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

<b>let </b><font color="Maroon" title="c1">f</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>;<br/>

<b>assume </b><a NAME="E1:115"/><i><font color="Green" title="E72">A1</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="c1">f</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>  <a href="asympt_1.html#K10" title="ASYMPT_1:func.10">Step1</a> <font color="Olive" title="b1">n</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c2">g</font> =  <a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> 1;<br/>
<a NAME="E2:115"/>
<font color="Maroon" title="c1">f</font> is <a href="asympt_0.html#V4" title="ASYMPT_0:attr.4">eventually-positive</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_1"><b>proof </b></a><div class="add">

<b>take </b>
1
; <i><font color="Red">:: according to </font></i><a class="ref" href="asympt_0.html#D6" title="ASYMPT_0:def.6">ASYMPT_0:def 6</a><br/>

<b>let </b><font color="Maroon" title="c3">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>assume </b><a NAME="E1:115_1"/><i><font color="Green" title="E73">A2</font></i>: 
<font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 ;<br/>

<a NAME="E2:115_1"/><i><font color="Green" title="E74">A3</font></i>: 
<font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="asympt_1.html#K10" title="ASYMPT_1:func.10">Step1</a> <font color="Maroon" title="c3">n</font>
 
<b>by </b><i><a class="txt" href="asympt_1.html#E1:115"><i><font color="Green" title="E72">A1</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c4">m</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="E4:115_1"/><i><font color="Green" title="E75">A4</font></i>: 
( <font color="Maroon" title="c4">m</font> <a href="newton.html#K12" title="NEWTON:func.12">!</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> &amp; <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="newton.html#K12" title="NEWTON:func.12">!</a>  &amp;  <a href="asympt_1.html#K10" title="ASYMPT_1:func.10">Step1</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">m</font> <a href="newton.html#K12" title="NEWTON:func.12">!</a>  )
 <b>by </b><i><a class="txt" href="asympt_1.html#E1:115_1"><i><font color="Green" title="E73">A2</font></i></a>, <a class="ref" href="asympt_1.html#D9" target="_self" title="ASYMPT_1:def.9">Def9</a></i>;<br/>
<b>thus </b><a NAME="E5:115_1"/>
<font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="txt" href="asympt_1.html#E2:115_1"><i><font color="Green" title="E74">A3</font></i></a>, <a class="txt" href="asympt_1.html#E4:115_1"><i><font color="Green" title="E75">A4</font></i></a>, <a class="ref" href="newton.html#T23" title="NEWTON:th.23">NEWTON:23</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c3">f</font> = <font color="Maroon" title="c1">f</font> as  <a href="asympt_0.html#V4" title="ASYMPT_0:attr.4">eventually-positive</a> <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a> ;<br/>
<b>take </b>
<font color="Maroon" title="c3">f</font>
;<br/>

<a NAME="E4:115"/><i><font color="Green" title="E73">A5</font></i>: 
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">k</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><a NAME="E1:115_2"/>
<font color="Maroon" title="c3">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="115_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:115_2_1_1"/>
( <font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Maroon" title="c4">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="115_2_1_1_1"><b>suppose </b></a><a NAME="E1:115_2_1_1_1"/><i><font color="Green" title="E74">A6</font></i>: 
<font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><i><font color="Green" title="E75">A7</font></i>: <font color="Maroon" title="c3">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 0 = 
 <a href="asympt_1.html#K10" title="ASYMPT_1:func.10">Step1</a> 0
<b>by </b><i><a class="txt" href="asympt_1.html#E1:115"><i><font color="Green" title="E72">A1</font></i></a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="asympt_1.html#D9" target="_self" title="ASYMPT_1:def.9">Def9</a></i>
;<br/><a NAME="E3:115_2_1_1_1"/>
<font color="Maroon" title="c3">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default">0 <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>  <a href="asympt_1.html#K10" title="ASYMPT_1:func.10">Step1</a> 1
 <b>by </b><i><a class="txt" href="asympt_1.html#E1:115"><i><font color="Green" title="E72">A1</font></i></a></i>;<br/><b>hence </b><a NAME="E4:115_2_1_1_1"/>
<font color="Maroon" title="c3">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E1:115_2_1_1_1"><i><font color="Green" title="E74">A6</font></i></a>, <a class="txt" href="asympt_1.html#E2:115_2_1_1_1"><i><font color="Green" title="E75">A7</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="115_2_1_1_2"><b>suppose </b></a><a NAME="E1:115_2_1_1_2"/><i><font color="Green" title="E74">A8</font></i>: 
<font color="Maroon" title="c4">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:115_2_1_1_2"/><i><font color="Green" title="E75">A9</font></i>: 
<font color="Maroon" title="c4">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">k</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#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><b>consider </b><font color="Maroon" title="c5">n1</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="E4:115_2_1_1_2"/><i><font color="Green" title="E76">A10</font></i>: 
( <font color="Maroon" title="c5">n1</font> <a href="newton.html#K12" title="NEWTON:func.12">!</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">k</font> &amp; <font color="Maroon" title="c4">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="newton.html#K12" title="NEWTON:func.12">!</a>  )
 <b>and </b><br/><a NAME="E5:115_2_1_1_2"/><i><font color="Green" title="E77">A11</font></i>: 
 <a href="asympt_1.html#K10" title="ASYMPT_1:func.10">Step1</a> <font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">n1</font> <a href="newton.html#K12" title="NEWTON:func.12">!</a> 
 <b>by </b><i><a class="txt" href="asympt_1.html#E1:115_2_1_1_2"><i><font color="Green" title="E74">A8</font></i></a>, <a class="ref" href="asympt_1.html#D9" target="_self" title="ASYMPT_1:def.9">Def9</a></i>;<br/><a NAME="E6:115_2_1_1_2"/><i><font color="Green" title="E78">A12</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">n1</font> <a href="newton.html#K12" title="NEWTON:func.12">!</a> 
 <b>by </b><i><a class="txt" href="asympt_1.html#E1:115"><i><font color="Green" title="E72">A1</font></i></a>, <a class="txt" href="asympt_1.html#E5:115_2_1_1_2"><i><font color="Green" title="E77">A11</font></i></a></i>;<br/><a NAME="E7:115_2_1_1_2"/><i><font color="Green" title="E79">A13</font></i>: 
<font color="Maroon" title="c4">k</font> <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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="newton.html#K12" title="NEWTON:func.12">!</a> 
 <b>by </b><i><a class="txt" href="asympt_1.html#E4:115_2_1_1_2"><i><font color="Green" title="E76">A10</font></i></a>, <a class="ref" href="int_1.html#T20" title="INT_1:th.20">INT_1:20</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="115_2_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:115_2_1_1_2_1"/>
( <font color="Maroon" title="c4">k</font> <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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="newton.html#K12" title="NEWTON:func.12">!</a>  or <font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="newton.html#K12" title="NEWTON:func.12">!</a>  )
 </span><b>by </b><i><a class="txt" href="asympt_1.html#E7:115_2_1_1_2"><i><font color="Green" title="E79">A13</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="115_2_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:115_2_1_1_2_1_1"/>
<font color="Maroon" title="c4">k</font> <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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="newton.html#K12" title="NEWTON:func.12">!</a> 
 ;<br/><div class="add"><a NAME="E2:115_2_1_1_2_1_1"/><b>then </b>
( <font color="Maroon" title="c5">n1</font> <a href="newton.html#K12" title="NEWTON:func.12">!</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c4">k</font> <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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="newton.html#K12" title="NEWTON:func.12">!</a>  )
 <b>by </b><i><a class="txt" href="asympt_1.html#E2:115_2_1_1_2"><i><font color="Green" title="E75">A9</font></i></a>, <a class="txt" href="asympt_1.html#E4:115_2_1_1_2"><i><font color="Green" title="E76">A10</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E3:115_2_1_1_2_1_1"/><b>then </b>
 <a href="asympt_1.html#K10" title="ASYMPT_1:func.10">Step1</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</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="c5">n1</font> <a href="newton.html#K12" title="NEWTON:func.12">!</a> 
 <b>by </b><i><a class="ref" href="asympt_1.html#D9" target="_self" title="ASYMPT_1:def.9">Def9</a></i>;<br/><b>hence </b><a NAME="E4:115_2_1_1_2_1_1"/>
<font color="Maroon" title="c3">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E1:115"><i><font color="Green" title="E72">A1</font></i></a>, <a class="txt" href="asympt_1.html#E6:115_2_1_1_2"><i><font color="Green" title="E78">A12</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="115_2_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:115_2_1_1_2_1_2"/><i><font color="Green" title="E80">A14</font></i>: 
<font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="newton.html#K12" title="NEWTON:func.12">!</a> 
 ;<br/><div class="add"><a NAME="E2:115_2_1_1_2_1_2"/><i><font color="Green" title="E81">A15</font></i>: 
<font color="Maroon" title="c5">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><a NAME="E3:115_2_1_1_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="newton.html#K12" title="NEWTON:func.12">!</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="ref" href="newton.html#T23" title="NEWTON:th.23">NEWTON:23</a></i>;<br/><a NAME="E4:115_2_1_1_2_1_2"/><b>then </b>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="newton.html#K12" title="NEWTON:func.12">!</a> </span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="newton.html#K12" title="NEWTON:func.12">!</a> </span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E2:115_2_1_1_2_1_2"><i><font color="Green" title="E81">A15</font></i></a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></i>;<br/><a NAME="E5:115_2_1_1_2_1_2"/><b>then </b><i><font color="Green" title="E82">A16</font></i>: 
<font color="Maroon" title="c4">k</font> <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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="newton.html#K12" title="NEWTON:func.12">!</a> 
 <b>by </b><i><a class="txt" href="asympt_1.html#E1:115_2_1_1_2_1_2"><i><font color="Green" title="E80">A14</font></i></a>, <a class="ref" href="newton.html#T21" title="NEWTON:th.21">NEWTON:21</a></i>;<br/><i><font color="Green" title="E83">A17</font></i>: <font color="Maroon" title="c3">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
 <a href="asympt_1.html#K10" title="ASYMPT_1:func.10">Step1</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="asympt_1.html#E1:115"><i><font color="Green" title="E72">A1</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="newton.html#K12" title="NEWTON:func.12">!</a> 
<b>by </b><i><a class="txt" href="asympt_1.html#E1:115_2_1_1_2_1_2"><i><font color="Green" title="E80">A14</font></i></a>, <a class="txt" href="asympt_1.html#E5:115_2_1_1_2_1_2"><i><font color="Green" title="E82">A16</font></i></a>, <a class="ref" href="asympt_1.html#D9" target="_self" title="ASYMPT_1:def.9">Def9</a></i>
;<br/><a NAME="E7:115_2_1_1_2_1_2"/>
<font color="Maroon" title="c5">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">n1</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#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><b>hence </b><a NAME="E8:115_2_1_1_2_1_2"/>
<font color="Maroon" title="c3">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E6:115_2_1_1_2"><i><font color="Green" title="E78">A12</font></i></a>, <a class="txt" href="asympt_1.html#E6:115_2_1_1_2_1_2"><i><font color="Green" title="E83">A17</font></i></a>, <a class="txt" href="asympt_1.html#E106"><i><font color="Green" title="E68">Lm59</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E6:115"/><b>then </b><i><font color="Green" title="E74">A18</font></i>: 
for <font color="Olive" title="b1">k</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>   st <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 holds <br/><font color="Maroon" title="c3">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
;<br/>
<div><i><font color="Green" title="E75">A19</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="115_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">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><a NAME="E1:115_3"/>
<font color="Maroon" title="c3">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> 1</span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">n</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_3_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="115_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:115_3_1_1"/>
( <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="115_3_1_1_1"><b>suppose </b></a><a NAME="E1:115_3_1_1_1"/><i><font color="Green" title="E76">A20</font></i>: 
<font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><font color="Maroon" title="c3">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 0 = 
 <a href="asympt_1.html#K10" title="ASYMPT_1:func.10">Step1</a> 0
<b>by </b><i><a class="txt" href="asympt_1.html#E1:115"><i><font color="Green" title="E72">A1</font></i></a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="asympt_1.html#D9" target="_self" title="ASYMPT_1:def.9">Def9</a></i>
;<br/><b>hence </b><a NAME="E3:115_3_1_1_1"/>
<font color="Maroon" title="c3">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> 1</span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">n</font>
 <b>by </b><i><a class="txt" href="asympt_1.html#E1:115_3_1_1_1"><i><font color="Green" title="E76">A20</font></i></a>, <a class="ref" href="asympt_1.html#D3" target="_self" title="ASYMPT_1:def.3">Def3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="115_3_1_1_2"><b>suppose </b></a><a NAME="E1:115_3_1_1_2"/><i><font color="Green" title="E76">A21</font></i>: 
<font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c5">n1</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="E3:115_3_1_1_2"/><i><font color="Green" title="E77">A22</font></i>: 
( <font color="Maroon" title="c5">n1</font> <a href="newton.html#K12" title="NEWTON:func.12">!</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> &amp; <font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="newton.html#K12" title="NEWTON:func.12">!</a>  )
 <b>and </b><br/><a NAME="E4:115_3_1_1_2"/><i><font color="Green" title="E78">A23</font></i>: 
 <a href="asympt_1.html#K10" title="ASYMPT_1:func.10">Step1</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">n1</font> <a href="newton.html#K12" title="NEWTON:func.12">!</a> 
 <b>by </b><i><a class="ref" href="asympt_1.html#D9" target="_self" title="ASYMPT_1:def.9">Def9</a></i>;<br/><span class="p1">(<span class="default"><a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> 1</span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">n</font> = 
<font color="Maroon" title="c4">n</font> <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> 1
<b>by </b><i><a class="txt" href="asympt_1.html#E1:115_3_1_1_2"><i><font color="Green" title="E76">A21</font></i></a>, <a class="ref" href="asympt_1.html#D3" target="_self" title="ASYMPT_1:def.3">Def3</a></i>
<br/>.= 
<font color="Maroon" title="c4">n</font>
<b>by </b><i><a class="ref" href="power.html#T30" title="POWER:th.30">POWER:30</a></i>
;<br/><b>hence </b><a NAME="E6:115_3_1_1_2"/>
<font color="Maroon" title="c3">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> 1</span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">n</font>
 <b>by </b><i><a class="txt" href="asympt_1.html#E1:115"><i><font color="Green" title="E72">A1</font></i></a>, <a class="txt" href="asympt_1.html#E3:115_3_1_1_2"><i><font color="Green" title="E77">A22</font></i></a>, <a class="txt" href="asympt_1.html#E4:115_3_1_1_2"><i><font color="Green" title="E78">A23</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:115_4"/><i><font color="Green" title="E76">A24</font></i>: 
<font color="Maroon" title="c3">f</font> is <a href="asympt_0.html#V7" title="ASYMPT_0:attr.7">smooth</a>
 ;<br/><b>set </b><font color="Maroon" title="c4">h</font> = <font color="Maroon" title="c3">f</font> <a href="asympt_0.html#K15" title="ASYMPT_0:func.15">taken_every</a> 2;<br/><a NAME="E2:115_4"/>
<font color="Maroon" title="c3">f</font> <a href="asympt_0.html#R2" title="ASYMPT_0:pred.2">is_smooth_wrt</a> 2
 <b>by </b><i><a class="txt" href="asympt_1.html#E1:115_4"><i><font color="Green" title="E76">A24</font></i></a>, <a class="ref" href="asympt_0.html#D20" title="ASYMPT_0:def.20">ASYMPT_0:def 20</a></i>;<br/><a NAME="E3:115_4"/><b>then </b>
<font color="Maroon" title="c3">f</font> <a href="asympt_0.html#K15" title="ASYMPT_0:func.15">taken_every</a> 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_0.html#K9" title="ASYMPT_0:func.9">Big_Oh</a> <font color="Maroon" title="c3">f</font>
 <b>by </b><i><a class="ref" href="asympt_0.html#D19" title="ASYMPT_0:def.19">ASYMPT_0:def 19</a></i>;<br/><b>then consider </b><font color="Maroon" title="c5">t</font> being    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E5:115_4"/><i><font color="Green" title="E77">A25</font></i>: 
<font color="Maroon" title="c5">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="asympt_0.html#K15" title="ASYMPT_0:func.15">taken_every</a> 2
 <b>and </b><br/><a NAME="E6:115_4"/><i><font color="Green" title="E78">A26</font></i>: 
 ex <font color="Olive" title="b1">c</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ex <font color="Olive" title="b2">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>  st <br/>( <font color="Olive" title="b1">c</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp; ( for <font color="Olive" title="b3">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>   st <font color="Olive" title="b3">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b2">N</font> holds <br/>( <font color="Maroon" title="c5">t</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">c</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b3">n</font></span>)</span> &amp; <font color="Maroon" title="c5">t</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b3">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 ) ) )
 ;<br/><b>consider </b><font color="Maroon" title="c6">c</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>, <font color="Maroon" title="c7">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="E8:115_4"/>
<font color="Maroon" title="c6">c</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>and </b><br/><a NAME="E9:115_4"/><i><font color="Green" title="E79">A27</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>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">N</font> holds <br/>( <font color="Maroon" title="c5">t</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">c</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font></span>)</span> &amp; <font color="Maroon" title="c5">t</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 )
 <b>by </b><i><a class="txt" href="asympt_1.html#E6:115_4"><i><font color="Green" title="E78">A26</font></i></a></i>;<br/><b>set </b><font color="Maroon" title="c8">n2</font> =  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c7">N</font>,3</span>)</span>,<span class="p1">(<span class="default"><span class="p2"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><font color="Maroon" title="c6">c</font></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>;<br/><a NAME="E10:115_4"/><i><font color="Green" title="E80">A28</font></i>: 
(  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c7">N</font>,3</span>)</span>,<span class="p1">(<span class="default"><span class="p2"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><font color="Maroon" title="c6">c</font></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c7">N</font>,3 &amp;  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c7">N</font>,3</span>)</span>,<span class="p1">(<span class="default"><span class="p2"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><font color="Maroon" title="c6">c</font></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><font color="Maroon" title="c6">c</font></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></i>;<br/><a NAME="E11:115_4"/><i><font color="Green" title="E81">A29</font></i>: 
(  <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c7">N</font>,3 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">N</font> &amp;  <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c7">N</font>,3 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 3 )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></i>;<br/><a NAME="E12:115_4"/><b>then </b><i><font color="Green" title="E82">A30</font></i>: 
(  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c7">N</font>,3</span>)</span>,<span class="p1">(<span class="default"><span class="p2"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><font color="Maroon" title="c6">c</font></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">N</font> &amp;  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c7">N</font>,3</span>)</span>,<span class="p1">(<span class="default"><span class="p2"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><font color="Maroon" title="c6">c</font></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 3 )
 <b>by </b><i><a class="txt" href="asympt_1.html#E10:115_4"><i><font color="Green" title="E80">A28</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E13:115_4"/>
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c7">N</font>,3</span>)</span>,<span class="p1">(<span class="default"><span class="p2"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><font color="Maroon" title="c6">c</font></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> is   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T16" title="XXREAL_0:th.16">XXREAL_0:16</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c9">n2</font> =  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c7">N</font>,3</span>)</span>,<span class="p1">(<span class="default"><span class="p2"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><font color="Maroon" title="c6">c</font></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> as    <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>by </b><i><a class="txt" href="asympt_1.html#E10:115_4"><i><font color="Green" title="E80">A28</font></i></a>, <a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/><b>set </b><font color="Maroon" title="c10">n1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="newton.html#K12" title="NEWTON:func.12">!</a> </span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1;<br/><a NAME="E15:115_4"/><i><font color="Green" title="E83">A31</font></i>: 
<font color="Maroon" title="c9">n2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2
 <b>by </b><i><a class="txt" href="asympt_1.html#E12:115_4"><i><font color="Green" title="E82">A30</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E16:115_4"/><b>then </b><i><font color="Green" title="E84">A32</font></i>: 
<font color="Maroon" title="c9">n2</font> <a href="newton.html#K12" title="NEWTON:func.12">!</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 <b>by </b><i><a class="txt" href="asympt_1.html#E106"><i><font color="Green" title="E68">Lm59</font></i></a>, <a class="ref" href="newton.html#T20" title="NEWTON:th.20">NEWTON:20</a></i>;<br/><a NAME="E17:115_4"/><b>then </b><i><font color="Green" title="E85">A33</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="newton.html#K12" title="NEWTON:func.12">!</a> </span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <b>by </b><i><a class="txt" href="asympt_1.html#E4:115"><i><font color="Green" title="E73">A5</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c11">n1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="newton.html#K12" title="NEWTON:func.12">!</a> </span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 as    <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>by </b><i><a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/><b>set </b><font color="Maroon" title="c12">n3</font> = <font color="Maroon" title="c9">n2</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1;<br/><a NAME="E19:115_4"/><i><font color="Green" title="E86">A34</font></i>: 
<font color="Maroon" title="c9">n2</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <b>by </b><i><a class="txt" href="asympt_1.html#E4:115"><i><font color="Green" title="E73">A5</font></i></a>, <a class="txt" href="asympt_1.html#E15:115_4"><i><font color="Green" title="E83">A31</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c13">n3</font> = <font color="Maroon" title="c9">n2</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 as    <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>by </b><i><a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/><a NAME="E21:115_4"/>
<font color="Maroon" title="c9">n2</font> <a href="newton.html#K12" title="NEWTON:func.12">!</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c9">n2</font>
 <b>by </b><i><a class="txt" href="asympt_1.html#E12:115_4"><i><font color="Green" title="E82">A30</font></i></a>, <a class="txt" href="asympt_1.html#E109"><i><font color="Green" title="E71">Lm61</font></i></a></i>;<br/><a NAME="E22:115_4"/><b>then </b>
<font color="Maroon" title="c9">n2</font> <a href="newton.html#K12" title="NEWTON:func.12">!</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="int_1.html#T20" title="INT_1:th.20">INT_1:20</a></i>;<br/><a NAME="E23:115_4"/><b>then </b>
<font color="Maroon" title="c11">n1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c9">n2</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/><a NAME="E24:115_4"/><b>then </b>
<font color="Maroon" title="c11">n1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">N</font>
 <b>by </b><i><a class="txt" href="asympt_1.html#E12:115_4"><i><font color="Green" title="E82">A30</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E25:115_4"/><b>then </b><i><font color="Green" title="E87">A35</font></i>: 
<font color="Maroon" title="c5">t</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">c</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n1</font></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E9:115_4"><i><font color="Green" title="E79">A27</font></i></a></i>;<br/><a NAME="E26:115_4"/><i><font color="Green" title="E88">A36</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="newton.html#K12" title="NEWTON:func.12">!</a> </span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="newton.html#K12" title="NEWTON:func.12">!</a> </span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 0
 <b>by </b><i><a class="ref" href="real_1.html#T92" title="REAL_1:th.92">REAL_1:92</a></i>;<br/><a NAME="E27:115_4"/><i><font color="Green" title="E89">A37</font></i>: 
<font color="Maroon" title="c13">n3</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 0
 ;<br/><a NAME="E28:115_4"/><b>then </b><i><font color="Green" title="E90">A38</font></i>: 
<font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">n3</font> <a href="newton.html#K12" title="NEWTON:func.12">!</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">n2</font> <a href="newton.html#K12" title="NEWTON:func.12">!</a> 
 <b>by </b><i><a class="ref" href="newton.html#T21" title="NEWTON:th.21">NEWTON:21</a></i>;<br/><a NAME="E29:115_4"/><i><font color="Green" title="E91">A39</font></i>: 
<font color="Maroon" title="c13">n3</font> <a href="newton.html#K12" title="NEWTON:func.12">!</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <b>by </b><i><a class="txt" href="asympt_1.html#E19:115_4"><i><font color="Green" title="E86">A34</font></i></a>, <a class="txt" href="asympt_1.html#E106"><i><font color="Green" title="E68">Lm59</font></i></a>, <a class="ref" href="newton.html#T19" title="NEWTON:th.19">NEWTON:19</a></i>;<br/><a NAME="E30:115_4"/>
1 <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="c9">n2</font>
 <b>by </b><i><a class="txt" href="asympt_1.html#E12:115_4"><i><font color="Green" title="E82">A30</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E31:115_4"/><b>then </b>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n2</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/><a NAME="E32:115_4"/><b>then </b>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">n3</font> <a href="newton.html#K12" title="NEWTON:func.12">!</a> </span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E29:115_4"><i><font color="Green" title="E91">A39</font></i></a>, <a class="txt" href="asympt_1.html#E45"><i><font color="Green" title="E33">Lm26</font></i></a></i>;<br/><a NAME="E33:115_4"/><b>then </b>
<font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">n3</font> <a href="newton.html#K12" title="NEWTON:func.12">!</a> </span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c9">n2</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/><a NAME="E34:115_4"/><b>then </b>
<font color="Maroon" title="c9">n2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">n3</font> <a href="newton.html#K12" title="NEWTON:func.12">!</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">n3</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 ;<br/><a NAME="E35:115_4"/><b>then </b>
<font color="Maroon" title="c9">n2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="newton.html#K12" title="NEWTON:func.12">!</a> </span>)</span>
 <b>by </b><i><a class="ref" href="newton.html#T21" title="NEWTON:th.21">NEWTON:21</a></i>;<br/><a NAME="E36:115_4"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="newton.html#K12" title="NEWTON:func.12">!</a> </span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c9">n2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="newton.html#K12" title="NEWTON:func.12">!</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 1</span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="newton.html#K12" title="NEWTON:func.12">!</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E37:115_4"/><b>then </b>
<font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">n3</font> <a href="newton.html#K12" title="NEWTON:func.12">!</a> </span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="newton.html#K12" title="NEWTON:func.12">!</a> </span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c9">n2</font>
 <b>by </b><i><a class="txt" href="asympt_1.html#E28:115_4"><i><font color="Green" title="E90">A38</font></i></a>, <a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/><a NAME="E38:115_4"/><b>then </b>
<font color="Maroon" title="c13">n3</font> <a href="newton.html#K12" title="NEWTON:func.12">!</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="newton.html#K12" title="NEWTON:func.12">!</a> </span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E10:115_4"><i><font color="Green" title="E80">A28</font></i></a>, <a class="txt" href="asympt_1.html#E11:115_4"><i><font color="Green" title="E81">A29</font></i></a>, <a class="ref" href="real_2.html#T177" title="REAL_2:th.177">REAL_2:177</a></i>;<br/><a NAME="E39:115_4"/><b>then </b><i><font color="Green" title="E92">A40</font></i>: 
<font color="Maroon" title="c13">n3</font> <a href="newton.html#K12" title="NEWTON:func.12">!</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="newton.html#K12" title="NEWTON:func.12">!</a> </span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 1
 <b>by </b><i><a class="txt" href="asympt_1.html#E10:115_4"><i><font color="Green" title="E80">A28</font></i></a>, <a class="txt" href="asympt_1.html#E11:115_4"><i><font color="Green" title="E81">A29</font></i></a>, <a class="ref" href="xcmplx_1.html#T92" title="XCMPLX_1:th.92">XCMPLX_1:92</a></i>;<br/><i><font color="Green" title="E93">A41</font></i>: <font color="Maroon" title="c3">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n1</font> = 
 <a href="asympt_1.html#K10" title="ASYMPT_1:func.10">Step1</a> <font color="Maroon" title="c11">n1</font>
<b>by </b><i><a class="txt" href="asympt_1.html#E1:115"><i><font color="Green" title="E72">A1</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c13">n3</font> <a href="newton.html#K12" title="NEWTON:func.12">!</a> 
<b>by </b><i><a class="txt" href="asympt_1.html#E17:115_4"><i><font color="Green" title="E85">A33</font></i></a>, <a class="txt" href="asympt_1.html#E26:115_4"><i><font color="Green" title="E88">A36</font></i></a>, <a class="txt" href="asympt_1.html#E27:115_4"><i><font color="Green" title="E89">A37</font></i></a>, <a class="txt" href="asympt_1.html#E39:115_4"><i><font color="Green" title="E92">A40</font></i></a>, <a class="ref" href="asympt_1.html#D9" target="_self" title="ASYMPT_1:def.9">Def9</a></i>
;<br/><a NAME="E41:115_4"/><i><font color="Green" title="E94">A42</font></i>: 
<font color="Maroon" title="c5">t</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c11">n1</font></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E5:115_4"><i><font color="Green" title="E77">A25</font></i></a>, <a class="ref" href="asympt_0.html#D18" title="ASYMPT_0:def.18">ASYMPT_0:def 18</a></i>;<br/><a NAME="E42:115_4"/><i><font color="Green" title="E95">A43</font></i>: 
2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="newton.html#K12" title="NEWTON:func.12">!</a> </span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="newton.html#K12" title="NEWTON:func.12">!</a> </span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E15:115_4"><i><font color="Green" title="E83">A31</font></i></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/><a NAME="E43:115_4"/>
<font color="Maroon" title="c9">n2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c9">n2</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="E44:115_4"/><b>then </b>
<font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="newton.html#K12" title="NEWTON:func.12">!</a> </span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="newton.html#K12" title="NEWTON:func.12">!</a> </span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E16:115_4"><i><font color="Green" title="E84">A32</font></i></a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></i>;<br/><a NAME="E45:115_4"/><b>then </b>
<font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="newton.html#K12" title="NEWTON:func.12">!</a> </span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="newton.html#K12" title="NEWTON:func.12">!</a> 
 <b>by </b><i><a class="ref" href="newton.html#T21" title="NEWTON:th.21">NEWTON:21</a></i>;<br/><a NAME="E46:115_4"/><b>then </b><i><font color="Green" title="E96">A44</font></i>: 
2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="newton.html#K12" title="NEWTON:func.12">!</a> </span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="newton.html#K12" title="NEWTON:func.12">!</a> 
 <b>by </b><i><a class="txt" href="asympt_1.html#E42:115_4"><i><font color="Green" title="E95">A43</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E47:115_4"/>
2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c11">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="newton.html#K12" title="NEWTON:func.12">!</a> </span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E26:115_4"><i><font color="Green" title="E88">A36</font></i></a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></i>;<br/><a NAME="E48:115_4"/><b>then </b><i><font color="Green" title="E97">A45</font></i>: 
2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c11">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="newton.html#K12" title="NEWTON:func.12">!</a> 
 <b>by </b><i><a class="txt" href="asympt_1.html#E46:115_4"><i><font color="Green" title="E96">A44</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E49:115_4"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="newton.html#K12" title="NEWTON:func.12">!</a> </span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="newton.html#K12" title="NEWTON:func.12">!</a> </span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="newton.html#K12" title="NEWTON:func.12">!</a> </span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E16:115_4"><i><font color="Green" title="E84">A32</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E50:115_4"/><b>then </b><i><font color="Green" title="E98">A46</font></i>: 
<font color="Maroon" title="c9">n2</font> <a href="newton.html#K12" title="NEWTON:func.12">!</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="newton.html#K12" title="NEWTON:func.12">!</a> </span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/><i><font color="Green" title="E99">A47</font></i>: <font color="Maroon" title="c3">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c11">n1</font></span>)</span> = 
 <a href="asympt_1.html#K10" title="ASYMPT_1:func.10">Step1</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c11">n1</font></span>)</span>
<b>by </b><i><a class="txt" href="asympt_1.html#E1:115"><i><font color="Green" title="E72">A1</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">n3</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="newton.html#K12" title="NEWTON:func.12">!</a> 
<b>by </b><i><a class="txt" href="asympt_1.html#E48:115_4"><i><font color="Green" title="E97">A45</font></i></a>, <a class="txt" href="asympt_1.html#E50:115_4"><i><font color="Green" title="E98">A46</font></i></a>, <a class="ref" href="asympt_1.html#D9" target="_self" title="ASYMPT_1:def.9">Def9</a>, <a class="txt" href="asympt_1.html#E16:115_4"><i><font color="Green" title="E84">A32</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">n3</font> <a href="newton.html#K12" title="NEWTON:func.12">!</a> </span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T21" title="NEWTON:th.21">NEWTON:21</a></i>
;<br/><a NAME="E52:115_4"/><i><font color="Green" title="E100">A48</font></i>: 
<span class="p1"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><font color="Maroon" title="c6">c</font></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><font color="Maroon" title="c6">c</font></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><a NAME="E53:115_4"/>
<span class="p1"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><font color="Maroon" title="c6">c</font></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">c</font>
 <b>by </b><i><a class="ref" href="int_1.html#D5" title="INT_1:def.5">INT_1:def 5</a></i>;<br/><a NAME="E54:115_4"/><b>then </b>
<span class="p1"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><font color="Maroon" title="c6">c</font></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c6">c</font>
 <b>by </b><i><a class="txt" href="asympt_1.html#E52:115_4"><i><font color="Green" title="E100">A48</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E55:115_4"/><b>then </b><i><font color="Green" title="E101">A49</font></i>: 
<font color="Maroon" title="c9">n2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c6">c</font>
 <b>by </b><i><a class="txt" href="asympt_1.html#E10:115_4"><i><font color="Green" title="E80">A28</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E56:115_4"/>
<font color="Maroon" title="c13">n3</font> <a href="newton.html#K12" title="NEWTON:func.12">!</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="ref" href="newton.html#T23" title="NEWTON:th.23">NEWTON:23</a></i>;<br/><b>hence </b><a NAME="E57:115_4"/>
contradiction
 <b>by </b><i><a class="txt" href="asympt_1.html#E25:115_4"><i><font color="Green" title="E87">A35</font></i></a>, <a class="txt" href="asympt_1.html#E40:115_4"><i><font color="Green" title="E93">A41</font></i></a>, <a class="txt" href="asympt_1.html#E41:115_4"><i><font color="Green" title="E94">A42</font></i></a>, <a class="txt" href="asympt_1.html#E51:115_4"><i><font color="Green" title="E99">A47</font></i></a>, <a class="txt" href="asympt_1.html#E55:115_4"><i><font color="Green" title="E101">A49</font></i></a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E9:115"/>
( <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> &amp; <font color="Maroon" title="c1">f</font> is <a href="asympt_0.html#V6" title="ASYMPT_0:attr.6">eventually-nondecreasing</a> &amp; ( 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="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> 1</span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font> ) &amp; not <font color="Maroon" title="c3">f</font> is <a href="asympt_0.html#V7" title="ASYMPT_0:attr.7">smooth</a> )
 <b>by </b><i><a class="txt" href="asympt_1.html#E6:115"><i><font color="Green" title="E74">A18</font></i></a>, <a class="txt" href="asympt_1.html#E7:115"><i><font color="Green" title="E75">A19</font></i></a>, <a class="ref" href="asympt_0.html#D8" title="ASYMPT_0:def.8">ASYMPT_0:def 8</a></i>;<br/>


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