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<b>let </b><font color="Maroon">ff</font> be   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a> ;<br/>

<b>assume </b><a NAME="E1:56"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">ff</font> <a href="hidden.html#R1">=</a>  <a href="binop_2.html#K31">compreal</a> 
 ;<br/>

<a NAME="E2:56"/><i><font color="Green">E26</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K7">-</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">ff</font></span>)</span></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_cos.html#K27">exp_R</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">ff</font></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_cos.html#K27">exp_R</a>  <a href="seq_1.html#K7">-</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">ff</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Olive">p</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">ff</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">p</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K7">-</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">ff</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">p</font> ) )
 
<b>by </b><i><a class="ref" href="seq_1.html#D4">SEQ_1:def 4</a></i>;<br/>
<a NAME="E3:56"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K7">-</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">ff</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K1">REAL</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1"><b>proof </b></a><div class="add">

<a NAME="E1:56_1"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">ff</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i>, <a class="txt" href="sin_cos2.html#E4:56"><i><font color="Green">E27</font></i></a>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a>, <a class="ref" href="sin_cos.html#T51">SIN_COS:51</a></i>;<br/>
<b>hence </b><a NAME="E2:56_1"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K7">-</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">ff</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i>, <a class="ref" href="sin_cos.html#T51">SIN_COS:51</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:56"/><b>then </b><i><font color="Green">E27</font></i>: 
(  <a href="numbers.html#K1">REAL</a>  <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <a href="sin_cos2.html#K1" target="_self">sinh</a>  &amp;  <a href="numbers.html#K1">REAL</a>  <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K7">-</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">ff</font></span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="seq_1.html#D6">SEQ_1:def 6</a></i>;<br/>
<a NAME="E5:56"/>
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#K1" target="_self">sinh</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"><span class="p2">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K7">-</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">ff</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">p</font>
 
<b>by </b><i>, <a class="ref" href="sin_cos2.html#D1" target="_self">Def1</a></i>;<br/>
<b>hence </b><a NAME="E6:56"/>
 <a href="sin_cos2.html#K1" target="_self">sinh</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K7">-</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">ff</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="partfun1.html#T34">PARTFUN1:34</a></i>;<br/>

<b>thus </b><a NAME="E7:56"/>
verum
 ;<br/>


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