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

<b>let </b><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font> be  <a href="asympt_0.html#V1" title="ASYMPT_0:attr.1">logbase</a> <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>let </b><font color="Maroon" title="c3">f</font>, <font color="Maroon" title="c4">g</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:4"/><i><font color="Green" title="E3">A1</font></i>: 
<font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <b>and </b><br/><a NAME="E2:4"/><i><font color="Green" title="E4">A2</font></i>: 
<font color="Maroon" title="c2">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <b>and </b><br/><a NAME="E3:4"/><i><font color="Green" title="E5">A3</font></i>: 
( <font color="Maroon" title="c3">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; ( for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 holds <br/><font color="Maroon" title="c3">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">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="c1">a</font>,<font color="Olive" title="b1">n</font> ) )
 <b>and </b><br/><a NAME="E4:4"/><i><font color="Green" title="E6">A4</font></i>: 
( <font color="Maroon" title="c4">g</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; ( for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 holds <br/><font color="Maroon" title="c4">g</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="power.html#K6" title="POWER:func.6">log</a> <font color="Maroon" title="c2">b</font>,<font color="Olive" title="b1">n</font> ) )
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c5">f</font> = <font color="Maroon" title="c3">f</font> as  <a href="asympt_0.html#V4" title="ASYMPT_0:attr.4">eventually-positive</a> <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a> <b>by </b><i><a class="txt" href="asympt_1.html#E1:4"><i><font color="Green" title="E3">A1</font></i></a>, <a class="txt" href="asympt_1.html#E3:4"><i><font color="Green" title="E5">A3</font></i></a>, <a class="txt" href="asympt_1.html#E3"><i><font color="Green" title="E2">Lm2</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">g</font> = <font color="Maroon" title="c4">g</font> as  <a href="asympt_0.html#V4" title="ASYMPT_0:attr.4">eventually-positive</a> <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a> <b>by </b><i><a class="txt" href="asympt_1.html#E2:4"><i><font color="Green" title="E4">A2</font></i></a>, <a class="txt" href="asympt_1.html#E4:4"><i><font color="Green" title="E6">A4</font></i></a>, <a class="txt" href="asympt_1.html#E3"><i><font color="Green" title="E2">Lm2</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">f</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c6">g</font>
;<br/>

<a NAME="E7:4"/><i><font color="Green" title="E7">A5</font></i>: 
( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp; <font color="Maroon" title="c1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 
<b>by </b><i><a class="ref" href="asympt_0.html#D3" title="ASYMPT_0:def.3">ASYMPT_0:def 3</a></i>;<br/>
<a NAME="E8:4"/><i><font color="Green" title="E8">A6</font></i>: 
( <font color="Maroon" title="c2">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp; <font color="Maroon" title="c2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 
<b>by </b><i><a class="ref" href="asympt_0.html#D3" title="ASYMPT_0:def.3">ASYMPT_0:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:4_1_1"/>
<font color="Maroon" title="c7">x</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> <font color="Maroon" title="c5">f</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c8">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="E3:4_1_1"/><i><font color="Green" title="E9">A7</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">t</font>
 <b>and </b><br/><a NAME="E4:4_1_1"/><i><font color="Green" title="E10">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="c8">t</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">c</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b3">n</font></span>)</span> &amp; <font color="Maroon" title="c8">t</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b3">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 ) ) )
 ;<br/><b>consider </b><font color="Maroon" title="c9">c</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="E6:4_1_1"/><i><font color="Green" title="E11">A9</font></i>: 
<font color="Maroon" title="c9">c</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>and </b><br/><a NAME="E7:4_1_1"/><i><font color="Green" title="E12">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="c10">N</font> holds <br/>( <font color="Maroon" title="c8">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="c9">c</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font></span>)</span> &amp; <font color="Maroon" title="c8">t</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 )
 <b>by </b><i><a class="txt" href="asympt_1.html#E4:4_1_1"><i><font color="Green" title="E10">A8</font></i></a></i>;<br/><a NAME="E8:4_1_1"/>
 <a href="power.html#K6" title="POWER:func.6">log</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="power.html#K6" title="POWER:func.6">log</a> <font color="Maroon" title="c1">a</font>,1
 <b>by </b><i><a class="txt" href="asympt_1.html#E1:4"><i><font color="Green" title="E3">A1</font></i></a>, <a class="txt" href="asympt_1.html#E2:4"><i><font color="Green" title="E4">A2</font></i></a>, <a class="ref" href="power.html#T65" title="POWER:th.65">POWER:65</a></i>;<br/><a NAME="E9:4_1_1"/><b>then </b>
 <a href="power.html#K6" title="POWER:func.6">log</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</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#E7:4"><i><font color="Green" title="E7">A5</font></i></a>, <a class="ref" href="power.html#T59" title="POWER:th.59">POWER:59</a></i>;<br/><a NAME="E10:4_1_1"/><b>then </b><i><font color="Green" title="E13">A11</font></i>: 
<font color="Maroon" title="c9">c</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> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c9">c</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> 0
 <b>by </b><i><a class="txt" href="asympt_1.html#E6:4_1_1"><i><font color="Green" title="E11">A9</font></i></a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_1_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon" title="c11">N1</font> = <font color="Maroon" title="c10">N</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1;<br/><b>let </b><font color="Maroon" title="c12">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="E1:4_1_1_1"/><i><font color="Green" title="E14">A12</font></i>: 
<font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c11">N1</font>
 ;<br/><a NAME="E2:4_1_1_1"/>
<font color="Maroon" title="c10">N</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c10">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#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><a NAME="E3:4_1_1_1"/><b>then </b><i><font color="Green" title="E15">A13</font></i>: 
<font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c10">N</font>
 <b>by </b><i><a class="txt" href="asympt_1.html#E1:4_1_1_1"><i><font color="Green" title="E14">A12</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E4:4_1_1_1"/><b>then </b><i><font color="Green" title="E16">A14</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="c12">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">c</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c12">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E7:4_1_1"><i><font color="Green" title="E12">A10</font></i></a></i>;<br/><font color="Maroon" title="c5">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c12">n</font> = 
 <a href="power.html#K6" title="POWER:func.6">log</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c12">n</font>
<b>by </b><i><a class="txt" href="asympt_1.html#E3:4"><i><font color="Green" title="E5">A3</font></i></a>, <a class="txt" href="asympt_1.html#E1:4_1_1_1"><i><font color="Green" title="E14">A12</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</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="c2">b</font>,<font color="Maroon" title="c12">n</font></span>)</span>
<b>by </b><i><a class="txt" href="asympt_1.html#E7:4"><i><font color="Green" title="E7">A5</font></i></a>, <a class="txt" href="asympt_1.html#E8:4"><i><font color="Green" title="E8">A6</font></i></a>, <a class="txt" href="asympt_1.html#E1:4_1_1_1"><i><font color="Green" title="E14">A12</font></i></a>, <a class="ref" href="power.html#T64" title="POWER:th.64">POWER:64</a></i>
;<br/><a NAME="E6:4_1_1_1"/><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="c12">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="c9">c</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="c1">a</font>,<font color="Maroon" title="c2">b</font></span>)</span></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="c2">b</font>,<font color="Maroon" title="c12">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E4:4_1_1_1"><i><font color="Green" title="E16">A14</font></i></a></i>;<br/><b>hence </b><a NAME="E7:4_1_1_1"/>
<font color="Maroon" title="c8">t</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c12">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="c9">c</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="c1">a</font>,<font color="Maroon" title="c2">b</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c12">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E4:4"><i><font color="Green" title="E6">A4</font></i></a>, <a class="txt" href="asympt_1.html#E1:4_1_1_1"><i><font color="Green" title="E14">A12</font></i></a></i>;<br/><b>thus </b><a NAME="E8:4_1_1_1"/>
<font color="Maroon" title="c8">t</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c12">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#E7:4_1_1"><i><font color="Green" title="E12">A10</font></i></a>, <a class="txt" href="asympt_1.html#E3:4_1_1_1"><i><font color="Green" title="E15">A13</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E12:4_1_1"/>
<font color="Maroon" title="c7">x</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> <font color="Maroon" title="c6">g</font>
 <b>by </b><i><a class="txt" href="asympt_1.html#E3:4_1_1"><i><font color="Green" title="E9">A7</font></i></a>, <a class="txt" href="asympt_1.html#E10:4_1_1"><i><font color="Green" title="E13">A11</font></i></a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:4_1"/>
<font color="Maroon" title="c7">x</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> <font color="Maroon" title="c6">g</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c8">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="E4:4_1"/><i><font color="Green" title="E9">A15</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">t</font>
 <b>and </b><br/><a NAME="E5:4_1"/><i><font color="Green" title="E10">A16</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="c8">t</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">c</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b3">n</font></span>)</span> &amp; <font color="Maroon" title="c8">t</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b3">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 ) ) )
 ;<br/><b>consider </b><font color="Maroon" title="c9">c</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="E7:4_1"/><i><font color="Green" title="E11">A17</font></i>: 
<font color="Maroon" title="c9">c</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>and </b><br/><a NAME="E8:4_1"/><i><font color="Green" title="E12">A18</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="c8">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="c9">c</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font></span>)</span> &amp; <font color="Maroon" title="c8">t</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 )
 <b>by </b><i><a class="txt" href="asympt_1.html#E5:4_1"><i><font color="Green" title="E10">A16</font></i></a></i>;<br/><a NAME="E9:4_1"/>
 <a href="power.html#K6" title="POWER:func.6">log</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="power.html#K6" title="POWER:func.6">log</a> <font color="Maroon" title="c2">b</font>,1
 <b>by </b><i><a class="txt" href="asympt_1.html#E1:4"><i><font color="Green" title="E3">A1</font></i></a>, <a class="txt" href="asympt_1.html#E2:4"><i><font color="Green" title="E4">A2</font></i></a>, <a class="ref" href="power.html#T65" title="POWER:th.65">POWER:65</a></i>;<br/><a NAME="E10:4_1"/><b>then </b>
 <a href="power.html#K6" title="POWER:func.6">log</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c1">a</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#E8:4"><i><font color="Green" title="E8">A6</font></i></a>, <a class="ref" href="power.html#T59" title="POWER:th.59">POWER:59</a></i>;<br/><a NAME="E11:4_1"/><b>then </b><i><font color="Green" title="E13">A19</font></i>: 
<font color="Maroon" title="c9">c</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> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c1">a</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c9">c</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> 0
 <b>by </b><i><a class="txt" href="asympt_1.html#E7:4_1"><i><font color="Green" title="E11">A17</font></i></a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_2"><b>now </b></a><div class="add"><b>take </b><font color="Maroon" title="c11">N1</font> = <font color="Maroon" title="c10">N</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1;<br/><b>let </b><font color="Maroon" title="c12">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="E1:4_1_2"/><i><font color="Green" title="E14">A20</font></i>: 
<font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c11">N1</font>
 ;<br/><a NAME="E2:4_1_2"/>
<font color="Maroon" title="c10">N</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c10">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#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><a NAME="E3:4_1_2"/><b>then </b><i><font color="Green" title="E15">A21</font></i>: 
<font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c10">N</font>
 <b>by </b><i><a class="txt" href="asympt_1.html#E1:4_1_2"><i><font color="Green" title="E14">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="E4:4_1_2"/><b>then </b><i><font color="Green" title="E16">A22</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="c12">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">c</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c12">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E8:4_1"><i><font color="Green" title="E12">A18</font></i></a></i>;<br/><font color="Maroon" title="c6">g</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c12">n</font> = 
 <a href="power.html#K6" title="POWER:func.6">log</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c12">n</font>
<b>by </b><i><a class="txt" href="asympt_1.html#E4:4"><i><font color="Green" title="E6">A4</font></i></a>, <a class="txt" href="asympt_1.html#E1:4_1_2"><i><font color="Green" title="E14">A20</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c1">a</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="c1">a</font>,<font color="Maroon" title="c12">n</font></span>)</span>
<b>by </b><i><a class="txt" href="asympt_1.html#E7:4"><i><font color="Green" title="E7">A5</font></i></a>, <a class="txt" href="asympt_1.html#E8:4"><i><font color="Green" title="E8">A6</font></i></a>, <a class="txt" href="asympt_1.html#E1:4_1_2"><i><font color="Green" title="E14">A20</font></i></a>, <a class="ref" href="power.html#T64" title="POWER:th.64">POWER:64</a></i>
;<br/><a NAME="E6:4_1_2"/><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="c12">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="c9">c</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="c2">b</font>,<font color="Maroon" title="c1">a</font></span>)</span></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="c1">a</font>,<font color="Maroon" title="c12">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E4:4_1_2"><i><font color="Green" title="E16">A22</font></i></a></i>;<br/><b>hence </b><a NAME="E7:4_1_2"/>
<font color="Maroon" title="c8">t</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c12">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="c9">c</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="c2">b</font>,<font color="Maroon" title="c1">a</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c12">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E3:4"><i><font color="Green" title="E5">A3</font></i></a>, <a class="txt" href="asympt_1.html#E1:4_1_2"><i><font color="Green" title="E14">A20</font></i></a></i>;<br/><b>thus </b><a NAME="E8:4_1_2"/>
<font color="Maroon" title="c8">t</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c12">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#E8:4_1"><i><font color="Green" title="E12">A18</font></i></a>, <a class="txt" href="asympt_1.html#E3:4_1_2"><i><font color="Green" title="E15">A21</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E13:4_1"/>
<font color="Maroon" title="c7">x</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> <font color="Maroon" title="c5">f</font>
 <b>by </b><i><a class="txt" href="asympt_1.html#E4:4_1"><i><font color="Green" title="E9">A15</font></i></a>, <a class="txt" href="asympt_1.html#E11:4_1"><i><font color="Green" title="E13">A19</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:4"/>
( <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> &amp; <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> &amp;  <a href="asympt_0.html#K9" title="ASYMPT_0:func.9">Big_Oh</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="asympt_0.html#K9" title="ASYMPT_0:func.9">Big_Oh</a> <font color="Maroon" title="c6">g</font> )
 <b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>


</div>
