<?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="E1:107"/><i><font color="Green" title="E67">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 <br/> ( ( <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_1.html#K9" title="ASYMPT_1:func.9">POWEROF2SET</a>  implies <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> <font color="Olive" title="b1">n</font> ) &amp; ( not <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_1.html#K9" title="ASYMPT_1:func.9">POWEROF2SET</a>  implies <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> 2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</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/>
<b>set </b><font color="Maroon" title="c3">h</font> = <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="asympt_0.html#K15" title="ASYMPT_0:func.15">taken_every</a> 2;<br/>
<b>set </b><font color="Maroon" title="c4">p</font> =  <a href="asympt_1.html#K2" title="ASYMPT_1:func.2">seq_logn</a> ;<br/>
<b>set </b><font color="Maroon" title="c5">q</font> = <a href="asympt_1.html#K2" title="ASYMPT_1:func.2">seq_logn</a>  <a href="seq_1.html#K19" title="SEQ_1:func.19">/"</a> <span class="p1">(<span class="default"><a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> 1</span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c6">X</font> =  <a href="asympt_1.html#K9" title="ASYMPT_1:func.9">POWEROF2SET</a> ;<br/>
<a NAME="E2:107"/><i><font color="Green" title="E68">A2</font></i>: 
3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 
;<br/>
<a NAME="E3:107"/><i><font color="Green" title="E69">A3</font></i>: 
 <a href="asympt_0.html#K11" title="ASYMPT_0:func.11">Big_Theta</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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">t</font> where <font color="Olive" title="b1">t</font> is    <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>  :  ex <font color="Olive" title="b2">c</font>, <font color="Olive" title="b3">d</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ex <font color="Olive" title="b4">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="b2">c</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp; <font color="Olive" title="b3">d</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp; ( for <font color="Olive" title="b5">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="b5">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b4">N</font> holds <br/>( <font color="Olive" title="b3">d</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<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="b5">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">t</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b5">n</font> &amp; <font color="Olive" title="b1">t</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b5">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">c</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<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="b5">n</font></span>)</span> ) ) ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="asympt_0.html#T27" title="ASYMPT_0:th.27">ASYMPT_0:27</a></i>;<br/>
<a NAME="E4:107"/><i><font color="Green" title="E70">A4</font></i>: 
<font color="Maroon" title="c1">f</font> is    <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>by </b><i><a class="ref" href="funct_2.html#T11" title="FUNCT_2:th.11">FUNCT_2:11</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="107_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:107_1"/><i><font color="Green" title="E71">A5</font></i>: 
<font color="Maroon" title="c7">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <b>and </b><br/><a NAME="E2:107_1"/><i><font color="Green" title="E72">A6</font></i>: 
<font color="Maroon" title="c7">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_1.html#K9" title="ASYMPT_1:func.9">POWEROF2SET</a> 
 ;<br/><i><font color="Green" title="E73">A7</font></i>: <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="c7">n</font> = 
<font color="Maroon" title="c7">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#E1:107_1"><i><font color="Green" title="E71">A5</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="c7">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="E4:107_1"/>
1 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<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="c7">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">n</font>
 <b>by </b><i><a class="txt" href="asympt_1.html#E1:107"><i><font color="Green" title="E67">A1</font></i></a>, <a class="txt" href="asympt_1.html#E2:107_1"><i><font color="Green" title="E72">A6</font></i></a></i>;<br/><b>thus </b><a NAME="E5:107_1"/>
<font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<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="c7">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E1:107"><i><font color="Green" title="E67">A1</font></i></a>, <a class="txt" href="asympt_1.html#E2:107_1"><i><font color="Green" title="E72">A6</font></i></a>, <a class="txt" href="asympt_1.html#E3:107_1"><i><font color="Green" title="E73">A7</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:107"/>
<font color="Maroon" title="c1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_0.html#K14" title="ASYMPT_0:func.14">Big_Theta</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="asympt_1.html#K9" title="ASYMPT_1:func.9">POWEROF2SET</a> 
 <b>by </b><i><a class="txt" href="asympt_1.html#E4:107"><i><font color="Green" title="E70">A4</font></i></a></i>;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:107_2"/>
<font color="Maroon" title="c1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_0.html#K11" title="ASYMPT_0:func.11">Big_Theta</a> <span class="p1">(<span class="default"><a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> 1</span>)</span>
 ;<br/><b>then consider </b><font color="Maroon" title="c7">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="E3:107_2"/><i><font color="Green" title="E71">A8</font></i>: 
<font color="Maroon" title="c7">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font>
 <b>and </b><br/><a NAME="E4:107_2"/><i><font color="Green" title="E72">A9</font></i>: 
 ex <font color="Olive" title="b1">c</font>, <font color="Olive" title="b2">d</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ex <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 <br/>( <font color="Olive" title="b1">c</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp; <font color="Olive" title="b2">d</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp; ( for <font color="Olive" title="b4">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="b4">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b3">N</font> holds <br/>( <font color="Olive" title="b2">d</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<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="b4">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">t</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b4">n</font> &amp; <font color="Maroon" title="c7">t</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b4">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"><span class="p2">(<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="b4">n</font></span>)</span> ) ) )
 <b>by </b><i><a class="txt" href="asympt_1.html#E3:107"><i><font color="Green" title="E69">A3</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c8">c</font>, <font color="Maroon" title="c9">d</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>, <font color="Maroon" title="c10">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="E6:107_2"/><i><font color="Green" title="E73">A10</font></i>: 
( <font color="Maroon" title="c8">c</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp; <font color="Maroon" title="c9">d</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 )
 <b>and </b><br/><a NAME="E7:107_2"/><i><font color="Green" title="E74">A11</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="c10">N</font> holds <br/>( <font color="Maroon" title="c9">d</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<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></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">t</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font> &amp; <font color="Maroon" title="c7">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="c8">c</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<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></span>)</span> )
 <b>by </b><i><a class="txt" href="asympt_1.html#E4:107_2"><i><font color="Green" title="E72">A9</font></i></a></i>;<br/><a NAME="E8:107_2"/>
( <a href="asympt_1.html#K2" title="ASYMPT_1:func.2">seq_logn</a>  <a href="seq_1.html#K19" title="SEQ_1:func.19">/"</a> <span class="p1">(<span class="default"><a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> 1</span>)</span> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="asympt_1.html#K2" title="ASYMPT_1:func.2">seq_logn</a>  <a href="seq_1.html#K19" title="SEQ_1:func.19">/"</a> <span class="p2">(<span class="default"><a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 <b>by </b><i><a class="txt" href="asympt_1.html#E23"><i><font color="Green" title="E20">Lm16</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c11">N0</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="E10:107_2"/><i><font color="Green" title="E75">A12</font></i>: 
for <font color="Olive" title="b1">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>   st <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c11">N0</font> holds <br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="asympt_1.html#K2" title="ASYMPT_1:func.2">seq_logn</a>  <a href="seq_1.html#K19" title="SEQ_1:func.19">/"</a> <span class="p4">(<span class="default"><a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 0</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2
 <b>by </b><i><a class="ref" href="seq_2.html#D7" title="SEQ_2:def.7">SEQ_2:def 7</a></i>;<br/><b>set </b><font color="Maroon" title="c12">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="c11">N0</font>,<font color="Maroon" title="c10">N</font></span>)</span>,<span class="p1">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p2"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><font color="Maroon" title="c8">c</font></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>,2</span>)</span>;<br/><a NAME="E11:107_2"/><i><font color="Green" title="E76">A13</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="c11">N0</font>,<font color="Maroon" title="c10">N</font></span>)</span>,<span class="p1">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p2"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><font color="Maroon" title="c8">c</font></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>,2</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="c11">N0</font>,<font color="Maroon" title="c10">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="c11">N0</font>,<font color="Maroon" title="c10">N</font></span>)</span>,<span class="p1">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p2"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><font color="Maroon" title="c8">c</font></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>,2</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><font color="Maroon" title="c8">c</font></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>,2 )
 <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="E12:107_2"/>
(  <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c11">N0</font>,<font color="Maroon" title="c10">N</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c11">N0</font> &amp;  <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c11">N0</font>,<font color="Maroon" title="c10">N</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c10">N</font> )
 <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="E13:107_2"/><b>then </b><i><font color="Green" title="E77">A14</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="c11">N0</font>,<font color="Maroon" title="c10">N</font></span>)</span>,<span class="p1">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p2"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><font color="Maroon" title="c8">c</font></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>,2</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c11">N0</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="c11">N0</font>,<font color="Maroon" title="c10">N</font></span>)</span>,<span class="p1">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p2"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><font color="Maroon" title="c8">c</font></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>,2</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c10">N</font> )
 <b>by </b><i><a class="txt" href="asympt_1.html#E11:107_2"><i><font color="Green" title="E76">A13</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E14:107_2"/>
(  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><font color="Maroon" title="c8">c</font></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>,2 <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="c8">c</font></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span> &amp;  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><font color="Maroon" title="c8">c</font></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>,2 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 )
 <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="E15:107_2"/><b>then </b><i><font color="Green" title="E78">A15</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="c11">N0</font>,<font color="Maroon" title="c10">N</font></span>)</span>,<span class="p1">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p2"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><font color="Maroon" title="c8">c</font></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>,2</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="c8">c</font></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span> &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="c11">N0</font>,<font color="Maroon" title="c10">N</font></span>)</span>,<span class="p1">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p2"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><font color="Maroon" title="c8">c</font></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>,2</span>)</span> <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#E11:107_2"><i><font color="Green" title="E76">A13</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E16:107_2"/>
 <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="c11">N0</font>,<font color="Maroon" title="c10">N</font></span>)</span>,<span class="p1">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p2"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><font color="Maroon" title="c8">c</font></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>,2</span>)</span> is   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="107_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="107_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:107_2_1_1"/>
(  <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="c11">N0</font>,<font color="Maroon" title="c10">N</font></span>)</span>,<span class="p1">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p2"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><font color="Maroon" title="c8">c</font></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>,2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c11">N0</font>,<font color="Maroon" title="c10">N</font> or  <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="c11">N0</font>,<font color="Maroon" title="c10">N</font></span>)</span>,<span class="p1">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p2"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><font color="Maroon" title="c8">c</font></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>,2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><font color="Maroon" title="c8">c</font></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>,2 )
 </span><b>by </b><i><a class="ref" href="xxreal_0.html#T16" title="XXREAL_0:th.16">XXREAL_0:16</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="107_2_1_1_1"><b>suppose </b></a><a NAME="E1:107_2_1_1_1"/>
 <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="c11">N0</font>,<font color="Maroon" title="c10">N</font></span>)</span>,<span class="p1">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p2"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><font color="Maroon" title="c8">c</font></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>,2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c11">N0</font>,<font color="Maroon" title="c10">N</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:107_2_1_1_1"/>
 <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="c11">N0</font>,<font color="Maroon" title="c10">N</font></span>)</span>,<span class="p1">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p2"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><font color="Maroon" title="c8">c</font></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>,2</span>)</span> is   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="107_2_1_1_2"><b>suppose </b></a><a NAME="E1:107_2_1_1_2"/>
 <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="c11">N0</font>,<font color="Maroon" title="c10">N</font></span>)</span>,<span class="p1">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p2"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><font color="Maroon" title="c8">c</font></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>,2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><font color="Maroon" title="c8">c</font></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>,2
 ;<br/><div class="add"><b>hence </b><a NAME="E2:107_2_1_1_2"/>
 <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="c11">N0</font>,<font color="Maroon" title="c10">N</font></span>)</span>,<span class="p1">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p2"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><font color="Maroon" title="c8">c</font></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>,2</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/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c13">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="c11">N0</font>,<font color="Maroon" title="c10">N</font></span>)</span>,<span class="p1">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p2"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><font color="Maroon" title="c8">c</font></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>,2</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#E11:107_2"><i><font color="Green" title="E76">A13</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="c14">N3</font> = <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c13">N2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1;<br/><a NAME="E18:107_2"/>
2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c13">N2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="ref" href="power.html#T39" title="POWER:th.39">POWER:39</a></i>;<br/><a NAME="E19:107_2"/><b>then </b>
2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c13">N2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 <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="E20:107_2"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c13">N2</font></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> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c15">N3</font> = <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c13">N2</font></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/><a NAME="E22:107_2"/>
2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c13">N2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c13">N2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="asympt_1.html#E15:107_2"><i><font color="Green" title="E78">A15</font></i></a>, <a class="txt" href="asympt_1.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/><a NAME="E23:107_2"/><b>then </b><i><font color="Green" title="E79">A16</font></i>: 
<font color="Maroon" title="c15">N3</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c13">N2</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T22" title="XREAL_1:th.22">XREAL_1:22</a></i>;<br/><a NAME="E24:107_2"/><b>then </b><i><font color="Green" title="E80">A17</font></i>: 
<font color="Maroon" title="c15">N3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c10">N</font>
 <b>by </b><i><a class="txt" href="asympt_1.html#E13:107_2"><i><font color="Green" title="E77">A14</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E25:107_2"/>
not <font color="Maroon" title="c15">N3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_1.html#K9" title="ASYMPT_1:func.9">POWEROF2SET</a> 
 <b>by </b><i><a class="txt" href="asympt_1.html#E15:107_2"><i><font color="Green" title="E78">A15</font></i></a>, <a class="txt" href="asympt_1.html#E101"><i><font color="Green" title="E66">Lm57</font></i></a></i>;<br/><a NAME="E26:107_2"/><b>then </b><i><font color="Green" title="E81">A18</font></i>: 
<font color="Maroon" title="c7">t</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c15">N3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c15">N3</font>
 <b>by </b><i><a class="txt" href="asympt_1.html#E1:107"><i><font color="Green" title="E67">A1</font></i></a>, <a class="txt" href="asympt_1.html#E3:107_2"><i><font color="Green" title="E71">A8</font></i></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="c15">N3</font> = 
<font color="Maroon" title="c15">N3</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#E23:107_2"><i><font color="Green" title="E79">A16</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="c15">N3</font>
<b>by </b><i><a class="ref" href="power.html#T30" title="POWER:th.30">POWER:30</a></i>
;<br/><a NAME="E28:107_2"/><b>then </b><i><font color="Green" title="E82">A19</font></i>: 
2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c15">N3</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">c</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c15">N3</font>
 <b>by </b><i><a class="txt" href="asympt_1.html#E7:107_2"><i><font color="Green" title="E74">A11</font></i></a>, <a class="txt" href="asympt_1.html#E24:107_2"><i><font color="Green" title="E80">A17</font></i></a>, <a class="txt" href="asympt_1.html#E26:107_2"><i><font color="Green" title="E81">A18</font></i></a></i>;<br/><a NAME="E29:107_2"/>
2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c15">N3</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="ref" href="power.html#T39" title="POWER:th.39">POWER:39</a></i>;<br/><a NAME="E30:107_2"/><b>then </b>
 <a href="power.html#K6" title="POWER:func.6">log</a> 2,<span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c15">N3</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="power.html#K6" title="POWER:func.6">log</a> 2,<span class="p1">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c15">N3</font></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E28:107_2"><i><font color="Green" title="E82">A19</font></i></a>, <a class="ref" href="pre_ff.html#T12" title="PRE_FF:th.12">PRE_FF:12</a></i>;<br/><a NAME="E31:107_2"/><b>then </b>
<font color="Maroon" title="c15">N3</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,2</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="power.html#K6" title="POWER:func.6">log</a> 2,<span class="p1">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c15">N3</font></span>)</span>
 <b>by </b><i><a class="ref" href="power.html#T63" title="POWER:th.63">POWER:63</a></i>;<br/><a NAME="E32:107_2"/><b>then </b>
<font color="Maroon" title="c15">N3</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>  <a href="power.html#K6" title="POWER:func.6">log</a> 2,<span class="p1">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c15">N3</font></span>)</span>
 <b>by </b><i><a class="ref" href="power.html#T60" title="POWER:th.60">POWER:60</a></i>;<br/><a NAME="E33:107_2"/><b>then </b><i><font color="Green" title="E83">A20</font></i>: 
<font color="Maroon" title="c15">N3</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c8">c</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c15">N3</font></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E6:107_2"><i><font color="Green" title="E73">A10</font></i></a>, <a class="txt" href="asympt_1.html#E23:107_2"><i><font color="Green" title="E79">A16</font></i></a>, <a class="ref" href="power.html#T61" title="POWER:th.61">POWER:61</a></i>;<br/><a NAME="E34:107_2"/><i><font color="Green" title="E84">A21</font></i>: 
<font color="Maroon" title="c15">N3</font> <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="c8">c</font></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E15:107_2"><i><font color="Green" title="E78">A15</font></i></a>, <a class="txt" href="asympt_1.html#E23:107_2"><i><font color="Green" title="E79">A16</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E35:107_2"/>
<span class="p1"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><font color="Maroon" title="c8">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="c8">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="E36:107_2"/><b>then </b>
<font color="Maroon" title="c15">N3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c8">c</font>
 <b>by </b><i><a class="txt" href="asympt_1.html#E34:107_2"><i><font color="Green" title="E84">A21</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E37:107_2"/><b>then </b>
 <a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c15">N3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c8">c</font>
 <b>by </b><i><a class="txt" href="asympt_1.html#E6:107_2"><i><font color="Green" title="E73">A10</font></i></a>, <a class="ref" href="pre_ff.html#T12" title="PRE_FF:th.12">PRE_FF:12</a></i>;<br/><a NAME="E38:107_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c15">N3</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c15">N3</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c8">c</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c15">N3</font></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="E39:107_2"/><b>then </b>
<font color="Maroon" title="c15">N3</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c15">N3</font></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E33:107_2"><i><font color="Green" title="E83">A20</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E40:107_2"/><b>then </b>
<font color="Maroon" title="c15">N3</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c15">N3</font>
 <b>by </b><i><a class="ref" href="real_2.html#T177" title="REAL_2:th.177">REAL_2:177</a></i>;<br/><a NAME="E41:107_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c15">N3</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">N3</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c15">N3</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">N3</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span>
 <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="E42:107_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">N3</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c15">N3</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c15">N3</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">N3</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span>
 ;<br/><a NAME="E43:107_2"/><b>then </b>
1 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c15">N3</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">N3</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E23:107_2"><i><font color="Green" title="E79">A16</font></i></a>, <a class="ref" href="xcmplx_0.html#D7" title="XCMPLX_0:def.7">XCMPLX_0:def 7</a></i>;<br/><a NAME="E44:107_2"/><b>then </b><i><font color="Green" title="E85">A22</font></i>: 
<span class="p1">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c15">N3</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c15">N3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2
 <b>by </b><i><a class="ref" href="xcmplx_0.html#D9" title="XCMPLX_0:def.9">XCMPLX_0:def 9</a></i>;<br/><a NAME="E45:107_2"/>
<font color="Maroon" title="c15">N3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c11">N0</font>
 <b>by </b><i><a class="txt" href="asympt_1.html#E13:107_2"><i><font color="Green" title="E77">A14</font></i></a>, <a class="txt" href="asympt_1.html#E23:107_2"><i><font color="Green" title="E79">A16</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E46:107_2"/><b>then </b><i><font color="Green" title="E86">A23</font></i>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="asympt_1.html#K2" title="ASYMPT_1:func.2">seq_logn</a>  <a href="seq_1.html#K19" title="SEQ_1:func.19">/"</a> <span class="p4">(<span class="default"><a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c15">N3</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 0</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2
 <b>by </b><i><a class="txt" href="asympt_1.html#E10:107_2"><i><font color="Green" title="E75">A12</font></i></a></i>;<br/><span class="p1">(<span class="default"><a href="asympt_1.html#K2" title="ASYMPT_1:func.2">seq_logn</a>  <a href="seq_1.html#K19" title="SEQ_1:func.19">/"</a> <span class="p2">(<span class="default"><a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c15">N3</font> = 
<span class="p1">(<span class="default"><a href="asympt_1.html#K2" title="ASYMPT_1:func.2">seq_logn</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c15">N3</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><span class="p2">(<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="c15">N3</font></span>)</span>
<b>by </b><i><a class="txt" href="asympt_1.html#E10"><i><font color="Green" title="E7">Lm4</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c15">N3</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><span class="p2">(<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="c15">N3</font></span>)</span>
<b>by </b><i><a class="txt" href="asympt_1.html#E23:107_2"><i><font color="Green" title="E79">A16</font></i></a>, <a class="ref" href="asympt_1.html#D2" target="_self" title="ASYMPT_1:def.2">Def2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c15">N3</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">N3</font> <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="asympt_1.html#E23:107_2"><i><font color="Green" title="E79">A16</font></i></a>, <a class="ref" href="asympt_1.html#D3" target="_self" title="ASYMPT_1:def.3">Def3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c15">N3</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c15">N3</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="E48:107_2"/>
contradiction
 <b>by </b><i><a class="txt" href="asympt_1.html#E44:107_2"><i><font color="Green" title="E85">A22</font></i></a>, <a class="txt" href="asympt_1.html#E46:107_2"><i><font color="Green" title="E86">A23</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/></div>
<b>end;</b></div>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="107_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:107_3"/>
<font color="Maroon" title="c7">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 ;<br/><a NAME="E2:107_3"/><i><font color="Green" title="E71">A24</font></i>: 
<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="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">n</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="107_3_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="107_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:107_3_1_1"/>
( <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Maroon" title="c7">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="107_3_1_1_1"><b>suppose </b></a><a NAME="E1:107_3_1_1_1"/>
<font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:107_3_1_1_1"/>
<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="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">n</font>
 <b>by </b><i><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="107_3_1_1_2"><b>suppose </b></a><a NAME="E1:107_3_1_1_2"/>
<font color="Maroon" title="c7">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/><div class="add"><b>hence </b><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="c7">n</font> = 
<font color="Maroon" title="c7">n</font> <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> 1
<b>by </b><i><a class="ref" href="asympt_1.html#D3" target="_self" title="ASYMPT_1:def.3">Def3</a></i>
<br/>.= 
<font color="Maroon" title="c7">n</font>
<b>by </b><i><a class="ref" href="power.html#T30" title="POWER:th.30">POWER:30</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><i><font color="Green" title="E72">A25</font></i>: <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> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> 1
<b>by </b><i><a class="ref" href="asympt_1.html#D3" target="_self" title="ASYMPT_1:def.3">Def3</a></i>
<br/>.= 
<font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
<b>by </b><i><a class="ref" href="power.html#T30" title="POWER:th.30">POWER:30</a></i>
;<br/><a NAME="E4:107_3"/>
<font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><b>hence </b><a NAME="E5:107_3"/>
<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="c7">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> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</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#E2:107_3"><i><font color="Green" title="E71">A24</font></i></a>, <a class="txt" href="asympt_1.html#E3:107_3"><i><font color="Green" title="E72">A25</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E9:107"/><b>then </b><i><font color="Green" title="E71">A26</font></i>: 
 <a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> 1 is <a href="asympt_0.html#V6" title="ASYMPT_0:attr.6">eventually-nondecreasing</a>
 
<b>by </b><i><a class="ref" href="asympt_0.html#D8" title="ASYMPT_0:def.8">ASYMPT_0:def 8</a></i>;<br/>
<a NAME="E10:107"/><i><font color="Green" title="E72">A27</font></i>: 
<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="asympt_0.html#K15" title="ASYMPT_0:func.15">taken_every</a> 2 is    <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>by </b><i><a class="ref" href="funct_2.html#T11" title="FUNCT_2:th.11">FUNCT_2:11</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="107_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:107_4"/><i><font color="Green" title="E73">A28</font></i>: 
<font color="Maroon" title="c7">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 ;<br/><a NAME="E2:107_4"/><b>then </b><i><font color="Green" title="E74">A29</font></i>: 
2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c7">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></i>;<br/><i><font color="Green" title="E75">A30</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> 1</span>)</span> <a href="asympt_0.html#K15" title="ASYMPT_0:func.15">taken_every</a> 2</span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">n</font> = 
<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> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c7">n</font></span>)</span>
<b>by </b><i><a class="ref" href="asympt_0.html#D18" title="ASYMPT_0:def.18">ASYMPT_0:def 18</a></i>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c7">n</font></span>)</span> <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#E2:107_4"><i><font color="Green" title="E74">A29</font></i></a>, <a class="ref" href="asympt_1.html#D3" target="_self" title="ASYMPT_1:def.3">Def3</a></i>
<br/>.= 
2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c7">n</font>
<b>by </b><i><a class="ref" href="power.html#T30" title="POWER:th.30">POWER:30</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="c7">n</font> = 
<font color="Maroon" title="c7">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#E1:107_4"><i><font color="Green" title="E73">A28</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="c7">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="E5:107_4"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> 1</span>)</span> <a href="asympt_0.html#K15" title="ASYMPT_0:func.15">taken_every</a> 2</span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<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="c7">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E3:107_4"><i><font color="Green" title="E75">A30</font></i></a></i>;<br/><b>thus </b><a NAME="E6:107_4"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> 1</span>)</span> <a href="asympt_0.html#K15" title="ASYMPT_0:func.15">taken_every</a> 2</span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">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#E3:107_4"><i><font color="Green" title="E75">A30</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E12:107"/><b>then </b>
<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="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> <span class="p1">(<span class="default"><a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="asympt_1.html#E10:107"><i><font color="Green" title="E72">A27</font></i></a></i>;<br/>
<a NAME="E13:107"/><b>then </b>
 <a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> 1 <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#E9:107"><i><font color="Green" title="E71">A26</font></i></a>, <a class="ref" href="asympt_0.html#D19" title="ASYMPT_0:def.19">ASYMPT_0:def 19</a></i>;<br/>
<b>hence </b><a NAME="E14:107"/>
 <a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> 1 is <a href="asympt_0.html#V7" title="ASYMPT_0:attr.7">smooth</a>
 <b>by </b><i><a class="ref" href="asympt_0.html#T37" title="ASYMPT_0:th.37">ASYMPT_0:37</a></i>;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:107_5"/>
<font color="Maroon" title="c1">f</font> is <a href="asympt_0.html#V6" title="ASYMPT_0:attr.6">eventually-nondecreasing</a>
 ;<br/><b>then consider </b><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="E3:107_5"/><i><font color="Green" title="E73">A31</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="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> <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="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="asympt_0.html#D8" title="ASYMPT_0:def.8">ASYMPT_0:def 8</a></i>;<br/><b>set </b><font color="Maroon" title="c8">N1</font> = <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">N</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1;<br/><a NAME="E4:107_5"/><i><font color="Green" title="E74">A32</font></i>: 
<font color="Maroon" title="c7">N</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 <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="E5:107_5"/><b>then </b><i><font color="Green" title="E75">A33</font></i>: 
2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">N</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> 2
 <b>by </b><i><a class="ref" href="pre_ff.html#T10" title="PRE_FF:th.10">PRE_FF:10</a></i>;<br/>2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> 2 = 
2 <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
<b>by </b><i><a class="ref" href="power.html#T53" title="POWER:th.53">POWER:53</a></i>
<br/>.= 
4

;<br/><a NAME="E7:107_5"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">N</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></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> 3
 <b>by </b><i><a class="txt" href="asympt_1.html#E2:107"><i><font color="Green" title="E68">A2</font></i></a>, <a class="txt" href="asympt_1.html#E5:107_5"><i><font color="Green" title="E75">A33</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="c9">N1</font> = <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">N</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></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/><a NAME="E9:107_5"/>
not <font color="Maroon" title="c9">N1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_1.html#K9" title="ASYMPT_1:func.9">POWEROF2SET</a> 
 <b>by </b><i><a class="txt" href="asympt_1.html#E4:107_5"><i><font color="Green" title="E74">A32</font></i></a>, <a class="txt" href="asympt_1.html#E101"><i><font color="Green" title="E66">Lm57</font></i></a></i>;<br/><a NAME="E10:107_5"/><b>then </b><i><font color="Green" title="E76">A34</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="c9">N1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c9">N1</font>
 <b>by </b><i><a class="txt" href="asympt_1.html#E1:107"><i><font color="Green" title="E67">A1</font></i></a></i>;<br/><a NAME="E11:107_5"/>
<font color="Maroon" title="c9">N1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_1.html#K9" title="ASYMPT_1:func.9">POWEROF2SET</a> 
 ;<br/><a NAME="E12:107_5"/><b>then </b><i><font color="Green" title="E77">A35</font></i>: 
<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="c9">N1</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="c9">N1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="asympt_1.html#E1:107"><i><font color="Green" title="E67">A1</font></i></a></i>;<br/><a NAME="E13:107_5"/>
2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">N</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">N</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="asympt_1.html#E4:107_5"><i><font color="Green" title="E74">A32</font></i></a>, <a class="txt" href="asympt_1.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/><a NAME="E14:107_5"/><b>then </b><i><font color="Green" title="E78">A36</font></i>: 
<font color="Maroon" title="c9">N1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c7">N</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T22" title="XREAL_1:th.22">XREAL_1:22</a></i>;<br/><a NAME="E15:107_5"/><b>then </b><i><font color="Green" title="E79">A37</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="c9">N1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <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="c9">N1</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#E10:107_5"><i><font color="Green" title="E76">A34</font></i></a>, <a class="txt" href="asympt_1.html#E12:107_5"><i><font color="Green" title="E77">A35</font></i></a>, <a class="ref" href="power.html#T44" title="POWER:th.44">POWER:44</a></i>;<br/><a NAME="E16:107_5"/>
<font color="Maroon" title="c7">N</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">N</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 0
 <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="E17:107_5"/><b>then </b>
<font color="Maroon" title="c9">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#E14:107_5"><i><font color="Green" title="E78">A36</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E18:107_5"/>
contradiction
 <b>by </b><i><a class="txt" href="asympt_1.html#E3:107_5"><i><font color="Green" title="E73">A31</font></i></a>, <a class="txt" href="asympt_1.html#E15:107_5"><i><font color="Green" title="E79">A37</font></i></a></i>;<br/></div>
<b>end;</b></div>


</div>
