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<b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">R</font> be   <a href="fdiff_1.html#NM1">REST</a>;<br/>



<a NAME="E1:15"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">R</font> is <a href="partfun1.html#V1">total</a>
 
<b>by </b><i/>;<br/>
<a NAME="E2:15"/><b>then </b><i><font color="Green">E55</font></i>: 
<font color="Maroon">r</font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">R</font> is <a href="partfun1.html#V1">total</a>
 
<b>by </b><i><a class="ref" href="rfunct_1.html#T67">RFUNCT_1:67</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">h</font> be  <a href="fdiff_1.html#V1" target="_self">convergent_to_0</a> <a href="seq_1.html#NM1">Real_Sequence</a>;<br/><a NAME="E1:15_1"/><i><font color="Green">E56</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">h</font></span>)</span> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">h</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i/>;<br/><i><font color="Green">E57</font></i>: <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">R</font></span>)</span> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">h</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K14">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">h</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="rfunct_2.html#T33">RFUNCT_2:33</a></i>
<br/>.= 
<font color="Maroon">r</font> <a href="seq_1.html#K14">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">h</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T27">SEQ_1:27</a></i>
;<br/><b>hence </b><a NAME="E3:15_1"/>
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">R</font></span>)</span> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">h</font></span>)</span> is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i>, <a class="ref" href="seq_2.html#T21">SEQ_2:21</a></i>;<br/><b>thus </b> <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">R</font></span>)</span> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">h</font></span>)</span></span>)</span> = 
<font color="Maroon">r</font> <a href="real_1.html#K4">*</a> 0
<b>by </b><i>, , <a class="ref" href="seq_2.html#T22">SEQ_2:22</a></i>
<br/>.= 
0

;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:15"/>
<font color="Maroon">r</font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">R</font> is   <a href="fdiff_1.html#NM1">REST</a>
 <b>by </b><i>, </i>;<br/>


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