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

<b>let </b><font color="Maroon" title="c1">R1</font> be   <a href="fdiff_1.html#NM1" title="FDIFF_1:NM.1">REST</a>;<br/>

<a NAME="E1:26"/><i><font color="Green" title="E27">A1</font></i>: 
<font color="Maroon" title="c1">R1</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" title="FDIFF_1:def.3">FDIFF_1:def 3</a></i>;<br/>
<a NAME="E2:26"/><i><font color="Green" title="E28">A2</font></i>: 
 <a href="seq_1.html#K16" title="SEQ_1:func.16">-</a> <font color="Maroon" title="c1">R1</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>
 
<b>by </b><i><a class="txt" href="integra8.html#E1:26"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="rfunct_1.html#T68" title="RFUNCT_1:th.68">RFUNCT_1:68</a></i>;<br/>
<a NAME="E3:26"/><b>then </b><i><font color="Green" title="E29">B1</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c1">R1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="seq_1.html#K16" title="SEQ_1:func.16">-</a> <font color="Maroon" title="c1">R1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  )
 
<b>by </b><i><a class="txt" href="integra8.html#E1:26"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a></i>;<br/>
<a NAME="E4:26"/><i><font color="Green" title="E30">Lma</font></i>: 
for <font color="Olive" title="b1">h</font> being <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> holds   <a href="seq_1.html#K17" title="SEQ_1:func.17">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">R1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Olive" title="b1">h</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><a href="seq_1.html#K16" title="SEQ_1:func.16">-</a> <font color="Maroon" title="c1">R1</font></span>)</span> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Olive" title="b1">h</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">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="E1:26_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">h</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c1">R1</font>
 
<b>by </b><i><a class="txt" href="integra8.html#E3:26"><i><font color="Green" title="E29">B1</font></i></a>, <a class="ref" href="seq_1.html#D1" title="SEQ_1:def.1">SEQ_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E2:26_1"/>
 <a href="seq_1.html#K17" title="SEQ_1:func.17">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">R1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c2">h</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><a href="seq_1.html#K16" title="SEQ_1:func.16">-</a> <font color="Maroon" title="c1">R1</font></span>)</span> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c2">h</font>
 <b>by </b><i><a class="ref" href="rfunct_2.html#T25" title="RFUNCT_2:th.25">RFUNCT_2:25</a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c2">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="E1:26_2"/><i><font color="Green" title="E31">A3</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">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="c1">R1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c2">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="c2">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="c1">R1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c2">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" title="FDIFF_1:def.3">FDIFF_1:def 3</a></i>;<br/><i><font color="Green" title="E32">A5</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c2">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"><a href="seq_1.html#K16" title="SEQ_1:func.16">-</a> <font color="Maroon" title="c1">R1</font></span>)</span> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c2">h</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">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"><a href="seq_1.html#K17" title="SEQ_1:func.17">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">R1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c2">h</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra8.html#E4:26"><i><font color="Green" title="E30">Lma</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">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"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="seq_1.html#K14" title="SEQ_1:func.14">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">R1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c2">h</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T25" title="SEQ_1:th.25">SEQ_1:25</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="seq_1.html#K14" title="SEQ_1:func.14">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">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="c1">R1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c2">h</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T27" title="SEQ_1:th.27">SEQ_1:27</a></i>
<br/>.= 
 <a href="seq_1.html#K17" title="SEQ_1:func.17">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">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="c1">R1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c2">h</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T25" title="SEQ_1:th.25">SEQ_1:25</a></i>
;<br/><b>hence </b><a NAME="E3:26_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">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"><a href="seq_1.html#K16" title="SEQ_1:func.16">-</a> <font color="Maroon" title="c1">R1</font></span>)</span> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c2">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="integra8.html#E1:26_2"><i><font color="Green" title="E31">A3</font></i></a>, <a class="ref" href="seq_2.html#T23" title="SEQ_2:th.23">SEQ_2:23</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="c2">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"><a href="seq_1.html#K16" title="SEQ_1:func.16">-</a> <font color="Maroon" title="c1">R1</font></span>)</span> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c2">h</font></span>)</span></span>)</span> = 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">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="c1">R1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c2">h</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra8.html#E1:26_2"><i><font color="Green" title="E31">A3</font></i></a>, <a class="txt" href="integra8.html#E2:26_2"><i><font color="Green" title="E32">A5</font></i></a>, <a class="ref" href="seq_2.html#T24" title="SEQ_2:th.24">SEQ_2:24</a></i>
<br/>.= 
0
<b>by </b><i><a class="txt" href="integra8.html#E1:26_2"><i><font color="Green" title="E31">A3</font></i></a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:26"/>
 <a href="seq_1.html#K16" title="SEQ_1:func.16">-</a> <font color="Maroon" title="c1">R1</font> is   <a href="fdiff_1.html#NM1" title="FDIFF_1:NM.1">REST</a>
 <b>by </b><i><a class="txt" href="integra8.html#E2:26"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="fdiff_1.html#D3" title="FDIFF_1:def.3">FDIFF_1:def 3</a></i>;<br/>


</div>
