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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:60"/><i><font color="Green">E39</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E2:60"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 1
 ;<br/>

<b>set </b><font color="Maroon">c<sub>2</sub></font> =  <a href="asympt_1.html#K2" target="_self">seq_logn</a> ;<br/>
<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="asympt_1.html#K3" target="_self">seq_n^</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> = <a href="asympt_1.html#K2" target="_self">seq_logn</a>  <a href="seq_1.html#K19">/"</a> <span class="p1">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>
<a NAME="E3:60"/><i><font color="Green">E41</font></i>: 
( <a href="asympt_1.html#K2" target="_self">seq_logn</a>  <a href="seq_1.html#K19">/"</a> <span class="p1">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="asympt_1.html#K2" target="_self">seq_logn</a>  <a href="seq_1.html#K19">/"</a> <span class="p2">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i><a class="txt" href="asympt_1.html#E1:60"><i><font color="Green">E39</font></i></a>, <a class="txt" href="asympt_1.html#E23"><i><font color="Green">Lemma20</font></i></a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="power.html#K6">log</a> 2,<span class="p1">(<span class="default">1 <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>
<a NAME="E4:60"/>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K3">+</a> 1
 
<b>by </b><i><a class="txt" href="asympt_1.html#E1:60"><i><font color="Green">E39</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E5:60"/><b>then </b>
 <a href="power.html#K6">log</a> 2,1 <a href="xxreal_0.html#R1">&lt;</a>  <a href="power.html#K6">log</a> 2,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K3">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="power.html#T65">POWER:65</a></i>;<br/>
<a NAME="E6:60"/><b>then </b><i><font color="Green">E42</font></i>: 
 <a href="power.html#K6">log</a> 2,<span class="p1">(<span class="default">1 <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="power.html#T59">POWER:59</a></i>;<br/>
<a NAME="E7:60"/><b>then </b>
<span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,<span class="p2">(<span class="default">1 <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 16</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,<span class="p2">(<span class="default">1 <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E9:60"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>6</sub></font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="asympt_1.html#K2" target="_self">seq_logn</a>  <a href="seq_1.html#K19">/"</a> <span class="p4">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 0</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,<span class="p2">(<span class="default">1 <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 16
 <b>by </b><i><a class="txt" href="asympt_1.html#E3:60"><i><font color="Green">E41</font></i></a>, <a class="ref" href="seq_2.html#D7">SEQ_2:def 7</a></i>;<br/>
<a NAME="E10:60"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="power.html#K6">log</a> 2,<span class="p3">(<span class="default">1 <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">8 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Olive">b<sub>2</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 8 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Olive">b<sub>2</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">c<sub>7</sub></font> =  <a href="asympt_0.html#K4">max</a> 2,<font color="Maroon">c<sub>6</sub></font>;<br/>
<a NAME="E1:60_1"/><i><font color="Green">E44</font></i>: 
(  <a href="asympt_0.html#K4">max</a> 2,<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 &amp;  <a href="asympt_0.html#K4">max</a> 2,<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">c<sub>8</sub></font> =  <a href="asympt_0.html#K4">max</a> 2,<font color="Maroon">c<sub>6</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:60_1_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><a NAME="E2:60_1_1"/><b>then </b><i><font color="Green">E46</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 &amp; <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="asympt_1.html#E1:60_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E3:60_1_1"/><b>then </b><i><font color="Green">E47</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="asympt_1.html#K2" target="_self">seq_logn</a>  <a href="seq_1.html#K19">/"</a> <span class="p4">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 0</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,<span class="p2">(<span class="default">1 <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 16
 <b>by </b><i><a class="txt" href="asympt_1.html#E9:60"><i><font color="Green">E43</font></i></a></i>;<br/><a NAME="E4:60_1_1"/>
 <a href="power.html#K6">log</a> 2,2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="power.html#K6">log</a> 2,<font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="asympt_1.html#E2:60_1_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="pre_ff.html#T12">PRE_FF:12</a></i>;<br/><a NAME="E5:60_1_1"/><b>then </b><i><font color="Green">E48</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="power.html#K6">log</a> 2,<font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="power.html#T60">POWER:60</a></i>;<br/><i><font color="Green">E49</font></i>: <span class="p1">(<span class="default"><a href="asympt_1.html#K2" target="_self">seq_logn</a>  <a href="seq_1.html#K19">/"</a> <span class="p2">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>9</sub></font> = 
<span class="p1">(<span class="default"><a href="asympt_1.html#K2" target="_self">seq_logn</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="asympt_1.html#E10"><i><font color="Green">Lemma7</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="asympt_1.html#E1:60_1"><i><font color="Green">E44</font></i></a>, <a class="txt" href="asympt_1.html#E1:60_1_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="asympt_1.html#D2" target="_self">Def2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="asympt_1.html#E1:60_1"><i><font color="Green">E44</font></i></a>, <a class="txt" href="asympt_1.html#E1:60_1_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="asympt_1.html#D3" target="_self">Def3</a></i>
;<br/><a NAME="E7:60_1_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="asympt_1.html#E1:60_1"><i><font color="Green">E44</font></i></a>, <a class="txt" href="asympt_1.html#E1:60_1_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="power.html#T39">POWER:39</a></i>;<br/><a NAME="E8:60_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K2">"</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T124">XREAL_1:124</a></i>;<br/><a NAME="E9:60_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E5:60_1_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E10:60_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="asympt_1.html#K2" target="_self">seq_logn</a>  <a href="seq_1.html#K19">/"</a> <span class="p2">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="asympt_1.html#E6:60_1_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="xcmplx_0.html#D9">XCMPLX_0:def 9</a></i>;<br/><a NAME="E11:60_1_1"/><b>then </b><i><font color="Green">E51</font></i>: 
<span class="p1">(<span class="default"><a href="asympt_1.html#K2" target="_self">seq_logn</a>  <a href="seq_1.html#K19">/"</a> <span class="p2">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,<span class="p2">(<span class="default">1 <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 16
 <b>by </b><i><a class="txt" href="asympt_1.html#E3:60_1_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><a NAME="E12:60_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 <b>by </b><i><a class="txt" href="asympt_1.html#E2:60_1_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E13:60_1_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="series_1.html#K3">to_power</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>9</sub></font> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="asympt_1.html#E2:60"><i><font color="Green">E40</font></i></a>, <a class="ref" href="power.html#T44">POWER:44</a></i>;<br/><a NAME="E14:60_1_1"/><b>then </b>
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="series_1.html#K3">to_power</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E7:60_1_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="real_2.html#T151">REAL_2:151</a></i>;<br/><a NAME="E15:60_1_1"/><b>then </b>
1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="power.html#T30">POWER:30</a></i>;<br/><a NAME="E16:60_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E5:60_1_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E17:60_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T100">XCMPLX_1:100</a></i>;<br/><a NAME="E18:60_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="asympt_1.html#K2" target="_self">seq_logn</a>  <a href="seq_1.html#K19">/"</a> <span class="p2">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="asympt_1.html#E6:60_1_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="xcmplx_1.html#T100">XCMPLX_1:100</a></i>;<br/><a NAME="E19:60_1_1"/><b>then </b><i><font color="Green">E52</font></i>: 
<span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,<span class="p2">(<span class="default">1 <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 16
 <b>by </b><i><a class="txt" href="asympt_1.html#E11:60_1_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E20:60_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="real_1.html#K2">"</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="asympt_1.html#E1:60_1"><i><font color="Green">E44</font></i></a>, <a class="txt" href="asympt_1.html#E1:60_1_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="xreal_1.html#T124">XREAL_1:124</a></i>;<br/><a NAME="E21:60_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="real_1.html#K2">"</a> </span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E5:60_1_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E22:60_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="xcmplx_0.html#D9">XCMPLX_0:def 9</a></i>;<br/><a NAME="E23:60_1_1"/><b>then </b>
1 <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">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <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,<span class="p3">(<span class="default">1 <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 16</span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E19:60_1_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="real_2.html#T151">REAL_2:151</a></i>;<br/><a NAME="E24:60_1_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <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,<span class="p3">(<span class="default">1 <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 16</span>)</span>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T57">XCMPLX_1:57</a></i>;<br/><a NAME="E25:60_1_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 16 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,<span class="p2">(<span class="default">1 <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T57">XCMPLX_1:57</a></i>;<br/><a NAME="E26:60_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,<span class="p2">(<span class="default">1 <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default">16 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><a href="power.html#K6">log</a> 2,<span class="p3">(<span class="default">1 <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,<span class="p2">(<span class="default">1 <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E6:60"><i><font color="Green">E42</font></i></a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E27:60_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,<span class="p2">(<span class="default">1 <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 16
 <b>by </b><i><a class="txt" href="asympt_1.html#E6:60"><i><font color="Green">E42</font></i></a>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/><a NAME="E28:60_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="power.html#K6">log</a> 2,<span class="p3">(<span class="default">1 <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 16 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">c<sub>9</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E5:60_1_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E29:60_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,<span class="p2">(<span class="default">1 <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">c<sub>9</sub></font></span>)</span></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">c<sub>9</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 16 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">c<sub>9</sub></font></span>)</span>
 ;<br/><a NAME="E30:60_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,<span class="p2">(<span class="default">1 <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default">8 <a href="nat_1.html#K1">+</a> 8</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">c<sub>9</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E5:60_1_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/><a NAME="E31:60_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="power.html#K6">log</a> 2,<span class="p3">(<span class="default">1 <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">8 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">c<sub>9</sub></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">8 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default">8 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">8 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><b>hence </b><a NAME="E32:60_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="power.html#K6">log</a> 2,<span class="p3">(<span class="default">1 <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">8 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 8 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">c<sub>9</sub></font></span>)</span>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:60_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="power.html#K6">log</a> 2,<span class="p3">(<span class="default">1 <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">8 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Olive">b<sub>2</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 8 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Olive">b<sub>2</sub></font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E11:60"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="power.html#K6">log</a> 2,<span class="p3">(<span class="default">1 <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">8 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Olive">b<sub>2</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 8 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Olive">b<sub>2</sub></font></span>)</span>
 ;<br/>


</div>
