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

<a NAME="E1:72"/>
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/><font color="Olive">b<sub>2</sub></font> <a href="real_1.html#K6">/</a> 6 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_1_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">c<sub>1</sub></font> = 7;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:72_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><a NAME="E2:72_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 6
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E3:72_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K6">/</a> 6 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 6 <a href="real_1.html#K6">/</a> 6
 <b>by </b><i><a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/><b>hence </b><a NAME="E4:72_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K6">/</a> 6 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:72_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/><font color="Olive">b<sub>2</sub></font> <a href="real_1.html#K6">/</a> 6 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 ;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">c<sub>1</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:72"/><i><font color="Green">E46</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>1</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="real_1.html#K6">/</a> 6 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 ;<br/>
<a NAME="E4:72"/>
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/><font color="Olive">b<sub>2</sub></font> <a href="real_1.html#K6">/</a> 3 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 2 <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="72_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_2_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">c<sub>2</sub></font> = 30;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:72_2_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><a NAME="E2:72_2_1"/><b>then </b><i><font color="Green">E48</font></i>: 
2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>3</sub></font> <a href="series_1.html#K3">to_power</a> 6
 <b>by </b><i><a class="txt" href="asympt_1.html#E66"><i><font color="Green">Lemma43</font></i></a></i>;<br/><a NAME="E3:72_2_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="series_1.html#K3">to_power</a> 6 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="asympt_1.html#E1:72_2_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="power.html#T39">POWER:39</a></i>;<br/><a NAME="E4:72_2_1"/><b>then </b>
 <a href="power.html#K6">log</a> 2,<span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="power.html#K6">log</a> 2,<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="series_1.html#K3">to_power</a> 6</span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E2:72_2_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="power.html#T65">POWER:65</a></i>;<br/><a NAME="E5:72_2_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,2</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="power.html#K6">log</a> 2,<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="series_1.html#K3">to_power</a> 6</span>)</span>
 <b>by </b><i><a class="ref" href="power.html#T63">POWER:63</a></i>;<br/><a NAME="E6:72_2_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K2">*</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="power.html#K6">log</a> 2,<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="series_1.html#K3">to_power</a> 6</span>)</span>
 <b>by </b><i><a class="ref" href="power.html#T60">POWER:60</a></i>;<br/><a NAME="E7:72_2_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K2">*</a> 2</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>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E1:72_2_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="power.html#T63">POWER:63</a></i>;<br/><a NAME="E8:72_2_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 3 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <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>3</sub></font></span>)</span></span>)</span>
 ;<br/><b>hence </b><a NAME="E9:72_2_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K6">/</a> 3 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 2 <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>3</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="real_2.html#T178">REAL_2:178</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:72_2"/>
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/><font color="Olive">b<sub>2</sub></font> <a href="real_1.html#K6">/</a> 3 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 2 <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>then consider </b><font color="Maroon">c<sub>2</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E6:72"/><i><font color="Green">E47</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>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="real_1.html#K6">/</a> 3 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 2 <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>1</sub></font></span>)</span>
 ;<br/>
<a NAME="E7:72"/>
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/><font color="Olive">b<sub>2</sub></font> <a href="real_1.html#K6">/</a> 2 <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="Olive">b<sub>2</sub></font></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="Olive">b<sub>2</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_3"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> = 2 <a href="series_1.html#K3">to_power</a> 10 as   <a href="nat_1.html#NM1">Nat</a> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_3_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:72_3_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="power.html#K6">log</a> 2,<font color="Maroon">c<sub>5</sub></font>;<br/><a NAME="E2:72_3_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="asympt_1.html#E1:72_3_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="power.html#T39">POWER:39</a></i>;<br/><a NAME="E3:72_3_1"/><i><font color="Green">E50</font></i>: 
2 <a href="series_1.html#K3">to_power</a> 9 <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="E4:72_3_1"/>
2 <a href="series_1.html#K3">to_power</a> 10 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 2 <a href="series_1.html#K3">to_power</a> 9
 <b>by </b><i><a class="ref" href="power.html#T44">POWER:44</a></i>;<br/><a NAME="E5:72_3_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 2 <a href="series_1.html#K3">to_power</a> 9
 <b>by </b><i><a class="txt" href="asympt_1.html#E1:72_3_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E6:72_3_1"/><b>then </b>
 <a href="power.html#K6">log</a> 2,<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="power.html#K6">log</a> 2,<span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> 9</span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E3:72_3_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="power.html#T65">POWER:65</a></i>;<br/><a NAME="E7:72_3_1"/><b>then </b>
 <a href="power.html#K6">log</a> 2,<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 9 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,2</span>)</span>
 <b>by </b><i><a class="ref" href="power.html#T63">POWER:63</a></i>;<br/><a NAME="E8:72_3_1"/><b>then </b><i><font color="Green">E51</font></i>: 
 <a href="power.html#K6">log</a> 2,<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 9 <a href="nat_1.html#K2">*</a> 1
 <b>by </b><i><a class="ref" href="power.html#T60">POWER:60</a></i>;<br/><a NAME="E9:72_3_1"/><b>then </b><i><font color="Green">E52</font></i>: 
2 <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>5</sub></font></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"><a href="power.html#K6">log</a> 2,<font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E10:72_3_1"/><b>then </b><i><font color="Green">E53</font></i>: 
<span class="p1">(<span class="default">2 <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>5</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <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>5</sub></font></span>)</span></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">2 <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>5</sub></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="E11:72_3_1"/>
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<sub>5</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default">2 <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>5</sub></font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> 
 <b>by </b><i><a class="txt" href="asympt_1.html#E8:72_3_1"><i><font color="Green">E51</font></i></a>, <a class="txt" href="asympt_1.html#E67"><i><font color="Green">Lemma44</font></i></a></i>;<br/><a NAME="E12:72_3_1"/><b>then </b>
 <a href="power.html#K6">log</a> 2,<span class="p1">(<span class="default">2 <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="power.html#K6">log</a> 2,<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <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>5</sub></font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E10:72_3_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="power.html#T65">POWER:65</a></i>;<br/><a NAME="E13:72_3_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,2</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="power.html#K6">log</a> 2,<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <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>5</sub></font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>
 <b>by </b><i><a class="ref" href="power.html#T63">POWER:63</a></i>;<br/><a NAME="E14:72_3_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="real_1.html#K4">*</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="power.html#K6">log</a> 2,<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <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>5</sub></font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>
 <b>by </b><i><a class="ref" href="power.html#T60">POWER:60</a></i>;<br/><a NAME="E15:72_3_1"/><b>then </b>
 <a href="power.html#K6">log</a> 2,<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="power.html#K6">log</a> 2,<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <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>5</sub></font></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="power.html#T53">POWER:53</a></i>;<br/><a NAME="E16:72_3_1"/><b>then </b>
 <a href="power.html#K6">log</a> 2,<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 2 <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">2 <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>5</sub></font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E9:72_3_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="power.html#T63">POWER:63</a></i>;<br/><a NAME="E17:72_3_1"/><b>then </b>
 <a href="power.html#K6">log</a> 2,<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 2 <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"><a href="power.html#K6">log</a> 2,<span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="series_1.html#K3">to_power</a> 2</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E2:72_3_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="power.html#T63">POWER:63</a></i>;<br/><a NAME="E18:72_3_1"/><b>then </b><i><font color="Green">E54</font></i>: 
 <a href="power.html#K6">log</a> 2,<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 2 <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"><a href="power.html#K6">log</a> 2,<span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="power.html#T53">POWER:53</a></i>;<br/><a NAME="E19:72_3_1"/>
2 <a href="series_1.html#K3">to_power</a> 10 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 2 <a href="series_1.html#K3">to_power</a> 1
 <b>by </b><i><a class="ref" href="power.html#T44">POWER:44</a></i>;<br/><a NAME="E20:72_3_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 2 <a href="series_1.html#K3">to_power</a> 1
 <b>by </b><i><a class="txt" href="asympt_1.html#E1:72_3_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E21:72_3_1"/><b>then </b><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 2
 <b>by </b><i><a class="ref" href="power.html#T30">POWER:30</a></i>;<br/><a NAME="E22:72_3_1"/><b>then </b><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E23:72_3_1"/>
2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 2 <a href="nat_1.html#K2">*</a> 2
 <b>by </b><i><a class="txt" href="asympt_1.html#E21:72_3_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E24:72_3_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 2 <a href="nat_1.html#K2">*</a> 2
 <b>by </b><i><a class="txt" href="asympt_1.html#E22:72_3_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E25:72_3_1"/><b>then </b>
 <a href="power.html#K6">log</a> 2,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="power.html#K6">log</a> 2,<span class="p1">(<span class="default">2 <a href="square_1.html#K5">^2</a> </span>)</span>
 <b>by </b><i><a class="ref" href="power.html#T65">POWER:65</a></i>;<br/><a NAME="E26:72_3_1"/><b>then </b>
 <a href="power.html#K6">log</a> 2,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="power.html#K6">log</a> 2,<span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="power.html#T53">POWER:53</a></i>;<br/><a NAME="E27:72_3_1"/><b>then </b>
 <a href="power.html#K6">log</a> 2,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,2</span>)</span>
 <b>by </b><i><a class="ref" href="power.html#T63">POWER:63</a></i>;<br/><a NAME="E28:72_3_1"/><b>then </b><i><font color="Green">E57</font></i>: 
 <a href="power.html#K6">log</a> 2,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 2 <a href="nat_1.html#K2">*</a> 1
 <b>by </b><i><a class="ref" href="power.html#T60">POWER:60</a></i>;<br/><a NAME="E29:72_3_1"/>
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<sub>5</sub></font></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">2 <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"><a href="power.html#K6">log</a> 2,<span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E18:72_3_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="power.html#T44">POWER:44</a></i>;<br/><a NAME="E30:72_3_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <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">2 <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"><a href="power.html#K6">log</a> 2,<span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E2:72_3_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="power.html#D3">POWER:def 3</a></i>;<br/><a NAME="E31:72_3_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <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,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="power.html#K6">log</a> 2,<span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="power.html#K4">to_power</a> 2</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E28:72_3_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="power.html#T63">POWER:63</a></i>;<br/><a NAME="E32:72_3_1"/><b>then </b><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <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,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="power.html#K6">log</a> 2,<span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="power.html#T53">POWER:53</a></i>;<br/><a NAME="E33:72_3_1"/>
<span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="asympt_1.html#E28:72_3_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="square_1.html#T74">SQUARE_1:74</a></i>;<br/><a NAME="E34:72_3_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></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,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> 
 <b>by </b><i><a class="txt" href="asympt_1.html#E32:72_3_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="power.html#D3">POWER:def 3</a></i>;<br/><a NAME="E35:72_3_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></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,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="series_1.html#K3">to_power</a> 2</span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> 
 <b>by </b><i><a class="ref" href="power.html#T53">POWER:53</a></i>;<br/><a NAME="E36:72_3_1"/><b>then </b><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default">2 <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>5</sub></font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> 
 <b>by </b><i><a class="txt" href="asympt_1.html#E2:72_3_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="power.html#T63">POWER:63</a></i>;<br/><a NAME="E37:72_3_1"/>
2 <a href="series_1.html#K3">to_power</a> 10 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 2 <a href="series_1.html#K3">to_power</a> 0
 <b>by </b><i><a class="ref" href="power.html#T44">POWER:44</a></i>;<br/><a NAME="E38:72_3_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 2 <a href="series_1.html#K3">to_power</a> 0
 <b>by </b><i><a class="txt" href="asympt_1.html#E1:72_3_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E39:72_3_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 <b>by </b><i><a class="ref" href="power.html#T29">POWER:29</a></i>;<br/><a NAME="E40:72_3_1"/><b>then </b>
 <a href="power.html#K6">log</a> 2,<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="power.html#K6">log</a> 2,1
 <b>by </b><i><a class="ref" href="power.html#T65">POWER:65</a></i>;<br/><a NAME="E41:72_3_1"/><b>then </b>
 <a href="power.html#K6">log</a> 2,<font color="Maroon">c<sub>5</sub></font> <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="E42:72_3_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">c<sub>5</sub></font></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>5</sub></font></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"><a href="power.html#K6">log</a> 2,<font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E43:72_3_1"/><b>then </b>
4 <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">c<sub>5</sub></font></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>5</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 2 <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">c<sub>5</sub></font></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>5</sub></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="E44:72_3_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 2 <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">c<sub>5</sub></font></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>5</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E36:72_3_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E45:72_3_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K6">/</a> 2 <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<sub>5</sub></font></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>5</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="real_2.html#T178">REAL_2:178</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:72_3"/>
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/><font color="Olive">b<sub>2</sub></font> <a href="real_1.html#K6">/</a> 2 <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="Olive">b<sub>2</sub></font></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="Olive">b<sub>2</sub></font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E9:72"/><i><font color="Green">E48</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>3</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="real_1.html#K6">/</a> 2 <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="Olive">b<sub>1</sub></font></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="Olive">b<sub>1</sub></font></span>)</span>
 ;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="asympt_0.html#K4">max</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K4">max</a> <font color="Maroon">c<sub>1</sub></font>,2</span>)</span>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4">max</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>;<br/>
<a NAME="E10:72"/><i><font color="Green">E49</font></i>: 
(  <a href="asympt_0.html#K4">max</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K4">max</a> <font color="Maroon">c<sub>1</sub></font>,2</span>)</span>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4">max</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="asympt_0.html#K4">max</a> <font color="Maroon">c<sub>1</sub></font>,2 &amp;  <a href="asympt_0.html#K4">max</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K4">max</a> <font color="Maroon">c<sub>1</sub></font>,2</span>)</span>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4">max</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="asympt_0.html#K4">max</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/>
<a NAME="E11:72"/><i><font color="Green">E50</font></i>: 
(  <a href="asympt_0.html#K4">max</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="asympt_0.html#K4">max</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/>
<a NAME="E12:72"/>
(  <a href="asympt_0.html#K4">max</a> <font color="Maroon">c<sub>1</sub></font>,2 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="asympt_0.html#K4">max</a> <font color="Maroon">c<sub>1</sub></font>,2 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/>
<a NAME="E13:72"/><b>then </b><i><font color="Green">E51</font></i>: 
(  <a href="asympt_0.html#K4">max</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K4">max</a> <font color="Maroon">c<sub>1</sub></font>,2</span>)</span>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4">max</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="asympt_0.html#K4">max</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K4">max</a> <font color="Maroon">c<sub>1</sub></font>,2</span>)</span>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4">max</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="asympt_0.html#K4">max</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K4">max</a> <font color="Maroon">c<sub>1</sub></font>,2</span>)</span>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4">max</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="asympt_0.html#K4">max</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K4">max</a> <font color="Maroon">c<sub>1</sub></font>,2</span>)</span>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4">max</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 )
 
<b>by </b><i><a class="txt" href="asympt_1.html#E10:72"><i><font color="Green">E49</font></i></a>, <a class="txt" href="asympt_1.html#E11:72"><i><font color="Green">E50</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:72_4"/>
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="asympt_0.html#K4">max</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K4">max</a> <font color="Maroon">c<sub>1</sub></font>,2</span>)</span>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4">max</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
 ;<br/><a NAME="E2:72_4"/><b>then </b><i><font color="Green">E52</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 )
 <b>by </b><i><a class="txt" href="asympt_1.html#E13:72"><i><font color="Green">E51</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E3:72_4"/><b>then </b><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K6">/</a> 6 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 <b>by </b><i><a class="txt" href="asympt_1.html#E3:72"><i><font color="Green">E46</font></i></a></i>;<br/><a NAME="E4:72_4"/>
<font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K6">/</a> 3 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 2 <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>5</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E6:72"><i><font color="Green">E47</font></i></a>, <a class="txt" href="asympt_1.html#E2:72_4"><i><font color="Green">E52</font></i></a></i>;<br/><a NAME="E5:72_4"/><b>then </b><i><font color="Green">E54</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K6">/</a> 6</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">2 <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>5</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E3:72_4"><i><font color="Green">E53</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E6:72_4"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K6">/</a> 2 <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<sub>5</sub></font></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>5</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E9:72"><i><font color="Green">E48</font></i></a>, <a class="txt" href="asympt_1.html#E2:72_4"><i><font color="Green">E52</font></i></a></i>;<br/><a NAME="E7:72_4"/><i><font color="Green">E56</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K6">/</a> 6</span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K6">/</a> 3</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><a NAME="E8:72_4"/>
<span class="p1">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default">2 <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>5</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</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>5</sub></font></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>5</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> 
 ;<br/><a NAME="E9:72_4"/><b>then </b><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> 
 <b>by </b><i><a class="txt" href="asympt_1.html#E5:72_4"><i><font color="Green">E54</font></i></a>, <a class="txt" href="asympt_1.html#E6:72_4"><i><font color="Green">E55</font></i></a>, <a class="txt" href="asympt_1.html#E7:72_4"><i><font color="Green">E56</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E10:72_4"/>
 <a href="power.html#K6">log</a> 2,<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="power.html#K6">log</a> 2,2
 <b>by </b><i><a class="txt" href="asympt_1.html#E2:72_4"><i><font color="Green">E52</font></i></a>, <a class="ref" href="pre_ff.html#T12">PRE_FF:12</a></i>;<br/><a NAME="E11:72_4"/><b>then </b>
 <a href="power.html#K6">log</a> 2,<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="power.html#T60">POWER:60</a></i>;<br/><a NAME="E12:72_4"/><b>then </b><i><font color="Green">E58</font></i>: 
1 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="nat_1.html#K1">+</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E13:72_4"/>
<span class="p1">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T65">XREAL_1:65</a></i>;<br/><a NAME="E14:72_4"/><b>then </b>
 <a href="square_1.html#K7">sqrt</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E9:72_4"><i><font color="Green">E57</font></i></a>, <a class="ref" href="square_1.html#T95">SQUARE_1:95</a></i>;<br/><a NAME="E15:72_4"/><b>then </b>
 <a href="square_1.html#K7">sqrt</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E12:72_4"><i><font color="Green">E58</font></i></a>, <a class="ref" href="square_1.html#T89">SQUARE_1:89</a></i>;<br/><b>hence </b><a NAME="E16:72_4"/>
<span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">c<sub>5</sub></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">c<sub>5</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E15:72"/>
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"><a href="square_1.html#K7">sqrt</a> <font color="Olive">b<sub>2</sub></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="Olive">b<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 ;<br/>


</div>
