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

<b>let </b><font color="Maroon" title="c3">R1</font>, <font color="Maroon" title="c4">R2</font> be   <a href="fdiff_1.html#NM1" title="FDIFF_1:NM.1">REST</a>;<br/>



<a NAME="E1:14"/><i><font color="Green" title="E8">A1</font></i>: 
( <font color="Maroon" title="c3">R1</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> &amp; <font color="Maroon" title="c4">R2</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> )
 
<b>by </b><i><a class="ref" href="fdiff_1.html#D3" target="_self" title="FDIFF_1:def.3">Def3</a></i>;<br/>
<a NAME="E2:14"/><b>then </b><i><font color="Green" title="E9">A2</font></i>: 
<font color="Maroon" title="c3">R1</font> <a href="seq_1.html#K6" title="SEQ_1:func.6">+</a> <font color="Maroon" title="c4">R2</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>
 
<b>by </b><i><a class="ref" href="rfunct_1.html#T66" title="RFUNCT_1:th.66">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" title="c5">h</font> be  <a href="fdiff_1.html#V1" title="FDIFF_1:attr.1">convergent_to_0</a> <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>;<br/><a NAME="E1:14_1"/><i><font color="Green" title="E10">A3</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">R1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c5">h</font></span>)</span> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">R1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c5">h</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 <b>by </b><i><a class="ref" href="fdiff_1.html#D3" target="_self" title="FDIFF_1:def.3">Def3</a></i>;<br/><a NAME="E2:14_1"/><i><font color="Green" title="E11">A4</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">R2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c5">h</font></span>)</span> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">R2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c5">h</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 <b>by </b><i><a class="ref" href="fdiff_1.html#D3" target="_self" title="FDIFF_1:def.3">Def3</a></i>;<br/><i><font color="Green" title="E12">A5</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">R1</font> <a href="seq_1.html#K6" title="SEQ_1:func.6">+</a> <font color="Maroon" title="c4">R2</font></span>)</span> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c5">h</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">R1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c5">h</font></span>)</span> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">R2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c5">h</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_1.html#E1:14"><i><font color="Green" title="E8">A1</font></i></a>, <a class="ref" href="rfunct_2.html#T32" title="RFUNCT_2:th.32">RFUNCT_2:32</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">R1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c5">h</font></span>)</span></span>)</span> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">R2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c5">h</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T24" title="SEQ_1:th.24">SEQ_1:24</a></i>
;<br/><b>hence </b><a NAME="E4:14_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">R1</font> <a href="seq_1.html#K6" title="SEQ_1:func.6">+</a> <font color="Maroon" title="c4">R2</font></span>)</span> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c5">h</font></span>)</span> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 <b>by </b><i><a class="txt" href="fdiff_1.html#E1:14_1"><i><font color="Green" title="E10">A3</font></i></a>, <a class="txt" href="fdiff_1.html#E2:14_1"><i><font color="Green" title="E11">A4</font></i></a>, <a class="ref" href="seq_2.html#T19" title="SEQ_2:th.19">SEQ_2:19</a></i>;<br/><b>thus </b> <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">R1</font> <a href="seq_1.html#K6" title="SEQ_1:func.6">+</a> <font color="Maroon" title="c4">R2</font></span>)</span> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c5">h</font></span>)</span></span>)</span> = 
0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 0
<b>by </b><i><a class="txt" href="fdiff_1.html#E1:14_1"><i><font color="Green" title="E10">A3</font></i></a>, <a class="txt" href="fdiff_1.html#E2:14_1"><i><font color="Green" title="E11">A4</font></i></a>, <a class="txt" href="fdiff_1.html#E3:14_1"><i><font color="Green" title="E12">A5</font></i></a>, <a class="ref" href="seq_2.html#T20" title="SEQ_2:th.20">SEQ_2:20</a></i>
<br/>.= 
0

;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:14"/>
<font color="Maroon" title="c3">R1</font> <a href="seq_1.html#K6" title="SEQ_1:func.6">+</a> <font color="Maroon" title="c4">R2</font> is   <a href="fdiff_1.html#NM1" title="FDIFF_1:NM.1">REST</a>
 <b>by </b><i><a class="txt" href="fdiff_1.html#E2:14"><i><font color="Green" title="E9">A2</font></i></a>, <a class="ref" href="fdiff_1.html#D3" target="_self" title="FDIFF_1:def.3">Def3</a></i>;<br/>

<a NAME="E5:14"/><i><font color="Green" title="E10">A6</font></i>: 
<font color="Maroon" title="c3">R1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c4">R2</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>
 
<b>by </b><i><a class="txt" href="fdiff_1.html#E1:14"><i><font color="Green" title="E8">A1</font></i></a>, <a class="ref" href="rfunct_1.html#T66" title="RFUNCT_1:th.66">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" title="c5">h</font> be  <a href="fdiff_1.html#V1" title="FDIFF_1:attr.1">convergent_to_0</a> <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>;<br/><a NAME="E1:14_2"/><i><font color="Green" title="E11">A7</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">R1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c5">h</font></span>)</span> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">R1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c5">h</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 <b>by </b><i><a class="ref" href="fdiff_1.html#D3" target="_self" title="FDIFF_1:def.3">Def3</a></i>;<br/><a NAME="E2:14_2"/><i><font color="Green" title="E12">A8</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">R2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c5">h</font></span>)</span> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">R2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c5">h</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 <b>by </b><i><a class="ref" href="fdiff_1.html#D3" target="_self" title="FDIFF_1:def.3">Def3</a></i>;<br/><i><font color="Green" title="E13">A9</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">R1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c4">R2</font></span>)</span> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c5">h</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">R1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c5">h</font></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">R2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c5">h</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_1.html#E1:14"><i><font color="Green" title="E8">A1</font></i></a>, <a class="ref" href="rfunct_2.html#T32" title="RFUNCT_2:th.32">RFUNCT_2:32</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">R1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c5">h</font></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">R2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c5">h</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T29" title="SEQ_1:th.29">SEQ_1:29</a></i>
;<br/><b>hence </b><a NAME="E4:14_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">R1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c4">R2</font></span>)</span> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c5">h</font></span>)</span> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 <b>by </b><i><a class="txt" href="fdiff_1.html#E1:14_2"><i><font color="Green" title="E11">A7</font></i></a>, <a class="txt" href="fdiff_1.html#E2:14_2"><i><font color="Green" title="E12">A8</font></i></a>, <a class="ref" href="seq_2.html#T25" title="SEQ_2:th.25">SEQ_2:25</a></i>;<br/><b>thus </b> <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">R1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c4">R2</font></span>)</span> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c5">h</font></span>)</span></span>)</span> = 
0 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 0
<b>by </b><i><a class="txt" href="fdiff_1.html#E1:14_2"><i><font color="Green" title="E11">A7</font></i></a>, <a class="txt" href="fdiff_1.html#E2:14_2"><i><font color="Green" title="E12">A8</font></i></a>, <a class="txt" href="fdiff_1.html#E3:14_2"><i><font color="Green" title="E13">A9</font></i></a>, <a class="ref" href="seq_2.html#T26" title="SEQ_2:th.26">SEQ_2:26</a></i>
<br/>.= 
0

;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:14"/>
<font color="Maroon" title="c3">R1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c4">R2</font> is   <a href="fdiff_1.html#NM1" title="FDIFF_1:NM.1">REST</a>
 <b>by </b><i><a class="txt" href="fdiff_1.html#E5:14"><i><font color="Green" title="E10">A6</font></i></a>, <a class="ref" href="fdiff_1.html#D3" target="_self" title="FDIFF_1:def.3">Def3</a></i>;<br/>

<a NAME="E8:14"/><i><font color="Green" title="E11">A10</font></i>: 
<font color="Maroon" title="c3">R1</font> <a href="seq_1.html#K8" title="SEQ_1:func.8">(#)</a> <font color="Maroon" title="c4">R2</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>
 
<b>by </b><i><a class="txt" href="fdiff_1.html#E1:14"><i><font color="Green" title="E8">A1</font></i></a>, <a class="ref" href="rfunct_1.html#T66" title="RFUNCT_1:th.66">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" title="c5">h</font> be  <a href="fdiff_1.html#V1" title="FDIFF_1:attr.1">convergent_to_0</a> <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>;<br/><a NAME="E1:14_3"/><i><font color="Green" title="E12">A11</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">R1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c5">h</font></span>)</span> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">R1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c5">h</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 <b>by </b><i><a class="ref" href="fdiff_1.html#D3" target="_self" title="FDIFF_1:def.3">Def3</a></i>;<br/><a NAME="E2:14_3"/><i><font color="Green" title="E13">A12</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">R2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c5">h</font></span>)</span> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">R2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c5">h</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 <b>by </b><i><a class="ref" href="fdiff_1.html#D3" target="_self" title="FDIFF_1:def.3">Def3</a></i>;<br/><a NAME="E3:14_3"/><i><font color="Green" title="E14">A13</font></i>: 
( <font color="Maroon" title="c5">h</font> <a href="seq_1.html#NR1" title="SEQ_1:NR.1">is_not_0</a>  &amp; <font color="Maroon" title="c5">h</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c5">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 <b>by </b><i><a class="ref" href="fdiff_1.html#D1" target="_self" title="FDIFF_1:def.1">Def1</a></i>;<br/><a NAME="E4:14_3"/><b>then </b><i><font color="Green" title="E15">A14</font></i>: 
<font color="Maroon" title="c5">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a>  <a href="seq_1.html#NR1" title="SEQ_1:NR.1">is_not_0</a> 
 <b>by </b><i><a class="ref" href="seq_1.html#T41" title="SEQ_1:th.41">SEQ_1:41</a></i>;<br/><a NAME="E5:14_3"/><i><font color="Green" title="E16">A15</font></i>: 
<font color="Maroon" title="c5">h</font> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">R1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c5">h</font></span>)</span></span>)</span> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 <b>by </b><i><a class="txt" href="fdiff_1.html#E1:14_3"><i><font color="Green" title="E12">A11</font></i></a>, <a class="txt" href="fdiff_1.html#E3:14_3"><i><font color="Green" title="E14">A13</font></i></a>, <a class="ref" href="seq_2.html#T28" title="SEQ_2:th.28">SEQ_2:28</a></i>;<br/><i><font color="Green" title="E17">A16</font></i>:  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">R1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c5">h</font></span>)</span></span>)</span></span>)</span> = 
0 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 0
<b>by </b><i><a class="txt" href="fdiff_1.html#E1:14_3"><i><font color="Green" title="E12">A11</font></i></a>, <a class="txt" href="fdiff_1.html#E3:14_3"><i><font color="Green" title="E14">A13</font></i></a>, <a class="ref" href="seq_2.html#T29" title="SEQ_2:th.29">SEQ_2:29</a></i>
<br/>.= 
0

;<br/><i><font color="Green" title="E18">A17</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">R1</font> <a href="seq_1.html#K8" title="SEQ_1:func.8">(#)</a> <font color="Maroon" title="c4">R2</font></span>)</span> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c5">h</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">R1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c5">h</font></span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">R2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c5">h</font></span>)</span></span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span>
<b>by </b><i><a class="txt" href="fdiff_1.html#E1:14"><i><font color="Green" title="E8">A1</font></i></a>, <a class="ref" href="rfunct_2.html#T32" title="RFUNCT_2:th.32">RFUNCT_2:32</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">R1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c5">h</font></span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">R2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c5">h</font></span>)</span></span>)</span> <a href="seq_1.html#K19" title="SEQ_1:func.19">/"</a> <font color="Maroon" title="c5">h</font>
<b>by </b><i><a class="ref" href="seq_1.html#D9" title="SEQ_1:def.9">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" title="c3">R1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c5">h</font></span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">R2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c5">h</font></span>)</span></span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span></span>)</span> <a href="seq_1.html#K19" title="SEQ_1:func.19">/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_1.html#E4:14_3"><i><font color="Green" title="E15">A14</font></i></a>, <a class="ref" href="seq_1.html#T51" title="SEQ_1:th.51">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" title="c3">R1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c5">h</font></span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">R2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c5">h</font></span>)</span></span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span></span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <font color="Maroon" title="c5">h</font></span>)</span> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#D9" title="SEQ_1:def.9">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" title="c3">R1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c5">h</font></span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">R2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c5">h</font></span>)</span></span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span></span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T44" title="SEQ_1:th.44">SEQ_1:44</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span></span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">R1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c5">h</font></span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">R2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c5">h</font></span>)</span></span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T42" title="SEQ_1:th.42">SEQ_1:42</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span></span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">R1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c5">h</font></span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">R2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c5">h</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T22" title="SEQ_1:th.22">SEQ_1:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span></span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">R1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c5">h</font></span>)</span></span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">R2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c5">h</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T22" title="SEQ_1:th.22">SEQ_1:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">R1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c5">h</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">R2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c5">h</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T22" title="SEQ_1:th.22">SEQ_1:22</a></i>
;<br/><b>hence </b><a NAME="E8:14_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">R1</font> <a href="seq_1.html#K8" title="SEQ_1:func.8">(#)</a> <font color="Maroon" title="c4">R2</font></span>)</span> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c5">h</font></span>)</span> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 <b>by </b><i><a class="txt" href="fdiff_1.html#E2:14_3"><i><font color="Green" title="E13">A12</font></i></a>, <a class="txt" href="fdiff_1.html#E5:14_3"><i><font color="Green" title="E16">A15</font></i></a>, <a class="ref" href="seq_2.html#T28" title="SEQ_2:th.28">SEQ_2:28</a></i>;<br/><b>thus </b> <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">R1</font> <a href="seq_1.html#K8" title="SEQ_1:func.8">(#)</a> <font color="Maroon" title="c4">R2</font></span>)</span> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c5">h</font></span>)</span></span>)</span> = 
0 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 0
<b>by </b><i><a class="txt" href="fdiff_1.html#E2:14_3"><i><font color="Green" title="E13">A12</font></i></a>, <a class="txt" href="fdiff_1.html#E5:14_3"><i><font color="Green" title="E16">A15</font></i></a>, <a class="txt" href="fdiff_1.html#E6:14_3"><i><font color="Green" title="E17">A16</font></i></a>, <a class="txt" href="fdiff_1.html#E7:14_3"><i><font color="Green" title="E18">A17</font></i></a>, <a class="ref" href="seq_2.html#T29" title="SEQ_2:th.29">SEQ_2:29</a></i>
<br/>.= 
0

;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:14"/>
<font color="Maroon" title="c3">R1</font> <a href="seq_1.html#K8" title="SEQ_1:func.8">(#)</a> <font color="Maroon" title="c4">R2</font> is   <a href="fdiff_1.html#NM1" title="FDIFF_1:NM.1">REST</a>
 <b>by </b><i><a class="txt" href="fdiff_1.html#E8:14"><i><font color="Green" title="E11">A10</font></i></a>, <a class="ref" href="fdiff_1.html#D3" target="_self" title="FDIFF_1:def.3">Def3</a></i>;<br/>


</div>
