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

<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="fdiff_1.html#NM1">REST</a>;<br/>



<a NAME="E1:14"/><i><font color="Green">E10</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> is <a href="partfun1.html#V1">total</a> &amp; <font color="Maroon">c<sub>4</sub></font> is <a href="partfun1.html#V1">total</a> )
 
<b>by </b><i><a class="ref" href="fdiff_1.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E2:14"/><b>then </b><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K6">+</a> <font color="Maroon">c<sub>4</sub></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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></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="E1:14_1"/><i><font color="Green">E12</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></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">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>5</sub></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">c<sub>5</sub></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">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="ref" href="fdiff_1.html#D3" target="_self">Def3</a></i>;<br/><a NAME="E2:14_1"/><i><font color="Green">E13</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></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">c<sub>4</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>5</sub></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">c<sub>5</sub></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">c<sub>4</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="ref" href="fdiff_1.html#D3" target="_self">Def3</a></i>;<br/><i><font color="Green">E14</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></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">c<sub>3</sub></font> <a href="seq_1.html#K6">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></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">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="seq_1.html#K9">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_1.html#E1:14"><i><font color="Green">E10</font></i></a>, <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">c<sub>5</sub></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">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></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">c<sub>4</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T24">SEQ_1:24</a></i>
;<br/><b>hence </b><a NAME="E4:14_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></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">c<sub>3</sub></font> <a href="seq_1.html#K6">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i><a class="txt" href="fdiff_1.html#E1:14_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="fdiff_1.html#E2:14_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="seq_2.html#T19">SEQ_2:19</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">c<sub>5</sub></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">c<sub>3</sub></font> <a href="seq_1.html#K6">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> = 
0 <a href="nat_1.html#K1">+</a> 0
<b>by </b><i><a class="txt" href="fdiff_1.html#E1:14_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="fdiff_1.html#E2:14_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="fdiff_1.html#E3:14_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="seq_2.html#T20">SEQ_2:20</a></i>
<br/>.= 
0

;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:14"/>
<font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K6">+</a> <font color="Maroon">c<sub>4</sub></font> is   <a href="fdiff_1.html#NM1">REST</a>
 <b>by </b><i><a class="txt" href="fdiff_1.html#E2:14"><i><font color="Green">E11</font></i></a>, <a class="ref" href="fdiff_1.html#D3" target="_self">Def3</a></i>;<br/>

<a NAME="E5:14"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K7">-</a> <font color="Maroon">c<sub>4</sub></font> is <a href="partfun1.html#V1">total</a>
 
<b>by </b><i><a class="txt" href="fdiff_1.html#E1:14"><i><font color="Green">E10</font></i></a>, <a class="ref" href="rfunct_1.html#T66">RFUNCT_1:66</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></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="E1:14_2"/><i><font color="Green">E13</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></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">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>5</sub></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">c<sub>5</sub></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">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="ref" href="fdiff_1.html#D3" target="_self">Def3</a></i>;<br/><a NAME="E2:14_2"/><i><font color="Green">E14</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></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">c<sub>4</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>5</sub></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">c<sub>5</sub></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">c<sub>4</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="ref" href="fdiff_1.html#D3" target="_self">Def3</a></i>;<br/><i><font color="Green">E15</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></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">c<sub>3</sub></font> <a href="seq_1.html#K7">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></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">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_1.html#E1:14"><i><font color="Green">E10</font></i></a>, <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">c<sub>5</sub></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">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></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">c<sub>4</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T29">SEQ_1:29</a></i>
;<br/><b>hence </b><a NAME="E4:14_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></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">c<sub>3</sub></font> <a href="seq_1.html#K7">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i><a class="txt" href="fdiff_1.html#E1:14_2"><i><font color="Green">E13</font></i></a>, <a class="txt" href="fdiff_1.html#E2:14_2"><i><font color="Green">E14</font></i></a>, <a class="ref" href="seq_2.html#T25">SEQ_2:25</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">c<sub>5</sub></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">c<sub>3</sub></font> <a href="seq_1.html#K7">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> = 
0 <a href="real_1.html#K5">-</a> 0
<b>by </b><i><a class="txt" href="fdiff_1.html#E1:14_2"><i><font color="Green">E13</font></i></a>, <a class="txt" href="fdiff_1.html#E2:14_2"><i><font color="Green">E14</font></i></a>, <a class="txt" href="fdiff_1.html#E3:14_2"><i><font color="Green">E15</font></i></a>, <a class="ref" href="seq_2.html#T26">SEQ_2:26</a></i>
<br/>.= 
0

;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:14"/>
<font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K7">-</a> <font color="Maroon">c<sub>4</sub></font> is   <a href="fdiff_1.html#NM1">REST</a>
 <b>by </b><i><a class="txt" href="fdiff_1.html#E5:14"><i><font color="Green">E12</font></i></a>, <a class="ref" href="fdiff_1.html#D3" target="_self">Def3</a></i>;<br/>

<a NAME="E8:14"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">c<sub>4</sub></font> is <a href="partfun1.html#V1">total</a>
 
<b>by </b><i><a class="txt" href="fdiff_1.html#E1:14"><i><font color="Green">E10</font></i></a>, <a class="ref" href="rfunct_1.html#T66">RFUNCT_1:66</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></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="E1:14_3"/><i><font color="Green">E14</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></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">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>5</sub></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">c<sub>5</sub></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">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="ref" href="fdiff_1.html#D3" target="_self">Def3</a></i>;<br/><a NAME="E2:14_3"/><i><font color="Green">E15</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></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">c<sub>4</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>5</sub></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">c<sub>5</sub></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">c<sub>4</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="ref" href="fdiff_1.html#D3" target="_self">Def3</a></i>;<br/><a NAME="E3:14_3"/><i><font color="Green">E16</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#NR1" target="_self">is_not_0</a>  &amp; <font color="Maroon">c<sub>5</sub></font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="ref" href="fdiff_1.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E4:14_3"/><b>then </b><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K18">"</a>  <a href="seq_1.html#NR1" target="_self">is_not_0</a> 
 <b>by </b><i><a class="ref" href="seq_1.html#T41">SEQ_1:41</a></i>;<br/><a NAME="E5:14_3"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K11">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></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">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i><a class="txt" href="fdiff_1.html#E1:14_3"><i><font color="Green">E14</font></i></a>, <a class="txt" href="fdiff_1.html#E3:14_3"><i><font color="Green">E16</font></i></a>, <a class="ref" href="seq_2.html#T28">SEQ_2:28</a></i>;<br/><i><font color="Green">E19</font></i>:  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span> = 
0 <a href="nat_1.html#K2">*</a> 0
<b>by </b><i><a class="txt" href="fdiff_1.html#E1:14_3"><i><font color="Green">E14</font></i></a>, <a class="txt" href="fdiff_1.html#E3:14_3"><i><font color="Green">E16</font></i></a>, <a class="ref" href="seq_2.html#T29">SEQ_2:29</a></i>
<br/>.= 
0

;<br/><i><font color="Green">E20</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></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">c<sub>3</sub></font> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K18">"</a> </span>)</span>
<b>by </b><i><a class="txt" href="fdiff_1.html#E1:14"><i><font color="Green">E10</font></i></a>, <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">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K19">/"</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="seq_1.html#D9">SEQ_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K18">"</a> </span>)</span></span>)</span> <a href="seq_1.html#K19">/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K18">"</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_1.html#E4:14_3"><i><font color="Green">E17</font></i></a>, <a class="ref" href="seq_1.html#T51">SEQ_1:51</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K18">"</a> </span>)</span></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="seq_1.html#K18">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#D9">SEQ_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K18">"</a> </span>)</span></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K18">"</a> </span>)</span> <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">c<sub>5</sub></font> <a href="seq_1.html#K18">"</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T44">SEQ_1:44</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K18">"</a> </span>)</span></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K18">"</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T42">SEQ_1:42</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K18">"</a> </span>)</span></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T22">SEQ_1:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K11">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K18">"</a> </span>)</span></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></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">c<sub>4</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T22">SEQ_1:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></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">c<sub>4</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T22">SEQ_1:22</a></i>
;<br/><b>hence </b><a NAME="E8:14_3"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></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">c<sub>3</sub></font> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i><a class="txt" href="fdiff_1.html#E2:14_3"><i><font color="Green">E15</font></i></a>, <a class="txt" href="fdiff_1.html#E5:14_3"><i><font color="Green">E18</font></i></a>, <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">c<sub>5</sub></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">c<sub>3</sub></font> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> = 
0 <a href="nat_1.html#K2">*</a> 0
<b>by </b><i><a class="txt" href="fdiff_1.html#E2:14_3"><i><font color="Green">E15</font></i></a>, <a class="txt" href="fdiff_1.html#E5:14_3"><i><font color="Green">E18</font></i></a>, <a class="txt" href="fdiff_1.html#E6:14_3"><i><font color="Green">E19</font></i></a>, <a class="txt" href="fdiff_1.html#E7:14_3"><i><font color="Green">E20</font></i></a>, <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="E10:14"/>
<font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">c<sub>4</sub></font> is   <a href="fdiff_1.html#NM1">REST</a>
 <b>by </b><i><a class="txt" href="fdiff_1.html#E8:14"><i><font color="Green">E13</font></i></a>, <a class="ref" href="fdiff_1.html#D3" target="_self">Def3</a></i>;<br/>


</div>
