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<b>let </b><font color="Maroon">R</font> be   <a href="fdiff_1.html#NM1">REST</a>;<br/><b>let </b><font color="Maroon">L</font> be   <a href="fdiff_1.html#NM2">LINEAR</a>;<br/>



<a NAME="E1:17"/><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="E2:17"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">L</font> is <a href="partfun1.html#V1">total</a>
 
<b>by </b><i/>;<br/>
<a NAME="E3:17"/><b>then </b><i><font color="Green">E56</font></i>: 
<font color="Maroon">R</font> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">L</font> is <a href="partfun1.html#V1">total</a>
 
<b>by </b><i>, <a class="ref" href="rfunct_1.html#T66">RFUNCT_1:66</a></i>;<br/>
<b>consider </b><font color="Maroon">x1</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E5:17"/><i><font color="Green">E57</font></i>: 
for <font color="Olive">p</font> being  <a href="real_1.html#NM1">Real</a> holds  <font color="Maroon">L</font> <a href="seq_1.html#K2">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> <a href="real_1.html#K4">*</a> <font color="Olive">p</font>
 <b>by </b><i/>;<br/>
<div><i><font color="Green">E67</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="17_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/><i><font color="Green">E70</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#K8">(#)</a> <font color="Maroon">L</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"><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> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">L</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#T32">RFUNCT_2:32</a></i>
<br/>.= 
<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="seq_1.html#K11">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">h</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T22">SEQ_1:22</a></i>
;<br/><a NAME="E2:17_1"/><i><font color="Green">E71</font></i>: 
( <font color="Maroon">h</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">h</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> = 
<font color="Maroon">L</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i>, <a class="ref" href="rfunct_2.html#T30">RFUNCT_2:30</a></i>
<br/>.= 
<font color="Maroon">x1</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">h</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="seq_1.html#T13">SEQ_1:13</a></i>
;<br/></div><b>end;</b></div><a NAME="E4:17_1"/><b>then </b><i><font color="Green">E72</font></i>: 
<font color="Maroon">L</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">h</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">x1</font> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">h</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/><a NAME="E5:17_1"/><b>then </b><i><font color="Green">E73</font></i>: 
<font color="Maroon">L</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">h</font> 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/><i><font color="Green">E74</font></i>:  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">h</font></span>)</span> = 
<font color="Maroon">x1</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/><a NAME="E7:17_1"/><i><font color="Green">E75</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/><b>hence </b><a NAME="E8:17_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#K8">(#)</a> <font color="Maroon">L</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#T28">SEQ_2:28</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#K8">(#)</a> <font color="Maroon">L</font></span>)</span> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">h</font></span>)</span></span>)</span> = 
0 <a href="nat_1.html#K2">*</a> 0
<b>by </b><i>, , , , <a class="ref" href="seq_2.html#T29">SEQ_2:29</a></i>
<br/>.= 
0

;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:17"/>
<font color="Maroon">R</font> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">L</font> is   <a href="fdiff_1.html#NM1">REST</a>
 <b>by </b><i>, </i>;<br/>

<b>thus </b><a NAME="E8:17"/>
<font color="Maroon">L</font> <a href="seq_1.html#K8">(#)</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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