<?xml version="1.0"?>
<div class="add">

<a NAME="E1:63"/><i><font color="Green">E25</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="sin_cos2.html#K1" target="_self">sinh</a>  <a href="rfunct_1.html#K2">/</a> <a href="sin_cos2.html#K4" target="_self">cosh</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <a href="sin_cos2.html#K1" target="_self">sinh</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <a href="sin_cos2.html#K4" target="_self">cosh</a> </span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><a href="sin_cos2.html#K4" target="_self">cosh</a>  <a href="relset_1.html#K11">"</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> &amp; ( for <font color="Olive">p</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">p</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="sin_cos2.html#K1" target="_self">sinh</a>  <a href="rfunct_1.html#K2">/</a> <a href="sin_cos2.html#K4" target="_self">cosh</a> </span>)</span> holds <br/><span class="p1">(<span class="default"><a href="sin_cos2.html#K1" target="_self">sinh</a>  <a href="rfunct_1.html#K2">/</a> <a href="sin_cos2.html#K4" target="_self">cosh</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="sin_cos2.html#K1" target="_self">sinh</a>  <a href="seq_1.html#K2">.</a> <font color="Olive">p</font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos2.html#K4" target="_self">cosh</a>  <a href="seq_1.html#K2">.</a> <font color="Olive">p</font></span>)</span> <a href="binop_2.html#K8">"</a> </span>)</span> ) )
 
<b>by </b><i><a class="ref" href="rfunct_1.html#D4">RFUNCT_1:def 4</a></i>;<br/>
<a NAME="E2:63"/>
not 0 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <a href="sin_cos2.html#K4" target="_self">cosh</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:63_1"/>
0 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <a href="sin_cos2.html#K4" target="_self">cosh</a> 
 ;<br/>

<a NAME="E2:63_1"/><b>then </b>
 ex <font color="Olive">p</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">p</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <a href="sin_cos2.html#K4" target="_self">cosh</a>  &amp; 0 <a href="hidden.html#R1">=</a> <a href="sin_cos2.html#K4" target="_self">cosh</a>  <a href="seq_1.html#K2">.</a> <font color="Olive">p</font> )
 
<b>by </b><i><a class="ref" href="partfun1.html#T26">PARTFUN1:26</a></i>;<br/>
<b>hence </b><a NAME="E3:63_1"/>
contradiction
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E3:63"/><b>then </b><i><font color="Green">E26</font></i>: 
<a href="sin_cos2.html#K4" target="_self">cosh</a>  <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="funct_1.html#T142">FUNCT_1:142</a></i>;<br/>
<a NAME="E4:63"/>
for <font color="Olive">p</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>   st <font color="Olive">p</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a>  holds <br/><a href="sin_cos2.html#K7" target="_self">tanh</a>  <a href="seq_1.html#K2">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="sin_cos2.html#K1" target="_self">sinh</a>  <a href="rfunct_1.html#K2">/</a> <a href="sin_cos2.html#K4" target="_self">cosh</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_2"><b>proof </b></a><div class="add">

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

<a NAME="E1:63_2"/>
<font color="Maroon">p</font> is   <a href="real_1.html#NM1">Real</a>
 
<b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><a href="sin_cos2.html#K1" target="_self">sinh</a>  <a href="rfunct_1.html#K2">/</a> <a href="sin_cos2.html#K4" target="_self">cosh</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font> = 
<span class="p1">(<span class="default"><a href="sin_cos2.html#K1" target="_self">sinh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos2.html#K4" target="_self">cosh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span> <a href="binop_2.html#K8">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="sin_cos2.html#D5" target="_self">Def5</a>, <a class="ref" href="sin_cos2.html#T12" target="_self">Th12</a>, <a class="txt" href="sin_cos2.html#E1:63"><i><font color="Green">E25</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="sin_cos2.html#K1" target="_self">sinh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><a href="sin_cos2.html#K4" target="_self">cosh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_0.html#D9">XCMPLX_0:def 9</a></i>
<br/>.= 
<a href="sin_cos2.html#K7" target="_self">tanh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E3:63_2"/>
( <font color="Maroon">p</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a>  implies <a href="sin_cos2.html#K7" target="_self">tanh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="sin_cos2.html#K1" target="_self">sinh</a>  <a href="rfunct_1.html#K2">/</a> <a href="sin_cos2.html#K4" target="_self">cosh</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font> )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:63"/>
 <a href="sin_cos2.html#K7" target="_self">tanh</a>  <a href="hidden.html#R1">=</a> <a href="sin_cos2.html#K1" target="_self">sinh</a>  <a href="rfunct_1.html#K2">/</a> <a href="sin_cos2.html#K4" target="_self">cosh</a> 
 <b>by </b><i><a class="ref" href="sin_cos2.html#D5" target="_self">Def5</a>, <a class="ref" href="sin_cos2.html#T12" target="_self">Th12</a>, <a class="txt" href="sin_cos2.html#E1:63"><i><font color="Green">E25</font></i></a>, <a class="ref" href="partfun1.html#T34">PARTFUN1:34</a></i>;<br/>


</div>
