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

<b>let </b><font color="Maroon" title="c1">f</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> , <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/><b>let </b><font color="Maroon" title="c2">x0</font>, <font color="Maroon" title="c3">g2</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>





<b>assume </b><a NAME="E1:9"/><i><font color="Green" title="E13">A1</font></i>: 
( <font color="Maroon" title="c1">f</font> <a href="fdiff_3.html#R2" title="FDIFF_3:pred.2">is_Rcontinuous_in</a> <font color="Maroon" title="c2">x0</font> &amp; <font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">x0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">g2</font> )
 ;<br/>

<b>given </b><font color="Maroon" title="c4">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><a NAME="E2:9"/><i><font color="Green" title="E14">A2</font></i>: 
( <font color="Maroon" title="c4">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp; <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c2">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">f</font> )
 ;<br/>

<b>assume </b><a NAME="E3:9"/><i><font color="Green" title="E15">A3</font></i>: 
for <font color="Olive" title="b1">r1</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">r1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp; <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c2">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Olive" title="b1">r1</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">f</font> holds <br/> ex <font color="Olive" title="b2">g</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b2">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c2">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Olive" title="b1">r1</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g2</font> )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <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 href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>( $2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c2">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p4">(<span class="default">$1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; $2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">f</font> &amp; <font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g2</font> );<br/>
<a NAME="E4:9"/><i><font color="Green" title="E16">A4</font></i>: 
for <font color="Olive" title="b1">n</font> being   <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>   ex <font color="Olive" title="b2">g</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">g</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">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> ;<br/>

<a NAME="E1:9_1"/><i><font color="Green" title="E17">A5</font></i>: 
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">n</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a></i>;<br/>
<a NAME="E2:9_1"/><b>then </b><i><font color="Green" title="E18">A6</font></i>: 
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E3:9_1"/><b>then </b><i><font color="Green" title="E19">A7</font></i>: 
<font color="Maroon" title="c4">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="fdiff_3.html#E2:9"><i><font color="Green" title="E14">A2</font></i></a>, <a class="ref" href="xreal_1.html#T141" title="XREAL_1:th.141">XREAL_1:141</a></i>;<br/>
<a NAME="E4:9_1"/>
<font color="Maroon" title="c2">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font>
 
<b>by </b><i><a class="txt" href="fdiff_3.html#E2:9"><i><font color="Green" title="E14">A2</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<a NAME="E5:9_1"/><b>then </b><i><font color="Green" title="E20">A8</font></i>: 
<font color="Maroon" title="c2">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c2">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T15" title="RCOMP_1:th.15">RCOMP_1:15</a></i>;<br/>
<a NAME="E6:9_1"/>
0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="fdiff_3.html#E1:9_1"><i><font color="Green" title="E17">A5</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<a NAME="E7:9_1"/><b>then </b>
<font color="Maroon" title="c4">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 1
 
<b>by </b><i><a class="txt" href="fdiff_3.html#E2:9"><i><font color="Green" title="E14">A2</font></i></a>, <a class="ref" href="real_2.html#T201" title="REAL_2:th.201">REAL_2:201</a></i>;<br/>
<a NAME="E8:9_1"/><b>then </b><i><font color="Green" title="E21">A9</font></i>: 
<font color="Maroon" title="c2">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<a NAME="E9:9_1"/>
( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c2">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x0</font> )
 
<b>by </b><i><a class="txt" href="fdiff_3.html#E2:9"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="fdiff_3.html#E2:9_1"><i><font color="Green" title="E18">A6</font></i></a>, <a class="ref" href="xreal_1.html#T141" title="XREAL_1:th.141">XREAL_1:141</a></i>;<br/>
<a NAME="E10:9_1"/><b>then </b>
<font color="Maroon" title="c2">x0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<a NAME="E11:9_1"/><b>then </b>
<font color="Maroon" title="c2">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">g1</font> where <font color="Olive" title="b1">g1</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( <font color="Maroon" title="c2">x0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">g1</font> &amp; <font color="Olive" title="b1">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="fdiff_3.html#E8:9_1"><i><font color="Green" title="E21">A9</font></i></a></i>;<br/>
<a NAME="E12:9_1"/><b>then </b>
<font color="Maroon" title="c2">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c2">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></i>;<br/>
<a NAME="E13:9_1"/><b>then </b>
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c2">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c2">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="txt" href="fdiff_3.html#E5:9_1"><i><font color="Green" title="E20">A8</font></i></a>, <a class="ref" href="rcomp_1.html#T16" title="RCOMP_1:th.16">RCOMP_1:16</a></i>;<br/>
<a NAME="E14:9_1"/><b>then </b><i><font color="Green" title="E22">A10</font></i>: 
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c2">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">f</font>
 
<b>by </b><i><a class="txt" href="fdiff_3.html#E2:9"><i><font color="Green" title="E14">A2</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">g</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E16:9_1"/><i><font color="Green" title="E23">A11</font></i>: 
( <font color="Maroon" title="c6">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c2">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g2</font> )
 <b>by </b><i><a class="txt" href="fdiff_3.html#E3:9"><i><font color="Green" title="E15">A3</font></i></a>, <a class="txt" href="fdiff_3.html#E3:9_1"><i><font color="Green" title="E19">A7</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c6">g</font>
;<br/>

<b>thus </b><a NAME="E17:9_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c6">g</font>]
 <b>by </b><i><a class="txt" href="fdiff_3.html#E14:9_1"><i><font color="Green" title="E22">A10</font></i></a>, <a class="txt" href="fdiff_3.html#E16:9_1"><i><font color="Green" title="E23">A11</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c5">a</font> being   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a><b> such that </b><br/><a NAME="E6:9"/><i><font color="Green" title="E17">A12</font></i>: 
for <font color="Olive" title="b1">n</font> being   <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>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Maroon" title="c5">a</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font>]
 <b>from </b><i><a class="ref" href="rcomp_1.html#S1" title="RCOMP_1:sch.1">RCOMP_1:sch 1</a>(<a class="txt" href="fdiff_3.html#E4:9"><i><font color="Green" title="E16">A4</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon" title="c6">b</font> = <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="funcop_1.html#K2" title="FUNCOP_1:func.2">--&gt;</a> <font color="Maroon" title="c2">x0</font> as   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a> <b>by </b><i><a class="ref" href="funcop_1.html#T57" title="FUNCOP_1:th.57">FUNCOP_1:57</a></i>;<br/>
<a NAME="E8:9"/><i><font color="Green" title="E18">A13</font></i>: 
for <font color="Olive" title="b1">n</font> being   <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>  holds  <font color="Maroon" title="c6">b</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x0</font>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/>
<a NAME="E9:9"/><i><font color="Green" title="E19">A14</font></i>: 
<font color="Maroon" title="c6">b</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 
<b>by </b><i><a class="ref" href="seq_4.html#T39" title="SEQ_4:th.39">SEQ_4:39</a></i>;<br/>
<i><font color="Green" title="E20">A15</font></i>:  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c6">b</font> = 
<font color="Maroon" title="c6">b</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 0
<b>by </b><i><a class="ref" href="seq_4.html#T41" title="SEQ_4:th.41">SEQ_4:41</a></i>
<br/>.= 
<font color="Maroon" title="c2">x0</font>
<b>by </b><i><a class="txt" href="fdiff_3.html#E8:9"><i><font color="Green" title="E18">A13</font></i></a></i>
;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <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> ) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  = <font color="Maroon" title="c2">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default">$1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>;<br/>
<b>consider </b><font color="Maroon" title="c7">d</font> being   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a><b> such that </b><br/><a NAME="E12:9"/><i><font color="Green" title="E21">A16</font></i>: 
for <font color="Olive" title="b1">n</font> being   <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>  holds  <font color="Maroon" title="c7">d</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">n</font>)
 <b>from </b><i><a class="ref" href="seq_1.html#S1" title="SEQ_1:sch.1">SEQ_1:sch 1</a>();<br/></i>
<a NAME="E13:9"/><i><font color="Green" title="E22">A17</font></i>: 
( <font color="Maroon" title="c7">d</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="c7">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x0</font> )
 
<b>by </b><i><a class="txt" href="fdiff_3.html#E12:9"><i><font color="Green" title="E21">A16</font></i></a>, <a class="ref" href="fcont_3.html#T14" title="FCONT_3:th.14">FCONT_3:14</a></i>;<br/>
<div><i><font color="Green" title="E23">A18</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="9_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">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> ;<br/><a NAME="E1:9_3"/>
<font color="Maroon" title="c5">a</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c2">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <b>by </b><i><a class="txt" href="fdiff_3.html#E6:9"><i><font color="Green" title="E17">A12</font></i></a></i>;<br/><a NAME="E2:9_3"/><b>then </b>
<font color="Maroon" title="c5">a</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">g1</font> where <font color="Olive" title="b1">g1</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( <font color="Maroon" title="c2">x0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">g1</font> &amp; <font color="Olive" title="b1">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></i>;<br/><a NAME="E3:9_3"/><b>then </b>
 ex <font color="Olive" title="b1">g1</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">n</font> &amp; <font color="Maroon" title="c2">x0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">g1</font> &amp; <font color="Olive" title="b1">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> )
 ;<br/><b>hence </b><a NAME="E4:9_3"/>
( <font color="Maroon" title="c6">b</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">n</font> &amp; <font color="Maroon" title="c5">a</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">d</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">n</font> )
 <b>by </b><i><a class="txt" href="fdiff_3.html#E8:9"><i><font color="Green" title="E18">A13</font></i></a>, <a class="txt" href="fdiff_3.html#E12:9"><i><font color="Green" title="E21">A16</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E15:9"/><b>then </b><i><font color="Green" title="E24">A19</font></i>: 
<font color="Maroon" title="c5">a</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 
<b>by </b><i><a class="txt" href="fdiff_3.html#E9:9"><i><font color="Green" title="E19">A14</font></i></a>, <a class="txt" href="fdiff_3.html#E10:9"><i><font color="Green" title="E20">A15</font></i></a>, <a class="txt" href="fdiff_3.html#E13:9"><i><font color="Green" title="E22">A17</font></i></a>, <a class="ref" href="seq_2.html#T33" title="SEQ_2:th.33">SEQ_2:33</a></i>;<br/>
<a NAME="E16:9"/><i><font color="Green" title="E25">A20</font></i>: 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x0</font>
 
<b>by </b><i><a class="txt" href="fdiff_3.html#E9:9"><i><font color="Green" title="E19">A14</font></i></a>, <a class="txt" href="fdiff_3.html#E10:9"><i><font color="Green" title="E20">A15</font></i></a>, <a class="txt" href="fdiff_3.html#E13:9"><i><font color="Green" title="E22">A17</font></i></a>, <a class="txt" href="fdiff_3.html#E14:9"><i><font color="Green" title="E23">A18</font></i></a>, <a class="ref" href="seq_2.html#T34" title="SEQ_2:th.34">SEQ_2:34</a></i>;<br/>
<a NAME="E17:9"/><i><font color="Green" title="E26">A21</font></i>: 
(  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <font color="Maroon" title="c2">x0</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">f</font></span>)</span> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">f</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_4"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:9_4"/><i><font color="Green" title="E27">A22</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <font color="Maroon" title="c2">x0</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">f</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:9_4_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">a</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c9">n</font> being    <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> <b> such that </b><br/><a NAME="E3:9_4_1"/><i><font color="Green" title="E28">A23</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">n</font>
 <b>by </b><i><a class="ref" href="rfunct_2.html#T9" title="RFUNCT_2:th.9">RFUNCT_2:9</a></i>;<br/>
<a NAME="E4:9_4_1"/><i><font color="Green" title="E29">A24</font></i>: 
( <font color="Maroon" title="c5">a</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c2">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c5">a</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">f</font> )
 
<b>by </b><i><a class="txt" href="fdiff_3.html#E6:9"><i><font color="Green" title="E17">A12</font></i></a></i>;<br/>
<a NAME="E5:9_4_1"/><b>then </b>
<font color="Maroon" title="c5">a</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">g1</font> where <font color="Olive" title="b1">g1</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( <font color="Maroon" title="c2">x0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">g1</font> &amp; <font color="Olive" title="b1">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></i>;<br/>
<a NAME="E6:9_4_1"/><b>then </b><i><font color="Green" title="E30">A25</font></i>: 
 ex <font color="Olive" title="b1">g1</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">n</font> &amp; <font color="Maroon" title="c2">x0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">g1</font> &amp; <font color="Olive" title="b1">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> )
 
;<br/>
<a NAME="E7:9_4_1"/>
<font color="Maroon" title="c2">x0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">a</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">n</font>
 
<b>by </b><i><a class="txt" href="fdiff_3.html#E1:9"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="fdiff_3.html#E6:9"><i><font color="Green" title="E17">A12</font></i></a></i>;<br/>
<a NAME="E8:9_4_1"/><b>then </b>
<font color="Maroon" title="c2">x0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">a</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">n</font>
 
<b>by </b><i><a class="txt" href="fdiff_3.html#E6:9_4_1"><i><font color="Green" title="E30">A25</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/>
<a NAME="E9:9_4_1"/><b>then </b>
<font color="Maroon" title="c5">a</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">g1</font> where <font color="Olive" title="b1">g1</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : <font color="Maroon" title="c2">x0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">g1</font> </span>}</span> 
 
;<br/>
<a NAME="E10:9_4_1"/><b>then </b>
<font color="Maroon" title="c5">a</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <font color="Maroon" title="c2">x0</font>
 
<b>by </b><i><a class="ref" href="limfunc1.html#D3" title="LIMFUNC1:def.3">LIMFUNC1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E11:9_4_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <font color="Maroon" title="c2">x0</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="fdiff_3.html#E3:9_4_1"><i><font color="Green" title="E28">A23</font></i></a>, <a class="txt" href="fdiff_3.html#E4:9_4_1"><i><font color="Green" title="E29">A24</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>

<a NAME="E2:9_4"/>
<span class="p1">(<span class="default"><a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <font color="Maroon" title="c2">x0</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">f</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<b>hence </b><a NAME="E3:9_4"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">f</font>
 <b>by </b><i><a class="txt" href="fdiff_3.html#E1:9_4"><i><font color="Green" title="E27">A22</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E18:9"/><i><font color="Green" title="E27">A26</font></i>: 
for <font color="Olive" title="b1">n</font> being   <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>  holds  <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">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> ;<br/>

<b>thus </b><font color="Maroon" title="c3">g2</font> = 
<font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">n</font></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_3.html#E6:9"><i><font color="Green" title="E17">A12</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">n</font>
<b>by </b><i><a class="txt" href="fdiff_3.html#E17:9"><i><font color="Green" title="E26">A21</font></i></a>, <a class="ref" href="rfunct_2.html#T21" title="RFUNCT_2:th.21">RFUNCT_2:21</a></i>
;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">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> ;<br/><a NAME="E1:9_6"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g2</font> )
 <b>by </b><i><a class="txt" href="fdiff_3.html#E18:9"><i><font color="Green" title="E27">A26</font></i></a></i>;<br/><b>hence </b><a NAME="E2:9_6"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 ;<br/></div><b>end;</b></div>
<a NAME="E20:9"/><b>then </b>
<font color="Maroon" title="c1">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c5">a</font> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>
 
<b>by </b><i><a class="ref" href="seqm_3.html#T16" title="SEQM_3:th.16">SEQM_3:16</a></i>;<br/>
<b>then </b> <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c5">a</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 0
<b>by </b><i><a class="ref" href="seq_4.html#T41" title="SEQ_4:th.41">SEQ_4:41</a></i>
<br/>.= 
<font color="Maroon" title="c3">g2</font>
<b>by </b><i><a class="txt" href="fdiff_3.html#E18:9"><i><font color="Green" title="E27">A26</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E22:9"/>
contradiction
 <b>by </b><i><a class="txt" href="fdiff_3.html#E1:9"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="fdiff_3.html#E15:9"><i><font color="Green" title="E24">A19</font></i></a>, <a class="txt" href="fdiff_3.html#E16:9"><i><font color="Green" title="E25">A20</font></i></a>, <a class="txt" href="fdiff_3.html#E17:9"><i><font color="Green" title="E26">A21</font></i></a>, <a class="ref" href="fdiff_3.html#D2" target="_self" title="FDIFF_3:def.2">Def2</a></i>;<br/>


</div>
