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

<span class="kw">let </span><font color="Maroon" title="c1">R1</font> be   <a href="fdiff_1.html#NM1" title="FDIFF_1:NM.1">REST</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c1">R1</font> is   <a href="fdiff_1.html#NM1" title="FDIFF_1:NM.1">REST</a></span><br/>

<a NAME="E1:25"/><span class="lab"><font color="Green" title="E26">A1</font></span>: 
<font color="Maroon" title="c1">R1</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fdiff_1.html#D2" title="FDIFF_1:def.2">FDIFF_1:def 2</a></span>;<br/>
<a NAME="E2:25"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A2</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</a></span>;<br/>
<a NAME="E3:25"/><span class="lab"><font color="Green" title="E28">A3</font></span>: 
 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="valued_1.html#K32" title="VALUED_1:func.32">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">R1</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</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="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c1">R1</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Olive" title="b1">h</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">R1</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</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="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c1">R1</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c2">h</font></span><br/>

<a NAME="E1:25_1"/>
 <a href="relat_1.html#NK4" title="RELAT_1:NK.4">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#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">R1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:25"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="ref" href="valued_0.html#D3" title="VALUED_0:def.3">VALUED_0:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:25_1"/>
 <a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">R1</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</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="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c1">R1</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c2">h</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rfunct_2.html#T10" title="RFUNCT_2:th.10">RFUNCT_2:10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c1">R1</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</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="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c1">R1</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c2">h</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/><a NAME="E1:25_2"/><span class="lab"><font color="Green" title="E29">A4</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">R1</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c2">h</font></span>)</span> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fdiff_1.html#D2" title="FDIFF_1:def.2">FDIFF_1:def 2</a></span>;<br/><span class="lab"><font color="Green" title="E30">A5</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c1">R1</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</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="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">R1</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c2">h</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:25"><span class="lab"><font color="Green" title="E28">A3</font></span></a></span>
<br/>.= 
 <a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">R1</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c2">h</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_1.html#T19" title="SEQ_1:th.19">SEQ_1:19</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:25_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c1">R1</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c2">h</font></span>)</span> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:25_2"><span class="lab"><font color="Green" title="E29">A4</font></span></a>, <a class="ref" href="seq_2.html#T9" title="SEQ_2:th.9">SEQ_2:9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <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="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c1">R1</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c2">h</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/><a NAME="E4:25_2"/><span class="lab"><font color="Green" title="E31">A6</font></span>: 
 <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="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">R1</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c2">h</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fdiff_1.html#D2" title="FDIFF_1:def.2">FDIFF_1:def 2</a></span>;<br/><span class="kw">thus </span> <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="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c1">R1</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</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="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">R1</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c2">h</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:25_2"><span class="lab"><font color="Green" title="E29">A4</font></span></a>, <a class="txt" href="#E2:25_2"><span class="lab"><font color="Green" title="E30">A5</font></span></a>, <a class="ref" href="seq_2.html#T10" title="SEQ_2:th.10">SEQ_2:10</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:25_2"><span class="lab"><font color="Green" title="E31">A6</font></span></a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:25"/>
 <a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c1">R1</font> is   <a href="fdiff_1.html#NM1" title="FDIFF_1:NM.1">REST</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:25"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="ref" href="fdiff_1.html#D2" title="FDIFF_1:def.2">FDIFF_1:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
