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

<b>let </b><font color="Maroon">f1</font> be   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a> , <font color="Maroon">f2</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="E1:90_1_2"/><i><font color="Green">E59</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K1">REAL</a>  &amp; ( for <font color="Olive">d</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  holds  <font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <font color="Olive">d</font> <a href="hidden.html#R1">=</a>  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><font color="Olive">d</font> <a href="sin_cos.html#K7" target="_self">ExpSeq</a> </span>)</span> ) )
 ;<br/>

<b>assume </b><a NAME="E2:90_1_2"/><i><font color="Green">E61</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K1">REAL</a>  &amp; ( for <font color="Olive">d</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  holds  <font color="Maroon">f2</font> <a href="seq_1.html#K2">.</a> <font color="Olive">d</font> <a href="hidden.html#R1">=</a>  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><font color="Olive">d</font> <a href="sin_cos.html#K7" target="_self">ExpSeq</a> </span>)</span> ) )
 ;<br/>

<a NAME="E3:90_1_2"/>
for <font color="Olive">d</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>   st <font color="Olive">d</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a>  holds <br/><font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <font color="Olive">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="seq_1.html#K2">.</a> <font color="Olive">d</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="90_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">d</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> ;<br/>

<b>assume </b><a NAME="E1:90_1_2_1"/>
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 ;<br/>

<b>thus </b><font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">d</font> = 
 <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="sin_cos.html#K7" target="_self">ExpSeq</a> </span>)</span>
<b>by </b><i><a class="ref" href="sin_cos.html#T3" target="_self">Th3</a></i>
<br/>.= 
<font color="Maroon">f2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">d</font>
<b>by </b><i><a class="ref" href="sin_cos.html#D10" target="_self">Def10</a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:90_1_2"/>
<font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font>
 <b>by </b><i><a class="ref" href="sin_cos.html#T3" target="_self">Th3</a>, <a class="ref" href="sin_cos.html#D10" target="_self">Def10</a>, <a class="ref" href="partfun1.html#T34">PARTFUN1:34</a></i>;<br/>


</div>
