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

<b>let </b><font color="Maroon" title="c3">R</font> be   <a href="fdiff_1.html#NM1" title="FDIFF_1:NM.1">REST</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">L</font> being  <a href="fdiff_1.html#NM2" title="FDIFF_1:NM.2">LINEAR</a> holds <br/> ( <font color="Maroon" title="c3">R</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Olive" title="b1">L</font> is   <a href="fdiff_1.html#NM1" title="FDIFF_1:NM.1">REST</a> &amp; <font color="Olive" title="b1">L</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c3">R</font> is   <a href="fdiff_1.html#NM1" title="FDIFF_1:NM.1">REST</a> )</span><br/><b>let </b><font color="Maroon" title="c4">L</font> be   <a href="fdiff_1.html#NM2" title="FDIFF_1:NM.2">LINEAR</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">R</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c4">L</font> is   <a href="fdiff_1.html#NM1" title="FDIFF_1:NM.1">REST</a> &amp; <font color="Maroon" title="c4">L</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c3">R</font> is   <a href="fdiff_1.html#NM1" title="FDIFF_1:NM.1">REST</a> )</span><br/>



<a NAME="E1:15"/><i><font color="Green" title="E11">A1</font></i>: 
<font color="Maroon" title="c3">R</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="E2:15"/><i><font color="Green" title="E12">A2</font></i>: 
<font color="Maroon" title="c4">L</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>
 
<b>by </b><i><a class="ref" href="fdiff_1.html#D4" target="_self" title="FDIFF_1:def.4">Def4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">x1</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E4:15"/><i><font color="Green" title="E13">A4</font></i>: 
for <font color="Olive" title="b1">p</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> holds  <font color="Maroon" title="c4">L</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b1">p</font>
 <b>by </b><i><a class="ref" href="fdiff_1.html#D4" target="_self" title="FDIFF_1:def.4">Def4</a></i>;<br/>
<div><i><font color="Green" title="E14">A5</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c6">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"><font color="Maroon" title="c3">R</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c4">L</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c6">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="c6">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"><font color="Maroon" title="c3">R</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c4">L</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c6">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/><i><font color="Green" title="E15">A6</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c6">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"><font color="Maroon" title="c3">R</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c4">L</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c6">h</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">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"><font color="Maroon" title="c3">R</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c6">h</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">L</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c6">h</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E1:15"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="#E2:15"><i><font color="Green" title="E12">A2</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="c6">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="c3">R</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c6">h</font></span>)</span></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">L</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c6">h</font></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/><a NAME="E2:15_1"/><i><font color="Green" title="E16">A7</font></i>: 
( <font color="Maroon" title="c6">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="c6">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="ref" href="fdiff_1.html#D1" target="_self" title="FDIFF_1:def.1">Def1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">L</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c6">h</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x1</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c6">h</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">n</font></span><br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c4">L</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c6">h</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">n</font> = 
<font color="Maroon" title="c4">L</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span>
<b>by </b><i><a class="txt" href="#E2:15"><i><font color="Green" title="E12">A2</font></i></a>, <a class="ref" href="funct_2.html#T192" title="FUNCT_2:th.192">FUNCT_2:192</a></i>
<br/>.= 
<font color="Maroon" title="c5">x1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span>
<b>by </b><i><a class="txt" href="#E4:15"><i><font color="Green" title="E13">A4</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">x1</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c6">h</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">n</font>
<b>by </b><i><a class="ref" href="seq_1.html#T13" title="SEQ_1:th.13">SEQ_1:13</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E4:15_1"/><b>then </b><i><font color="Green" title="E17">A8</font></i>: 
<font color="Maroon" title="c4">L</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c6">h</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">x1</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c6">h</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>;<br/><a NAME="E5:15_1"/><b>then </b><i><font color="Green" title="E18">A9</font></i>: 
<font color="Maroon" title="c4">L</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c6">h</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 <b>by </b><i><a class="txt" href="#E2:15_1"><i><font color="Green" title="E16">A7</font></i></a>, <a class="ref" href="seq_2.html#T21" title="SEQ_2:th.21">SEQ_2:21</a></i>;<br/><i><font color="Green" title="E19">A10</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="c4">L</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c6">h</font></span>)</span> = 
<font color="Maroon" title="c5">x1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="txt" href="#E2:15_1"><i><font color="Green" title="E16">A7</font></i></a>, <a class="txt" href="#E4:15_1"><i><font color="Green" title="E17">A8</font></i></a>, <a class="ref" href="seq_2.html#T22" title="SEQ_2:th.22">SEQ_2:22</a></i>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 

;<br/><a NAME="E7:15_1"/><i><font color="Green" title="E20">A11</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c6">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="c3">R</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c6">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="c6">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="c3">R</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c6">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>  )
 <b>by </b><i><a class="ref" href="fdiff_1.html#D3" target="_self" title="FDIFF_1:def.3">Def3</a></i>;<br/><b>hence </b><a NAME="E8:15_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">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"><font color="Maroon" title="c3">R</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c4">L</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c6">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="#E1:15_1"><i><font color="Green" title="E15">A6</font></i></a>, <a class="txt" href="#E5:15_1"><i><font color="Green" title="E18">A9</font></i></a>, <a class="ref" href="seq_2.html#T28" title="SEQ_2:th.28">SEQ_2:28</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="c6">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"><font color="Maroon" title="c3">R</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c4">L</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c6">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/><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="c6">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"><font color="Maroon" title="c3">R</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c4">L</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c6">h</font></span>)</span></span>)</span> = 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="txt" href="#E1:15_1"><i><font color="Green" title="E15">A6</font></i></a>, <a class="txt" href="#E5:15_1"><i><font color="Green" title="E18">A9</font></i></a>, <a class="txt" href="#E6:15_1"><i><font color="Green" title="E19">A10</font></i></a>, <a class="txt" href="#E7:15_1"><i><font color="Green" title="E20">A11</font></i></a>, <a class="ref" href="seq_2.html#T29" title="SEQ_2:th.29">SEQ_2:29</a></i>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 

; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:15"/>
<font color="Maroon" title="c3">R</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c4">L</font> is   <a href="fdiff_1.html#NM1" title="FDIFF_1:NM.1">REST</a>
 <b>by </b><i><a class="txt" href="#E1:15"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="#E2:15"><i><font color="Green" title="E12">A2</font></i></a>, <a class="ref" href="fdiff_1.html#D3" target="_self" title="FDIFF_1:def.3">Def3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">L</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c3">R</font> is   <a href="fdiff_1.html#NM1" title="FDIFF_1:NM.1">REST</a></span><br/>

<b>thus </b><a NAME="E7:15"/>
<font color="Maroon" title="c4">L</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c3">R</font> is   <a href="fdiff_1.html#NM1" title="FDIFF_1:NM.1">REST</a>
 <b>by </b><i><a class="txt" href="#E1:15"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="#E2:15"><i><font color="Green" title="E12">A2</font></i></a>, <a class="txt" href="#E5:15"><i><font color="Green" title="E14">A5</font></i></a>, <a class="ref" href="fdiff_1.html#D3" target="_self" title="FDIFF_1:def.3">Def3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
