</span> = 
 <a href="sin_cos5.html#K1" target="_self">coth</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">x2</font></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos5.html#K1" target="_self">coth</a> <font color="Maroon">x1</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos5.html#K1" target="_self">coth</a> <span class="p4">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">x2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos5.html#K1" target="_self">coth</a> <font color="Maroon">x1</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="sin_cos5.html#K1" target="_self">coth</a> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">x2</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos5.html#K1" target="_self">coth</a> <font color="Maroon">x1</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p4">(<span class="default"><a href="sin_cos5.html#K1" target="_self">coth</a> <font color="Maroon">x2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos5.html#K1" target="_self">coth</a> <font color="Maroon">x1</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="sin_cos5.html#K1" target="_self">coth</a> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">x2</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos5.html#K1" target="_self">coth</a> <font color="Maroon">x1</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos5.html#K1" target="_self">coth</a> <font color="Maroon">x2</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos5.html#K1" target="_self">coth</a> <font color="Maroon">x1</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_cos5.html#K1" target="_self">coth</a> <font color="Maroon">x2</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos5.html#K1" target="_self">coth</a> <font color="Maroon">x1</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos5.html#K1" target="_self">coth</a> <font color="Maroon">x2</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos5.html#K1" target="_self">coth</a> <font color="Maroon">x1</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="sin_cos5.html#K1" target="_self">coth</a> <font color="Maroon">x2</font></span>)</span></span>)</span>

;<br/>
<b>hence </b><a NAME="E8:43"/>
 <a href="sin_cos5.html#K1" target="_self">coth</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos5.html#K1" target="_self">coth</a> <font color="Maroon">x1</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos5.html#K1" target="_self">coth</a> <font color="Maroon">x2</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos5.html#K1" target="_self">coth</a> <font color="Maroon">x1</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="sin_cos5.html#K1" target="_self">coth</a> <font color="Maroon">x2</font></span>)</span></span>)</span>
 ;<br/>


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