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

<b>let </b><font color="Maroon">a</font> be  <a href="xxreal_0.html#V2">positive</a> <a href="real_1.html#NM1">Real</a>, <font color="Maroon">b</font> be  <a href="xxreal_0.html#V2">positive</a> <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:8"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b</font>
 ;<br/>

<b>set </b><font color="Maroon">f</font> =  <a href="asympt_1.html#K1" target="_self">seq_a^</a> <font color="Maroon">b</font>,1,0;<br/>
<b>set </b><font color="Maroon">g</font> =  <a href="asympt_1.html#K1" target="_self">seq_a^</a> <font color="Maroon">a</font>,1,0;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:8_1"/>
 <a href="asympt_1.html#K1" target="_self">seq_a^</a> <font color="Maroon">b</font>,1,0 <a href="hidden.html#R2">in</a>  <a href="asympt_0.html#K9">Big_Oh</a> <span class="p1">(<span class="default"><a href="asympt_1.html#K1" target="_self">seq_a^</a> <font color="Maroon">a</font>,1,0</span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">s</font> being    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <a href="numbers.html#K5">NAT</a> ,<a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E3:8_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="asympt_1.html#K1" target="_self">seq_a^</a> <font color="Maroon">b</font>,1,0
 <b>and </b><br/><a NAME="E4:8_1"/><i><font color="Green">E51</font></i>: 
 ex <font color="Olive">c</font> being  <a href="real_1.html#NM1">Real</a> ex <font color="Olive">N</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">c</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">N</font> holds <br/>( <font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">c</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="asympt_1.html#K1" target="_self">seq_a^</a> <font color="Maroon">a</font>,1,0</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> &amp; <font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) ) )
 ;<br/><b>consider </b><font color="Maroon">c</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">N</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:8_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>and </b><br/><a NAME="E7:8_1"/><i><font color="Green">E59</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">N</font> holds <br/>( <font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="asympt_1.html#K1" target="_self">seq_a^</a> <font color="Maroon">a</font>,1,0</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> &amp; <font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 <b>by </b><i/>;<br/><b>set </b><font color="Maroon">d</font> = <span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">b</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">a</font></span>)</span>;<br/><a NAME="E8:8_1"/>
 <a href="power.html#K6">log</a> 2,<font color="Maroon">b</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">a</font></span>)</span> <a href="real_1.html#K3">+</a> 0
 <b>by </b><i>, <a class="ref" href="power.html#T65">POWER:65</a></i>;<br/><a NAME="E9:8_1"/><b>then </b><i><font color="Green">E60</font></i>: 
<span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">b</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/><b>set </b><font color="Maroon">N0</font> = <span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">c</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">b</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></span><a href="int_1.html#K2">\]</a></span>;<br/><b>set </b><font color="Maroon">N1</font> =  <a href="xxreal_0.html#K4">max</a> <font color="Maroon">N</font>,<span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">c</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">b</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></span><a href="int_1.html#K2">\]</a></span>;<br/><a NAME="E10:8_1"/><i><font color="Green">E61</font></i>: 
(  <a href="xxreal_0.html#K4">max</a> <font color="Maroon">N</font>,<span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">c</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">b</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></span><a href="int_1.html#K2">\]</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">N</font> &amp;  <a href="xxreal_0.html#K4">max</a> <font color="Maroon">N</font>,<span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">c</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">b</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></span><a href="int_1.html#K2">\]</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">c</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">b</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></span><a href="int_1.html#K2">\]</a></span> )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/><a NAME="E11:8_1"/>
(  <a href="xxreal_0.html#K4">max</a> <font color="Maroon">N</font>,<span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">c</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">b</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></span><a href="int_1.html#K2">\]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">N</font> or  <a href="xxreal_0.html#K4">max</a> <font color="Maroon">N</font>,<span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">c</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">b</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></span><a href="int_1.html#K2">\]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">c</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">b</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></span><a href="int_1.html#K2">\]</a></span> )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T16">XXREAL_0:16</a></i>;<br/><b>then reconsider </b><font color="Maroon">N1</font> =  <a href="xxreal_0.html#K4">max</a> <font color="Maroon">N</font>,<span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">c</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">b</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></span><a href="int_1.html#K2">\]</a></span> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, <a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/><b>set </b><font color="Maroon">n</font> = <font color="Maroon">N1</font> <a href="nat_1.html#K1">+</a> 1;<br/><b>set </b><font color="Maroon">e</font> = 2 <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">N1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">a</font></span>)</span></span>)</span>;<br/><a NAME="E13:8_1"/><i><font color="Green">E62</font></i>: 
2 <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">N1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">a</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="power.html#T39">POWER:39</a></i>;<br/><a NAME="E14:8_1"/>
<font color="Maroon">N1</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">N1</font> <a href="nat_1.html#K1">+</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E15:8_1"/><b>then </b><i><font color="Green">E63</font></i>: 
( <font color="Maroon">N1</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">N</font> &amp; <font color="Maroon">N1</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">c</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">b</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></span><a href="int_1.html#K2">\]</a></span> )
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E16:8_1"/>
<span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">c</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">b</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></span><a href="int_1.html#K2">\]</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">c</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">b</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">a</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="int_1.html#D5">INT_1:def 5</a></i>;<br/><a NAME="E17:8_1"/><b>then </b>
<font color="Maroon">N1</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">c</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">b</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">a</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E18:8_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">N1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">b</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">a</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">c</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">b</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">b</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">a</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E19:8_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">N1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">b</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">a</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="power.html#K6">log</a> 2,<font color="Maroon">c</font>
 <b>by </b><i>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/><a NAME="E20:8_1"/><b>then </b>
2 <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">N1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">b</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 2 <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">c</font></span>)</span>
 <b>by </b><i><a class="ref" href="power.html#T44">POWER:44</a></i>;<br/><a NAME="E21:8_1"/><b>then </b>
2 <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">N1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">b</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">N1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c</font>
 <b>by </b><i>, <a class="ref" href="power.html#D3">POWER:def 3</a></i>;<br/><a NAME="E22:8_1"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">N1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">b</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">N1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c</font>
 <b>by </b><i><a class="ref" href="power.html#T34">POWER:34</a></i>;<br/><a NAME="E23:8_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="power.html#K4">to_power</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">N1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">b</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="power.html#K4">to_power</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">N1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">N1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">N1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E24:8_1"/><b>then </b>
2 <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">N1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">b</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">N1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/><a NAME="E25:8_1"/><b>then </b>
<font color="Maroon">b</font> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">N1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">N1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E26:8_1"/><b>then </b><i><font color="Green">E64</font></i>: 
<font color="Maroon">b</font> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">N1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">N1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E27:8_1"/>
<span class="p1">(<span class="default"><a href="asympt_1.html#K1" target="_self">seq_a^</a> <font color="Maroon">b</font>,1,0</span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">N1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="asympt_1.html#K1" target="_self">seq_a^</a> <font color="Maroon">a</font>,1,0</span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">N1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, , </i>;<br/><a NAME="E28:8_1"/><b>then </b>
<font color="Maroon">b</font> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><font color="Maroon">N1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 0</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="asympt_1.html#K1" target="_self">seq_a^</a> <font color="Maroon">a</font>,1,0</span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">N1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E29:8_1"/>
contradiction
 <b>by </b><i>, </i>;<br/></div>
<b>end;</b></div>


</div>
