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

<b>let </b><font color="Maroon" title="c1">h</font> be  <a href="asympt_0.html#V2" title="ASYMPT_0:attr.2">eventually-nonnegative</a> <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>;<br/>

<b>reconsider </b><font color="Maroon" title="c2">F</font> = <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><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></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> as    <a href="fraenkel.html#M1" title="FRAENKEL:mode.1">FUNCTION_DOMAIN</a> of  <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="fraenkel.html#T10" title="FRAENKEL:th.10">FRAENKEL:10</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c2">F</font>
;<br/>

<b>thus </b><a NAME="E2:41"/>
<font color="Maroon" title="c2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><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></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ;<br/>

<b>set </b><font color="Maroon" title="c3">G</font> =  <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#K4" title="ASYMPT_1:func.4">seq_const</a> 1</span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c4">p</font> =  <a href="asympt_1.html#K4" title="ASYMPT_1:func.4">seq_const</a> 1;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><b>now </b></a><div class="add"><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:41_1_1"/>
<font color="Maroon" title="c1">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> <a href="asympt_0.html#K18" title="ASYMPT_0:func.18">to_power</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K9" title="ASYMPT_0:func.9">Big_Oh</a> <span class="p2">(<span class="default"><a href="asympt_1.html#K4" title="ASYMPT_1:func.4">seq_const</a> 1</span>)</span></span>)</span>
 ;<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="E3:41_1_1"/><i><font color="Green" title="E31">A1</font></i>: 
<font color="Maroon" title="c1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">t</font>
 <b>and </b><br/><a NAME="E4:41_1_1"/><i><font color="Green" title="E32">A2</font></i>: 
 ex <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</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>  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">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> &amp; <font color="Olive" title="b2">g</font> <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#K4" title="ASYMPT_1:func.4">seq_const</a> 1</span>)</span> &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="Maroon" title="c5">t</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b4">n</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">g</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b4">n</font></span>)</span> ) )
 ;<br/><b>consider </b><font color="Maroon" title="c6">f</font>, <font color="Maroon" title="c7">g</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> , <font color="Maroon" title="c8">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="E6:41_1_1"/><i><font color="Green" title="E33">A3</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font>
 <b>and </b><br/><a NAME="E7:41_1_1"/><i><font color="Green" title="E34">A4</font></i>: 
<font color="Maroon" title="c7">g</font> <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#K4" title="ASYMPT_1:func.4">seq_const</a> 1</span>)</span>
 <b>and </b><br/><a NAME="E8:41_1_1"/><i><font color="Green" title="E35">A5</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="c8">N0</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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g</font> <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#E4:41_1_1"><i><font color="Green" title="E32">A2</font></i></a></i>;<br/><a NAME="E9:41_1_1"/><i><font color="Green" title="E36">A6</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> 1
 <b>by </b><i><a class="txt" href="asympt_1.html#E6:41_1_1"><i><font color="Green" title="E33">A3</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>consider </b><font color="Maroon" title="c9">g'</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="E11:41_1_1"/><i><font color="Green" title="E37">A7</font></i>: 
<font color="Maroon" title="c7">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">g'</font>
 <b>and </b><br/><a NAME="E12:41_1_1"/><i><font color="Green" title="E38">A8</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="c9">g'</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"><span class="p2">(<span class="default"><a href="asympt_1.html#K4" title="ASYMPT_1:func.4">seq_const</a> 1</span>)</span> <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="c9">g'</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 ) ) )
 <b>by </b><i><a class="txt" href="asympt_1.html#E7:41_1_1"><i><font color="Green" title="E34">A4</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c10">c</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>, <font color="Maroon" title="c11">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="E14:41_1_1"/><i><font color="Green" title="E39">A9</font></i>: 
<font color="Maroon" title="c10">c</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>and </b><br/><a NAME="E15:41_1_1"/><i><font color="Green" title="E40">A10</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="c11">N1</font> holds <br/>( <font color="Maroon" title="c9">g'</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="c10">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#K4" title="ASYMPT_1:func.4">seq_const</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> &amp; <font color="Maroon" title="c9">g'</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#E12:41_1_1"><i><font color="Green" title="E38">A8</font></i></a></i>;<br/><b>set </b><font color="Maroon" title="c12">N</font> =  <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> 2,<span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c8">N0</font>,<font color="Maroon" title="c11">N1</font></span>)</span>;<br/><a NAME="E16:41_1_1"/>
(  <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c8">N0</font>,<font color="Maroon" title="c11">N1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c8">N0</font> &amp;  <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c8">N0</font>,<font color="Maroon" title="c11">N1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c11">N1</font> &amp;  <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> 2,<span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c8">N0</font>,<font color="Maroon" title="c11">N1</font></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="c8">N0</font>,<font color="Maroon" title="c11">N1</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="E17:41_1_1"/><b>then </b><i><font color="Green" title="E41">A11</font></i>: 
(  <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> 2,<span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c8">N0</font>,<font color="Maroon" title="c11">N1</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c8">N0</font> &amp;  <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> 2,<span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c8">N0</font>,<font color="Maroon" title="c11">N1</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c11">N1</font> &amp;  <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> 2,<span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c8">N0</font>,<font color="Maroon" title="c11">N1</font></span>)</span> <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#T2" title="XXREAL_0:th.2">XXREAL_0:2</a>, <a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></i>;<br/><b>set </b><font color="Maroon" title="c13">k</font> = <span class="p1"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><font color="Maroon" title="c10">c</font></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>;<br/><a NAME="E18:41_1_1"/><i><font color="Green" title="E42">A12</font></i>: 
<span class="p1"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><font color="Maroon" title="c10">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="c10">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="E19:41_1_1"/>
<span class="p1"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><font color="Maroon" title="c10">c</font></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span> <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#E14:41_1_1"><i><font color="Green" title="E39">A9</font></i></a>, <a class="ref" href="int_1.html#D5" title="INT_1:def.5">INT_1:def 5</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c14">k</font> = <span class="p1"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><font color="Maroon" title="c10">c</font></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></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="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c15">i</font> = 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>  ;<br/><b>take </b><font color="Maroon" title="c16">N</font> =  <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> 2,<span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c8">N0</font>,<font color="Maroon" title="c11">N1</font></span>)</span>;<br/><b>take </b><font color="Maroon" title="c17">i</font> = <font color="Maroon" title="c15">i</font>;<br/><b>take </b><font color="Maroon" title="c18">k</font> = <font color="Maroon" title="c14">k</font>;<br/><b>thus </b><a NAME="E22:41_1_1"/>
<font color="Maroon" title="c17">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/><b>let </b><font color="Maroon" title="c19">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="E23:41_1_1"/><i><font color="Green" title="E43">A13</font></i>: 
<font color="Maroon" title="c19">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c16">N</font>
 ;<br/><a NAME="E24:41_1_1"/><b>then </b><i><font color="Green" title="E44">A14</font></i>: 
( <font color="Maroon" title="c19">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c8">N0</font> &amp; <font color="Maroon" title="c19">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c11">N1</font> &amp; <font color="Maroon" title="c19">n</font> <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#E17:41_1_1"><i><font color="Green" title="E41">A11</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E25:41_1_1"/><b>then </b><i><font color="Green" title="E45">A15</font></i>: 
( <font color="Maroon" title="c9">g'</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c19">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">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#K4" title="ASYMPT_1:func.4">seq_const</a> 1</span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c19">n</font></span>)</span> &amp; <font color="Maroon" title="c9">g'</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c19">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#E15:41_1_1"><i><font color="Green" title="E40">A10</font></i></a></i>;<br/><a NAME="E26:41_1_1"/><i><font color="Green" title="E46">A16</font></i>: 
<span class="p1">(<span class="default"><a href="asympt_1.html#K4" title="ASYMPT_1:func.4">seq_const</a> 1</span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c19">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/><a NAME="E27:41_1_1"/><i><font color="Green" title="E47">A17</font></i>: 
( <font color="Maroon" title="c19">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp; <font color="Maroon" title="c19">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 )
 <b>by </b><i><a class="txt" href="asympt_1.html#E24:41_1_1"><i><font color="Green" title="E44">A14</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><font color="Maroon" title="c6">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c19">n</font> = 
<font color="Maroon" title="c19">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#E9:41_1_1"><i><font color="Green" title="E36">A6</font></i></a>, <a class="txt" href="asympt_1.html#E17:41_1_1"><i><font color="Green" title="E41">A11</font></i></a>, <a class="txt" href="asympt_1.html#E23:41_1_1"><i><font color="Green" title="E43">A13</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="c19">n</font>
<b>by </b><i><a class="ref" href="power.html#T30" title="POWER:th.30">POWER:30</a></i>
;<br/><a NAME="E29:41_1_1"/><b>then </b><i><font color="Green" title="E48">A18</font></i>: 
<font color="Maroon" title="c1">h</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c19">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">n</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c19">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E3:41_1_1"><i><font color="Green" title="E31">A1</font></i></a>, <a class="txt" href="asympt_1.html#E8:41_1_1"><i><font color="Green" title="E35">A5</font></i></a>, <a class="txt" href="asympt_1.html#E24:41_1_1"><i><font color="Green" title="E44">A14</font></i></a></i>;<br/><a NAME="E30:41_1_1"/>
<font color="Maroon" title="c19">n</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c19">n</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c19">n</font> <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> 0
 <b>by </b><i><a class="txt" href="asympt_1.html#E11:41_1_1"><i><font color="Green" title="E37">A7</font></i></a>, <a class="txt" href="asympt_1.html#E25:41_1_1"><i><font color="Green" title="E45">A15</font></i></a>, <a class="txt" href="asympt_1.html#E27:41_1_1"><i><font color="Green" title="E47">A17</font></i></a>, <a class="ref" href="pre_ff.html#T10" title="PRE_FF:th.10">PRE_FF:10</a></i>;<br/><b>hence </b><a NAME="E31:41_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">h</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c19">n</font>
 <b>by </b><i><a class="txt" href="asympt_1.html#E29:41_1_1"><i><font color="Green" title="E48">A18</font></i></a>, <a class="ref" href="power.html#T29" title="POWER:th.29">POWER:29</a></i>;<br/><a NAME="E32:41_1_1"/>
<font color="Maroon" title="c7">g</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c19">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">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#K4" title="ASYMPT_1:func.4">seq_const</a> 1</span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c19">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E11:41_1_1"><i><font color="Green" title="E37">A7</font></i></a>, <a class="txt" href="asympt_1.html#E15:41_1_1"><i><font color="Green" title="E40">A10</font></i></a>, <a class="txt" href="asympt_1.html#E24:41_1_1"><i><font color="Green" title="E44">A14</font></i></a></i>;<br/><a NAME="E33:41_1_1"/><b>then </b><i><font color="Green" title="E49">A19</font></i>: 
<font color="Maroon" title="c1">h</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c19">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">n</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">c</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E26:41_1_1"><i><font color="Green" title="E46">A16</font></i></a>, <a class="txt" href="asympt_1.html#E27:41_1_1"><i><font color="Green" title="E47">A17</font></i></a>, <a class="txt" href="asympt_1.html#E29:41_1_1"><i><font color="Green" title="E48">A18</font></i></a>, <a class="ref" href="pre_ff.html#T10" title="PRE_FF:th.10">PRE_FF:10</a></i>;<br/><a NAME="E34:41_1_1"/>
<font color="Maroon" title="c19">n</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c10">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">n</font> <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c18">k</font>
 <b>by </b><i><a class="txt" href="asympt_1.html#E18:41_1_1"><i><font color="Green" title="E42">A12</font></i></a>, <a class="txt" href="asympt_1.html#E27:41_1_1"><i><font color="Green" title="E47">A17</font></i></a>, <a class="ref" href="pre_ff.html#T10" title="PRE_FF:th.10">PRE_FF:10</a></i>;<br/><a NAME="E35:41_1_1"/><b>then </b>
<font color="Maroon" title="c1">h</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c19">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">n</font> <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c18">k</font>
 <b>by </b><i><a class="txt" href="asympt_1.html#E33:41_1_1"><i><font color="Green" title="E49">A19</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="E36:41_1_1"/>
<font color="Maroon" title="c1">h</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c19">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">i</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> <font color="Maroon" title="c18">k</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c19">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E17:41_1_1"><i><font color="Green" title="E41">A11</font></i></a>, <a class="txt" href="asympt_1.html#E23:41_1_1"><i><font color="Green" title="E43">A13</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><b>given </b><font color="Maroon" title="c5">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> , <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">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> <b> such that </b><a NAME="E2:41_1"/>
<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="E3:41_1"/><i><font color="Green" title="E31">A20</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="c5">N0</font> holds <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">h</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="c1">h</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"><span class="p2">(<span class="default"><a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font></span>)</span> )
 ;<br/><b>reconsider </b><font color="Maroon" title="c8">t</font> = <font color="Maroon" title="c1">h</font> as    <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/><b>set </b><font color="Maroon" title="c9">N</font> =  <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c5">N0</font>,2;<br/><a NAME="E5:41_1"/><i><font color="Green" title="E32">A21</font></i>: 
(  <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c5">N0</font>,2 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">N0</font> &amp;  <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c5">N0</font>,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="E6:41_1"/><b>then </b><i><font color="Green" title="E33">A22</font></i>: 
 <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c5">N0</font>,2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E7:41_1"/><i><font color="Green" title="E34">A23</font></i>: 
(  <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c5">N0</font>,2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp;  <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c5">N0</font>,2 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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/><b>set </b><font color="Maroon" title="c10">c1</font> =  <a href="asympt_0.html#K2" title="ASYMPT_0:func.2">max</a> <font color="Maroon" title="c6">c</font>,2;<br/><a NAME="E8:41_1"/><i><font color="Green" title="E35">A24</font></i>: 
(  <a href="asympt_0.html#K2" title="ASYMPT_0:func.2">max</a> <font color="Maroon" title="c6">c</font>,2 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">c</font> &amp;  <a href="asympt_0.html#K2" title="ASYMPT_0:func.2">max</a> <font color="Maroon" title="c6">c</font>,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="E9:41_1"/><b>then </b><i><font color="Green" title="E36">A25</font></i>: 
(  <a href="asympt_0.html#K2" title="ASYMPT_0:func.2">max</a> <font color="Maroon" title="c6">c</font>,2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp;  <a href="asympt_0.html#K2" title="ASYMPT_0:func.2">max</a> <font color="Maroon" title="c6">c</font>,2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>set </b><font color="Maroon" title="c11">a</font> =  <a href="power.html#K6" title="POWER:func.6">log</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c5">N0</font>,2</span>)</span>,<span class="p1">(<span class="default"><a href="asympt_0.html#K2" title="ASYMPT_0:func.2">max</a> <font color="Maroon" title="c6">c</font>,2</span>)</span>;<br/><b>set </b><font color="Maroon" title="c12">b</font> = <font color="Maroon" title="c7">k</font> <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> <span class="p2">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c5">N0</font>,2</span>)</span>,<span class="p2">(<span class="default"><a href="asympt_0.html#K2" title="ASYMPT_0:func.2">max</a> <font color="Maroon" title="c6">c</font>,2</span>)</span></span>)</span>;<br/><b>reconsider </b><font color="Maroon" title="c13">f</font> =  <a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> 1 as    <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/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>] <b>means </b>( ( $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c5">N0</font>,2 implies $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) &amp; ( $1 <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="c5">N0</font>,2 implies $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="power.html#K6" title="POWER:func.6">log</a> $1,<span class="p1">(<span class="default"><font color="Maroon" title="c8">t</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> $1</span>)</span> ) );<br/><a NAME="E11:41_1"/><i><font color="Green" title="E37">A26</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   ex <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">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/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:41_1_2_1"/>
( <font color="Maroon" title="c14">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c5">N0</font>,2 or <font color="Maroon" title="c14">n</font> <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="c5">N0</font>,2 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_2_1_1"><b>suppose </b></a><a NAME="E1:41_1_2_1_1"/>
<font color="Maroon" title="c14">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c5">N0</font>,2
 ;<br/><div class="add"><b>hence </b><a NAME="E2:41_1_2_1_1"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c14">n</font>,<font color="Olive" title="b1">y</font>]
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_2_1_2"><b>suppose </b></a><a NAME="E1:41_1_2_1_2"/>
<font color="Maroon" title="c14">n</font> <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="c5">N0</font>,2
 ;<br/><div class="add"><b>hence </b><a NAME="E2:41_1_2_1_2"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c14">n</font>,<font color="Olive" title="b1">y</font>]
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>consider </b><font color="Maroon" title="c14">g</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <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="E13:41_1"/><i><font color="Green" title="E38">A27</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c14">g</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">x</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="asympt_1.html#E11:41_1"><i><font color="Green" title="E37">A26</font></i></a>);<br/></i><a NAME="E14:41_1"/><i><font color="Green" title="E39">A28</font></i>: 
<font color="Maroon" title="c14">g</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><i><font color="Green" title="E40">A29</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c15">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="E1:41_1_3"/><i><font color="Green" title="E41">A30</font></i>: 
<font color="Maroon" title="c15">n</font> <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="c5">N0</font>,2
 ;<br/><a NAME="E2:41_1_3"/><b>then </b>
<font color="Maroon" title="c15">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">N0</font>
 <b>by </b><i><a class="txt" href="asympt_1.html#E5:41_1"><i><font color="Green" title="E32">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="E3:41_1_3"/><b>then </b><i><font color="Green" title="E42">A31</font></i>: 
<font color="Maroon" title="c8">t</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c15">n</font> <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#E3:41_1"><i><font color="Green" title="E31">A20</font></i></a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c13">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c15">n</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">n</font> <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> 1</span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c15">n</font></span>)</span>
<b>by </b><i><a class="txt" href="asympt_1.html#E5:41_1"><i><font color="Green" title="E32">A21</font></i></a>, <a class="txt" href="asympt_1.html#E1:41_1_3"><i><font color="Green" title="E41">A30</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">n</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c15">n</font></span>)</span>
<b>by </b><i><a class="ref" href="power.html#T30" title="POWER:th.30">POWER:30</a></i>
<br/>.= 
<font color="Maroon" title="c15">n</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> <font color="Maroon" title="c15">n</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c8">t</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="asympt_1.html#E13:41_1"><i><font color="Green" title="E38">A27</font></i></a>, <a class="txt" href="asympt_1.html#E1:41_1_3"><i><font color="Green" title="E41">A30</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c8">t</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c15">n</font>
<b>by </b><i><a class="txt" href="asympt_1.html#E6:41_1"><i><font color="Green" title="E33">A22</font></i></a>, <a class="txt" href="asympt_1.html#E1:41_1_3"><i><font color="Green" title="E41">A30</font></i></a>, <a class="txt" href="asympt_1.html#E3:41_1_3"><i><font color="Green" title="E42">A31</font></i></a>, <a class="ref" href="power.html#D3" title="POWER:def.3">POWER:def 3</a></i>
;<br/></div><b>end;</b></div><a NAME="E16:41_1"/><i><font color="Green" title="E41">A32</font></i>: 
<font color="Maroon" title="c13">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_4"><b>now </b></a><div class="add"><a NAME="E1:41_1_4"/>
 <a href="power.html#K6" title="POWER:func.6">log</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c5">N0</font>,2</span>)</span>,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="txt" href="asympt_1.html#E7:41_1"><i><font color="Green" title="E34">A23</font></i></a>, <a class="ref" href="power.html#T59" title="POWER:th.59">POWER:59</a></i>;<br/><a NAME="E2:41_1_4"/><b>then </b>
 <a href="power.html#K6" title="POWER:func.6">log</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c5">N0</font>,2</span>)</span>,<span class="p1">(<span class="default"><a href="asympt_0.html#K2" title="ASYMPT_0:func.2">max</a> <font color="Maroon" title="c6">c</font>,2</span>)</span> <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#E6:41_1"><i><font color="Green" title="E33">A22</font></i></a>, <a class="txt" href="asympt_1.html#E9:41_1"><i><font color="Green" title="E36">A25</font></i></a>, <a class="ref" href="power.html#T65" title="POWER:th.65">POWER:65</a></i>;<br/><a NAME="E3:41_1_4"/><b>then </b>
<font color="Maroon" title="c7">k</font> <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> <span class="p2">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c5">N0</font>,2</span>)</span>,<span class="p2">(<span class="default"><a href="asympt_0.html#K2" title="ASYMPT_0:func.2">max</a> <font color="Maroon" title="c6">c</font>,2</span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c7">k</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/><b>hence </b><a NAME="E4:41_1_4"/>
<font color="Maroon" title="c7">k</font> <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> <span class="p2">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c5">N0</font>,2</span>)</span>,<span class="p2">(<span class="default"><a href="asympt_0.html#K2" title="ASYMPT_0:func.2">max</a> <font color="Maroon" title="c6">c</font>,2</span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/><b>let </b><font color="Maroon" title="c15">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="E5:41_1_4"/><i><font color="Green" title="E42">A33</font></i>: 
<font color="Maroon" title="c15">n</font> <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="c5">N0</font>,2
 ;<br/><a NAME="E6:41_1_4"/><b>then </b><i><font color="Green" title="E43">A34</font></i>: 
( <font color="Maroon" title="c15">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">N0</font> &amp; <font color="Maroon" title="c15">n</font> <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#E5:41_1"><i><font color="Green" title="E32">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="E7:41_1_4"/><b>then </b><i><font color="Green" title="E44">A35</font></i>: 
<font color="Maroon" title="c15">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E8:41_1_4"/><i><font color="Green" title="E45">A36</font></i>: 
( <font color="Maroon" title="c15">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp; <font color="Maroon" title="c15">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 <b>by </b><i><a class="txt" href="asympt_1.html#E5:41_1"><i><font color="Green" title="E32">A21</font></i></a>, <a class="txt" href="asympt_1.html#E5:41_1_4"><i><font color="Green" title="E42">A33</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E9:41_1_4"/><i><font color="Green" title="E46">A37</font></i>: 
 <a href="power.html#K6" title="POWER:func.6">log</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c5">N0</font>,2</span>)</span>,<span class="p1">(<span class="default"><a href="asympt_0.html#K2" title="ASYMPT_0:func.2">max</a> <font color="Maroon" title="c6">c</font>,2</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="power.html#K6" title="POWER:func.6">log</a> <font color="Maroon" title="c15">n</font>,<span class="p1">(<span class="default"><a href="asympt_0.html#K2" title="ASYMPT_0:func.2">max</a> <font color="Maroon" title="c6">c</font>,2</span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E6:41_1"><i><font color="Green" title="E33">A22</font></i></a>, <a class="txt" href="asympt_1.html#E9:41_1"><i><font color="Green" title="E36">A25</font></i></a>, <a class="txt" href="asympt_1.html#E5:41_1_4"><i><font color="Green" title="E42">A33</font></i></a>, <a class="txt" href="asympt_1.html#E36"><i><font color="Green" title="E29">Lm24</font></i></a></i>;<br/><a NAME="E10:41_1_4"/><i><font color="Green" title="E47">A38</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">t</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c15">n</font> &amp; <font color="Maroon" title="c8">t</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c15">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"><span class="p2">(<span class="default"><a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c15">n</font></span>)</span> )
 <b>by </b><i><a class="txt" href="asympt_1.html#E3:41_1"><i><font color="Green" title="E31">A20</font></i></a>, <a class="txt" href="asympt_1.html#E6:41_1_4"><i><font color="Green" title="E43">A34</font></i></a></i>;<br/><a NAME="E11:41_1_4"/><i><font color="Green" title="E48">A39</font></i>: 
<span class="p1">(<span class="default"><a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c15">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">n</font> <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c7">k</font>
 <b>by </b><i><a class="txt" href="asympt_1.html#E5:41_1"><i><font color="Green" title="E32">A21</font></i></a>, <a class="txt" href="asympt_1.html#E5:41_1_4"><i><font color="Green" title="E42">A33</font></i></a>, <a class="ref" href="asympt_1.html#D3" target="_self" title="ASYMPT_1:def.3">Def3</a></i>;<br/><a NAME="E12:41_1_4"/><b>then </b><i><font color="Green" title="E49">A40</font></i>: 
<span class="p1">(<span class="default"><a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c15">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#E5:41_1"><i><font color="Green" title="E32">A21</font></i></a>, <a class="txt" href="asympt_1.html#E5:41_1_4"><i><font color="Green" title="E42">A33</font></i></a>, <a class="ref" href="power.html#T39" title="POWER:th.39">POWER:39</a></i>;<br/><a NAME="E13:41_1_4"/>
<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"><span class="p2">(<span class="default"><a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K2" title="ASYMPT_0:func.2">max</a> <font color="Maroon" title="c6">c</font>,2</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"><a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c15">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E8:41_1"><i><font color="Green" title="E35">A24</font></i></a>, <a class="txt" href="asympt_1.html#E11:41_1_4"><i><font color="Green" title="E48">A39</font></i></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/><a NAME="E14:41_1_4"/><b>then </b>
<font color="Maroon" title="c8">t</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c15">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K2" title="ASYMPT_0:func.2">max</a> <font color="Maroon" title="c6">c</font>,2</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"><a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c15">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E10:41_1_4"><i><font color="Green" title="E47">A38</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E15:41_1_4"/><b>then </b><i><font color="Green" title="E50">A41</font></i>: 
 <a href="power.html#K6" title="POWER:func.6">log</a> <font color="Maroon" title="c15">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">t</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c15">n</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> <font color="Maroon" title="c15">n</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="asympt_0.html#K2" title="ASYMPT_0:func.2">max</a> <font color="Maroon" title="c6">c</font>,2</span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E7:41_1_4"><i><font color="Green" title="E44">A35</font></i></a>, <a class="txt" href="asympt_1.html#E10:41_1_4"><i><font color="Green" title="E47">A38</font></i></a>, <a class="ref" href="pre_ff.html#T12" title="PRE_FF:th.12">PRE_FF:12</a></i>;<br/><font color="Maroon" title="c8">t</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c15">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c15">n</font></span>)</span>
<b>by </b><i><a class="txt" href="asympt_1.html#E15:41_1"><i><font color="Green" title="E40">A29</font></i></a>, <a class="txt" href="asympt_1.html#E5:41_1_4"><i><font color="Green" title="E42">A33</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">n</font> <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> 1</span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c15">n</font></span>)</span>
<b>by </b><i><a class="txt" href="asympt_1.html#E5:41_1"><i><font color="Green" title="E32">A21</font></i></a>, <a class="txt" href="asympt_1.html#E5:41_1_4"><i><font color="Green" title="E42">A33</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">n</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c15">n</font></span>)</span>
<b>by </b><i><a class="ref" href="power.html#T30" title="POWER:th.30">POWER:30</a></i>
;<br/><b>then </b><i><font color="Green" title="E51">A42</font></i>:  <a href="power.html#K6" title="POWER:func.6">log</a> <font color="Maroon" title="c15">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">t</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c15">n</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c15">n</font></span>)</span> <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> <font color="Maroon" title="c15">n</font>,<font color="Maroon" title="c15">n</font></span>)</span>
<b>by </b><i><a class="txt" href="asympt_1.html#E8:41_1_4"><i><font color="Green" title="E45">A36</font></i></a>, <a class="ref" href="power.html#T63" title="POWER:th.63">POWER:63</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> 1
<b>by </b><i><a class="txt" href="asympt_1.html#E8:41_1_4"><i><font color="Green" title="E45">A36</font></i></a>, <a class="ref" href="power.html#T60" title="POWER:th.60">POWER:60</a></i>
;<br/><i><font color="Green" title="E52">A43</font></i>:  <a href="power.html#K6" title="POWER:func.6">log</a> <font color="Maroon" title="c15">n</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="asympt_0.html#K2" title="ASYMPT_0:func.2">max</a> <font color="Maroon" title="c6">c</font>,2</span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> <font color="Maroon" title="c15">n</font>,<span class="p2">(<span class="default"><a href="asympt_0.html#K2" title="ASYMPT_0:func.2">max</a> <font color="Maroon" title="c6">c</font>,2</span>)</span></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> <font color="Maroon" title="c15">n</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">n</font> <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c7">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="asympt_1.html#E8:41_1"><i><font color="Green" title="E35">A24</font></i></a>, <a class="txt" href="asympt_1.html#E8:41_1_4"><i><font color="Green" title="E45">A36</font></i></a>, <a class="txt" href="asympt_1.html#E11:41_1_4"><i><font color="Green" title="E48">A39</font></i></a>, <a class="txt" href="asympt_1.html#E12:41_1_4"><i><font color="Green" title="E49">A40</font></i></a>, <a class="ref" href="power.html#T61" title="POWER:th.61">POWER:61</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> <font color="Maroon" title="c15">n</font>,<span class="p2">(<span class="default"><a href="asympt_0.html#K2" title="ASYMPT_0:func.2">max</a> <font color="Maroon" title="c6">c</font>,2</span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> <font color="Maroon" title="c15">n</font>,<font color="Maroon" title="c15">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="asympt_1.html#E8:41_1_4"><i><font color="Green" title="E45">A36</font></i></a>, <a class="ref" href="power.html#T63" title="POWER:th.63">POWER:63</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> <font color="Maroon" title="c15">n</font>,<span class="p2">(<span class="default"><a href="asympt_0.html#K2" title="ASYMPT_0:func.2">max</a> <font color="Maroon" title="c6">c</font>,2</span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</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#E8:41_1_4"><i><font color="Green" title="E45">A36</font></i></a>, <a class="ref" href="power.html#T60" title="POWER:th.60">POWER:60</a></i>
;<br/><a NAME="E19:41_1_4"/><i><font color="Green" title="E53">A44</font></i>: 
<span class="p1">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> <font color="Maroon" title="c15">n</font>,<span class="p2">(<span class="default"><a href="asympt_0.html#K2" title="ASYMPT_0:func.2">max</a> <font color="Maroon" title="c6">c</font>,2</span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c7">k</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> <span class="p2">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c5">N0</font>,2</span>)</span>,<span class="p2">(<span class="default"><a href="asympt_0.html#K2" title="ASYMPT_0:func.2">max</a> <font color="Maroon" title="c6">c</font>,2</span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c7">k</font>
 <b>by </b><i><a class="txt" href="asympt_1.html#E9:41_1_4"><i><font color="Green" title="E46">A37</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E20:41_1_4"/>
<span class="p1">(<span class="default"><a href="asympt_1.html#K4" title="ASYMPT_1:func.4">seq_const</a> 1</span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c15">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/><b>hence </b><a NAME="E21:41_1_4"/>
<font color="Maroon" title="c14">g</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c15">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="c7">k</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> <span class="p3">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c5">N0</font>,2</span>)</span>,<span class="p3">(<span class="default"><a href="asympt_0.html#K2" title="ASYMPT_0:func.2">max</a> <font color="Maroon" title="c6">c</font>,2</span>)</span></span>)</span></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"><a href="asympt_1.html#K4" title="ASYMPT_1:func.4">seq_const</a> 1</span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c15">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E15:41_1_4"><i><font color="Green" title="E50">A41</font></i></a>, <a class="txt" href="asympt_1.html#E17:41_1_4"><i><font color="Green" title="E51">A42</font></i></a>, <a class="txt" href="asympt_1.html#E18:41_1_4"><i><font color="Green" title="E52">A43</font></i></a>, <a class="txt" href="asympt_1.html#E19:41_1_4"><i><font color="Green" title="E53">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="E22:41_1_4"/>
<font color="Maroon" title="c14">g</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c15">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="power.html#K6" title="POWER:func.6">log</a> <font color="Maroon" title="c15">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">t</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c15">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E13:41_1"><i><font color="Green" title="E38">A27</font></i></a>, <a class="txt" href="asympt_1.html#E5:41_1_4"><i><font color="Green" title="E42">A33</font></i></a></i>;<br/><a NAME="E23:41_1_4"/><b>then </b>
<font color="Maroon" title="c14">g</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c15">n</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> <font color="Maroon" title="c15">n</font>,1
 <b>by </b><i><a class="txt" href="asympt_1.html#E7:41_1_4"><i><font color="Green" title="E44">A35</font></i></a>, <a class="txt" href="asympt_1.html#E10:41_1_4"><i><font color="Green" title="E47">A38</font></i></a>, <a class="ref" href="pre_ff.html#T12" title="PRE_FF:th.12">PRE_FF:12</a></i>;<br/><b>hence </b><a NAME="E24:41_1_4"/>
<font color="Maroon" title="c14">g</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c15">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#E8:41_1_4"><i><font color="Green" title="E45">A36</font></i></a>, <a class="ref" href="power.html#T59" title="POWER:th.59">POWER:59</a></i>;<br/></div><b>end;</b></div><a NAME="E18:41_1"/><b>then </b>
<font color="Maroon" title="c14">g</font> <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#K4" title="ASYMPT_1:func.4">seq_const</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E14:41_1"><i><font color="Green" title="E39">A28</font></i></a></i>;<br/><b>hence </b><a NAME="E19:41_1"/>
<font color="Maroon" title="c1">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> <a href="asympt_0.html#K18" title="ASYMPT_0:func.18">to_power</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K9" title="ASYMPT_0:func.9">Big_Oh</a> <span class="p2">(<span class="default"><a href="asympt_1.html#K4" title="ASYMPT_1:func.4">seq_const</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E14:41_1"><i><font color="Green" title="E39">A28</font></i></a>, <a class="txt" href="asympt_1.html#E15:41_1"><i><font color="Green" title="E40">A29</font></i></a>, <a class="txt" href="asympt_1.html#E16:41_1"><i><font color="Green" title="E41">A32</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:41"/>
( <font color="Maroon" title="c1">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> <a href="asympt_0.html#K18" title="ASYMPT_0:func.18">to_power</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K9" title="ASYMPT_0:func.9">Big_Oh</a> <span class="p2">(<span class="default"><a href="asympt_1.html#K4" title="ASYMPT_1:func.4">seq_const</a> 1</span>)</span></span>)</span> iff  ex <font color="Olive" title="b1">N</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ex <font color="Olive" title="b2">c</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ex <font color="Olive" title="b3">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 <br/>( <font color="Olive" title="b2">c</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="b1">N</font> holds <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">h</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="c1">h</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="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> <font color="Olive" title="b3">k</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b4">n</font></span>)</span> ) ) ) )
 ;<br/>


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