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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:108"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> 0 &amp; ( 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#NR4" target="_self">&gt;</a> 0 holds <br/><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="power.html#K3">to_power</a> <span class="p1">(<span class="default">2 <a href="power.html#K3">to_power</a> <span class="p2"><a href="int_1.html#K1">[\</a><span class="default"><span class="p3">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="int_1.html#K1">/]</a></span></span>)</span> ) )
 <b>and </b><br/><a NAME="E2:108"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> 0 &amp; ( 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#NR4" target="_self">&gt;</a> 0 holds <br/><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="series_1.html#K3">to_power</a> <font color="Olive">b<sub>1</sub></font> ) )
 ;<br/>

<b>set </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="asympt_0.html#K15">taken_every</a> 2;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="asympt_1.html#K9" target="_self">POWEROF2SET</a> ;<br/>
<a NAME="E3:108"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="asympt_0.html#V4">eventually-positive</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_1"><b>proof </b></a><div class="add">

<b>take </b>
1
; <i><font color="Red">:: according to </font></i><a class="ref" href="asympt_0.html#D6">ASYMPT_0:def 6</a><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="E1:108_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 ;<br/>

<a NAME="E2:108_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="asympt_1.html#E2:108"><i><font color="Green">E68</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:108_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <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#E1:108_1"><i><font color="Green">E69</font></i></a>, <a class="ref" href="power.html#T39">POWER:39</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>2</sub></font> as  <a href="asympt_0.html#V4">eventually-positive</a> <a href="seq_1.html#NM1">Real_Sequence</a> ;<br/>
<b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/>

<a NAME="E5:108"/><i><font color="Green">E69</font></i>: 
 <a href="asympt_0.html#K11">Big_Theta</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <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>  : ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="real_1.html#NM1">Real</a>ex <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; ( for <font color="Olive">b<sub>4</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>5</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">b<sub>4</sub></font> holds <br/>( <font color="Olive">b<sub>3</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>5</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>5</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>5</sub></font></span>)</span> ) ) ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="asympt_0.html#T27">ASYMPT_0:27</a></i>;<br/>
<a NAME="E6:108"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is    <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>by </b><i><a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/>
<div><i><font color="Green">E71</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="108_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:108_2"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>and </b><br/><a NAME="E2:108_2"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="asympt_1.html#K9" target="_self">POWEROF2SET</a> 
 ;<br/><a NAME="E3:108_2"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="power.html#K3">to_power</a> <span class="p1">(<span class="default">2 <a href="power.html#K3">to_power</a> <span class="p2"><a href="int_1.html#K1">[\</a><span class="default"><span class="p3">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="int_1.html#K1">/]</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E1:108"><i><font color="Green">E67</font></i></a>, <a class="txt" href="asympt_1.html#E1:108_2"><i><font color="Green">E72</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E5:108_2"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="asympt_1.html#E2:108_2"><i><font color="Green">E73</font></i></a></i>;<br/> <a href="power.html#K6">log</a> 2,<font color="Maroon">c<sub>6</sub></font> = 
<font color="Maroon">c<sub>7</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>
<b>by </b><i><a class="txt" href="asympt_1.html#E5:108_2"><i><font color="Green">E75</font></i></a>, <a class="ref" href="power.html#T63">POWER:63</a></i>
<br/>.= 
<font color="Maroon">c<sub>7</sub></font> <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="E7:108_2"/><b>then </b><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="asympt_1.html#E3:108_2"><i><font color="Green">E74</font></i></a>, <a class="txt" href="asympt_1.html#E5:108_2"><i><font color="Green">E75</font></i></a>, <a class="ref" href="int_1.html#T47">INT_1:47</a></i>;<br/><b>hence </b><a NAME="E8:108_2"/>
1 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="asympt_1.html#E2:108"><i><font color="Green">E68</font></i></a>, <a class="txt" href="asympt_1.html#E1:108_2"><i><font color="Green">E72</font></i></a></i>;<br/><b>thus </b><a NAME="E9:108_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E2:108"><i><font color="Green">E68</font></i></a>, <a class="txt" href="asympt_1.html#E1:108_2"><i><font color="Green">E72</font></i></a>, <a class="txt" href="asympt_1.html#E7:108_2"><i><font color="Green">E76</font></i></a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E72</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="108_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:108_3"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="asympt_0.html#K11">Big_Theta</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>6</sub></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="E3:108_3"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E4:108_3"/><i><font color="Green">E74</font></i>: 
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="real_1.html#NM1">Real</a>ex <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; ( for <font color="Olive">b<sub>4</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>4</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">b<sub>3</sub></font> holds <br/>( <font color="Olive">b<sub>2</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>4</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>4</sub></font></span>)</span> ) ) )
 <b>by </b><i><a class="txt" href="asympt_1.html#E5:108"><i><font color="Green">E69</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">c<sub>9</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E6:108_3"/><i><font color="Green">E75</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 <b>and </b><br/><a NAME="E7:108_3"/><i><font color="Green">E76</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>9</sub></font> holds <br/>( <font color="Maroon">c<sub>8</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="asympt_1.html#E4:108_3"><i><font color="Green">E74</font></i></a></i>;<br/><a NAME="E8:108_3"/>
<font color="Maroon">c<sub>8</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#E6:108_3"><i><font color="Green">E75</font></i></a>, <a class="ref" href="xreal_1.html#T124">XREAL_1:124</a></i>;<br/><a NAME="E9:108_3"/><b>then </b><i><font color="Green">E77</font></i>: 
1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T217">XCMPLX_1:217</a></i>;<br/><b>set </b><font color="Maroon">c<sub>10</sub></font> =  <a href="xxreal_0.html#K4">max</a> <span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="int_1.html#K2">\]</a></span>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4">max</a> <font color="Maroon">c<sub>9</sub></font>,2</span>)</span>;<br/><a NAME="E10:108_3"/><i><font color="Green">E78</font></i>: 
(  <a href="xxreal_0.html#K4">max</a> <span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="int_1.html#K2">\]</a></span>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4">max</a> <font color="Maroon">c<sub>9</sub></font>,2</span>)</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">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="int_1.html#K2">\]</a></span> &amp;  <a href="xxreal_0.html#K4">max</a> <span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="int_1.html#K2">\]</a></span>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4">max</a> <font color="Maroon">c<sub>9</sub></font>,2</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>9</sub></font>,2 )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/><a NAME="E11:108_3"/>
(  <a href="asympt_0.html#K4">max</a> <font color="Maroon">c<sub>9</sub></font>,2 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>9</sub></font> &amp;  <a href="asympt_0.html#K4">max</a> <font color="Maroon">c<sub>9</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="E12:108_3"/><b>then </b><i><font color="Green">E79</font></i>: 
(  <a href="xxreal_0.html#K4">max</a> <span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="int_1.html#K2">\]</a></span>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4">max</a> <font color="Maroon">c<sub>9</sub></font>,2</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>9</sub></font> &amp;  <a href="xxreal_0.html#K4">max</a> <span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="int_1.html#K2">\]</a></span>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4">max</a> <font color="Maroon">c<sub>9</sub></font>,2</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 )
 <b>by </b><i><a class="txt" href="asympt_1.html#E10:108_3"><i><font color="Green">E78</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E13:108_3"/>
 <a href="xxreal_0.html#K4">max</a> <span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="int_1.html#K2">\]</a></span>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4">max</a> <font color="Maroon">c<sub>9</sub></font>,2</span>)</span> is   <a href="int_1.html#NM1">Integer</a>
 <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">c<sub>11</sub></font> =  <a href="xxreal_0.html#K4">max</a> <span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="int_1.html#K2">\]</a></span>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4">max</a> <font color="Maroon">c<sub>9</sub></font>,2</span>)</span> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="asympt_1.html#E10:108_3"><i><font color="Green">E78</font></i></a>, <a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = 2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>11</sub></font> as   <a href="nat_1.html#NM1">Nat</a> ;<br/><a NAME="E16:108_3"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="asympt_1.html#E12:108_3"><i><font color="Green">E79</font></i></a>, <a class="txt" href="asympt_1.html#E1"><i><font color="Green">Lemma1</font></i></a></i>;<br/><a NAME="E17:108_3"/>
<font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>11</sub></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="E18:108_3"/><b>then </b><i><font color="Green">E81</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="asympt_1.html#E16:108_3"><i><font color="Green">E80</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E19:108_3"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>12</sub></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="E20:108_3"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="asympt_1.html#E18:108_3"><i><font color="Green">E81</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E21:108_3"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="asympt_1.html#E12:108_3"><i><font color="Green">E79</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E22:108_3"/><b>then </b><i><font color="Green">E83</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E7:108_3"><i><font color="Green">E76</font></i></a></i>;<br/><a NAME="E23:108_3"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="series_1.html#K3">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E2:108"><i><font color="Green">E68</font></i></a></i>;<br/><a NAME="E24:108_3"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="power.html#K3">to_power</a> <span class="p1">(<span class="default">2 <a href="power.html#K3">to_power</a> <span class="p2"><a href="int_1.html#K1">[\</a><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>12</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E1:108"><i><font color="Green">E67</font></i></a>, <a class="txt" href="asympt_1.html#E3:108_3"><i><font color="Green">E73</font></i></a></i>;<br/> <a href="power.html#K6">log</a> 2,<font color="Maroon">c<sub>12</sub></font> = 
<font color="Maroon">c<sub>11</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>
<b>by </b><i><a class="ref" href="power.html#T63">POWER:63</a></i>
<br/>.= 
<font color="Maroon">c<sub>11</sub></font> <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="E26:108_3"/><b>then </b><i><font color="Green">E86</font></i>: 
 <a href="power.html#K6">log</a> 2,<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></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<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="asympt_1.html#E16:108_3"><i><font color="Green">E80</font></i></a>, <a class="txt" href="asympt_1.html#E19:108_3"><i><font color="Green">E82</font></i></a>, <a class="ref" href="power.html#T65">POWER:65</a></i>;<br/><a NAME="E27:108_3"/>
<font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 <b>by </b><i><a class="txt" href="asympt_1.html#E12:108_3"><i><font color="Green">E79</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E28:108_3"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 <b>by </b><i><a class="txt" href="asympt_1.html#E100"><i><font color="Green">Lemma65</font></i></a></i>;<br/><a NAME="E29:108_3"/><b>then </b>
2 <a href="series_1.html#K3">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></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<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/><a NAME="E30:108_3"/><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> <span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1</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"><font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="power.html#T65">POWER:65</a></i>;<br/><a NAME="E31:108_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1</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"><font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="power.html#T63">POWER:63</a></i>;<br/><a NAME="E32:108_3"/><b>then </b><i><font color="Green">E87</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <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>12</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="power.html#T60">POWER:60</a></i>;<br/><a NAME="E33:108_3"/>
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><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>12</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="int_1.html#K1">/]</a></span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E26:108_3"><i><font color="Green">E86</font></i></a>, <a class="ref" href="pre_ff.html#T11">PRE_FF:11</a></i>;<br/><a NAME="E34:108_3"/><b>then </b><i><font color="Green">E88</font></i>: 
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><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>12</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="ref" href="int_1.html#T47">INT_1:47</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_3_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:108_3_2"/>
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><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>12</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/><a NAME="E2:108_3_2"/><b>then </b><i><font color="Green">E89</font></i>: 
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><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>12</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="int_1.html#T20">INT_1:20</a></i>;<br/><a NAME="E3:108_3_2"/>
 <a href="power.html#K6">log</a> 2,<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><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>12</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span>
 <b>by </b><i><a class="ref" href="int_1.html#D4">INT_1:def 4</a></i>;<br/><b>hence </b><a NAME="E4:108_3_2"/>
contradiction
 <b>by </b><i><a class="txt" href="asympt_1.html#E32:108_3"><i><font color="Green">E87</font></i></a>, <a class="txt" href="asympt_1.html#E2:108_3_2"><i><font color="Green">E89</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><b>then </b><i><font color="Green">E89</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="series_1.html#K3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="asympt_1.html#E23:108_3"><i><font color="Green">E84</font></i></a>, <a class="txt" href="asympt_1.html#E24:108_3"><i><font color="Green">E85</font></i></a>, <a class="txt" href="asympt_1.html#E34:108_3"><i><font color="Green">E88</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="power.html#T34">POWER:34</a></i>
<br/>.= 
<font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="ref" href="power.html#T30">POWER:30</a></i>
;<br/><a NAME="E37:108_3"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="asympt_1.html#E23:108_3"><i><font color="Green">E84</font></i></a>, <a class="ref" href="power.html#T39">POWER:39</a></i>;<br/><a NAME="E38:108_3"/>
<span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="int_1.html#K2">\]</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="int_1.html#D5">INT_1:def 5</a></i>;<br/><a NAME="E39:108_3"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="asympt_1.html#E10:108_3"><i><font color="Green">E78</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E40:108_3"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="asympt_1.html#E18:108_3"><i><font color="Green">E81</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E41:108_3"/><b>then </b>
<font color="Maroon">c<sub>12</sub></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">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="real_1.html#K3">+</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E42:108_3"/><b>then </b>
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E9:108_3"><i><font color="Green">E77</font></i></a>, <a class="txt" href="asympt_1.html#E36:108_3"><i><font color="Green">E89</font></i></a>, <a class="ref" href="real_2.html#T151">REAL_2:151</a></i>;<br/><a NAME="E43:108_3"/><b>then </b>
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T56">XCMPLX_1:56</a></i>;<br/><a NAME="E44:108_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T57">XCMPLX_1:57</a></i>;<br/><a NAME="E45:108_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>8</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E37:108_3"><i><font color="Green">E90</font></i></a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><b>hence </b><a NAME="E46:108_3"/>
contradiction
 <b>by </b><i><a class="txt" href="asympt_1.html#E22:108_3"><i><font color="Green">E83</font></i></a>, <a class="txt" href="asympt_1.html#E37:108_3"><i><font color="Green">E90</font></i></a>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E73</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="108_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:108_4"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2
 ;<br/><a NAME="E2:108_4"/><b>then </b><i><font color="Green">E75</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 0 &amp; <font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E3:108_4"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="power.html#K3">to_power</a> <span class="p1">(<span class="default">2 <a href="power.html#K3">to_power</a> <span class="p2"><a href="int_1.html#K1">[\</a><span class="default"><span class="p3">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="int_1.html#K1">/]</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E1:108"><i><font color="Green">E67</font></i></a>, <a class="txt" href="asympt_1.html#E1:108_4"><i><font color="Green">E74</font></i></a></i>;<br/><a NAME="E4:108_4"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="power.html#K3">to_power</a> <span class="p1">(<span class="default">2 <a href="power.html#K3">to_power</a> <span class="p2"><a href="int_1.html#K1">[\</a><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>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E1:108"><i><font color="Green">E67</font></i></a></i>;<br/><a NAME="E5:108_4"/>
 <a href="power.html#K6">log</a> 2,<font color="Maroon">c<sub>6</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#E1:108_4"><i><font color="Green">E74</font></i></a>, <a class="ref" href="pre_ff.html#T12">PRE_FF:12</a></i>;<br/><a NAME="E6:108_4"/><b>then </b>
 <a href="power.html#K6">log</a> 2,<font color="Maroon">c<sub>6</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="E7:108_4"/><b>then </b>
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="int_1.html#K1">/]</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default">1</span><a href="int_1.html#K1">/]</a></span>
 <b>by </b><i><a class="ref" href="pre_ff.html#T11">PRE_FF:11</a></i>;<br/><a NAME="E8:108_4"/><b>then </b>
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="int_1.html#K1">/]</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="int_1.html#T47">INT_1:47</a></i>;<br/><a NAME="E9:108_4"/><b>then </b>
2 <a href="power.html#K3">to_power</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="int_1.html#K1">/]</a></span> <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="E10:108_4"/><b>then </b><i><font color="Green">E78</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="power.html#K3">to_power</a> <span class="p1">(<span class="default">2 <a href="power.html#K3">to_power</a> <span class="p2"><a href="int_1.html#K1">[\</a><span class="default"><span class="p3">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="int_1.html#K1">/]</a></span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="power.html#K3">to_power</a> <span class="p1">(<span class="default">2 <a href="power.html#K3">to_power</a> <span class="p2"><a href="int_1.html#K1">[\</a><span class="default"><span class="p3">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="int_1.html#K1">/]</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E1:108_4"><i><font color="Green">E74</font></i></a>, <a class="txt" href="asympt_1.html#E2:108_4"><i><font color="Green">E75</font></i></a>, <a class="ref" href="power.html#T42">POWER:42</a></i>;<br/><a NAME="E11:108_4"/>
 <a href="power.html#K6">log</a> 2,<font color="Maroon">c<sub>6</sub></font> <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>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E1:108_4"><i><font color="Green">E74</font></i></a>, <a class="txt" href="asympt_1.html#E2:108_4"><i><font color="Green">E75</font></i></a>, <a class="ref" href="power.html#T65">POWER:65</a></i>;<br/><a NAME="E12:108_4"/><b>then </b>
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="int_1.html#K1">/]</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><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>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span>
 <b>by </b><i><a class="ref" href="pre_ff.html#T11">PRE_FF:11</a></i>;<br/><a NAME="E13:108_4"/><b>then </b>
2 <a href="power.html#K3">to_power</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="int_1.html#K1">/]</a></span> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="power.html#K3">to_power</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><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>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span>
 <b>by </b><i><a class="ref" href="pre_ff.html#T10">PRE_FF:10</a></i>;<br/><a NAME="E14:108_4"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="power.html#K3">to_power</a> <span class="p1">(<span class="default">2 <a href="power.html#K3">to_power</a> <span class="p2"><a href="int_1.html#K1">[\</a><span class="default"><span class="p3">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="int_1.html#K1">/]</a></span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="power.html#K3">to_power</a> <span class="p1">(<span class="default">2 <a href="power.html#K3">to_power</a> <span class="p2"><a href="int_1.html#K1">[\</a><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>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E2:108_4"><i><font color="Green">E75</font></i></a>, <a class="ref" href="pre_ff.html#T10">PRE_FF:10</a></i>;<br/><b>hence </b><a NAME="E15:108_4"/>
<font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E3:108_4"><i><font color="Green">E76</font></i></a>, <a class="txt" href="asympt_1.html#E4:108_4"><i><font color="Green">E77</font></i></a>, <a class="txt" href="asympt_1.html#E10:108_4"><i><font color="Green">E78</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E74</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="108_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:108_5"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 ;<br/><a NAME="E2:108_5"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>6</sub></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="E3:108_5"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="asympt_1.html#E2:108"><i><font color="Green">E68</font></i></a>, <a class="txt" href="asympt_1.html#E1:108_5"><i><font color="Green">E75</font></i></a></i>;<br/><a NAME="E4:108_5"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="series_1.html#K3">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E2:108"><i><font color="Green">E68</font></i></a></i>;<br/><a NAME="E5:108_5"/>
<font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="asympt_1.html#E1:108_5"><i><font color="Green">E75</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E6:108_5"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E7:108_5"/><b>then </b><i><font color="Green">E79</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="series_1.html#K3">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E2:108_5"><i><font color="Green">E76</font></i></a>, <a class="ref" href="power.html#T44">POWER:44</a></i>;<br/><a NAME="E8:108_5"/>
<font color="Maroon">c<sub>6</sub></font> <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="asympt_1.html#E1:108_5"><i><font color="Green">E75</font></i></a>, <a class="txt" href="asympt_1.html#E2:108_5"><i><font color="Green">E76</font></i></a>, <a class="ref" href="power.html#T42">POWER:42</a></i>;<br/><b>hence </b><a NAME="E9:108_5"/>
<font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E3:108_5"><i><font color="Green">E77</font></i></a>, <a class="txt" href="asympt_1.html#E4:108_5"><i><font color="Green">E78</font></i></a>, <a class="txt" href="asympt_1.html#E7:108_5"><i><font color="Green">E79</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_6"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:108_6"/>
<font color="Maroon">c<sub>5</sub></font> <a href="asympt_0.html#R2">is_smooth_wrt</a> 2
 ;<br/><a NAME="E2:108_6"/><b>then </b><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="asympt_0.html#K15">taken_every</a> 2 <a href="hidden.html#R2">in</a>  <a href="asympt_0.html#K9">Big_Oh</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="asympt_0.html#D19">ASYMPT_0:def 19</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>6</sub></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="E4:108_6"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="asympt_0.html#K15">taken_every</a> 2
 <b>and </b><br/><a NAME="E5:108_6"/><i><font color="Green">E77</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a>ex <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; ( for <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">b<sub>2</sub></font> holds <br/>( <font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) ) )
 <b>by </b><i><a class="txt" href="asympt_1.html#E2:108_6"><i><font color="Green">E75</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">c<sub>8</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E7:108_6"/>
<font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>and </b><br/><a NAME="E8:108_6"/><i><font color="Green">E78</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>8</sub></font> holds <br/>( <font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 <b>by </b><i><a class="txt" href="asympt_1.html#E5:108_6"><i><font color="Green">E77</font></i></a></i>;<br/><b>set </b><font color="Maroon">c<sub>9</sub></font> =  <a href="xxreal_0.html#K4">max</a> <span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="int_1.html#K2">\]</a></span>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4">max</a> <font color="Maroon">c<sub>8</sub></font>,2</span>)</span>;<br/><a NAME="E9:108_6"/><i><font color="Green">E79</font></i>: 
(  <a href="xxreal_0.html#K4">max</a> <span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="int_1.html#K2">\]</a></span>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4">max</a> <font color="Maroon">c<sub>8</sub></font>,2</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="int_1.html#K2">\]</a></span> &amp;  <a href="xxreal_0.html#K4">max</a> <span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="int_1.html#K2">\]</a></span>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4">max</a> <font color="Maroon">c<sub>8</sub></font>,2</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>8</sub></font>,2 )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/><a NAME="E10:108_6"/><i><font color="Green">E80</font></i>: 
(  <a href="asympt_0.html#K4">max</a> <font color="Maroon">c<sub>8</sub></font>,2 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>8</sub></font> &amp;  <a href="asympt_0.html#K4">max</a> <font color="Maroon">c<sub>8</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="E11:108_6"/><b>then </b><i><font color="Green">E81</font></i>: 
(  <a href="xxreal_0.html#K4">max</a> <span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="int_1.html#K2">\]</a></span>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4">max</a> <font color="Maroon">c<sub>8</sub></font>,2</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>8</sub></font> &amp;  <a href="xxreal_0.html#K4">max</a> <span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="int_1.html#K2">\]</a></span>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4">max</a> <font color="Maroon">c<sub>8</sub></font>,2</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 )
 <b>by </b><i><a class="txt" href="asympt_1.html#E9:108_6"><i><font color="Green">E79</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E12:108_6"/><b>then </b><i><font color="Green">E82</font></i>: 
(  <a href="xxreal_0.html#K4">max</a> <span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="int_1.html#K2">\]</a></span>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4">max</a> <font color="Maroon">c<sub>8</sub></font>,2</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="xxreal_0.html#K4">max</a> <span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="int_1.html#K2">\]</a></span>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4">max</a> <font color="Maroon">c<sub>8</sub></font>,2</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E13:108_6"/><i><font color="Green">E83</font></i>: 
2 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p2"><a href="int_1.html#K2">[/</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="int_1.html#K2">\]</a></span>,<span class="p2">(<span class="default"><a href="asympt_0.html#K4">max</a> <font color="Maroon">c<sub>8</sub></font>,2</span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p2"><a href="int_1.html#K2">[/</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="int_1.html#K2">\]</a></span>,<span class="p2">(<span class="default"><a href="asympt_0.html#K4">max</a> <font color="Maroon">c<sub>8</sub></font>,2</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E9:108_6"><i><font color="Green">E79</font></i></a>, <a class="txt" href="asympt_1.html#E10:108_6"><i><font color="Green">E80</font></i></a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E14:108_6"/>
 <a href="xxreal_0.html#K4">max</a> <span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="int_1.html#K2">\]</a></span>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4">max</a> <font color="Maroon">c<sub>8</sub></font>,2</span>)</span> is   <a href="int_1.html#NM1">Integer</a>
 <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">c<sub>10</sub></font> =  <a href="xxreal_0.html#K4">max</a> <span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="int_1.html#K2">\]</a></span>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4">max</a> <font color="Maroon">c<sub>8</sub></font>,2</span>)</span> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="asympt_1.html#E9:108_6"><i><font color="Green">E79</font></i></a>, <a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="asympt_0.html#K15">taken_every</a> 2</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font> = 
<font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="asympt_0.html#D18">ASYMPT_0:def 18</a></i>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="series_1.html#K3">to_power</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="asympt_1.html#E2:108"><i><font color="Green">E68</font></i></a>, <a class="txt" href="asympt_1.html#E13:108_6"><i><font color="Green">E83</font></i></a></i>
;<br/><a NAME="E17:108_6"/><b>then </b><i><font color="Green">E84</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="asympt_0.html#K15">taken_every</a> 2</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>10</sub></font> <a href="series_1.html#K3">to_power</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E9:108_6"><i><font color="Green">E79</font></i></a>, <a class="txt" href="asympt_1.html#E10:108_6"><i><font color="Green">E80</font></i></a>, <a class="txt" href="asympt_1.html#E13:108_6"><i><font color="Green">E83</font></i></a>, <a class="ref" href="power.html#T42">POWER:42</a></i>;<br/><a NAME="E18:108_6"/>
<font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="txt" href="asympt_1.html#E11:108_6"><i><font color="Green">E81</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E19:108_6"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E20:108_6"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="series_1.html#K3">to_power</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>10</sub></font> <a href="series_1.html#K3">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E12:108_6"><i><font color="Green">E82</font></i></a>, <a class="ref" href="pre_ff.html#T10">PRE_FF:10</a></i>;<br/><a NAME="E21:108_6"/><b>then </b><i><font color="Green">E85</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="asympt_0.html#K15">taken_every</a> 2</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>10</sub></font> <a href="series_1.html#K3">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E17:108_6"><i><font color="Green">E84</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><i><font color="Green">E86</font></i>: <font color="Maroon">c<sub>10</sub></font> <a href="series_1.html#K3">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="series_1.html#K3">to_power</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="asympt_1.html#E9:108_6"><i><font color="Green">E79</font></i></a>, <a class="txt" href="asympt_1.html#E10:108_6"><i><font color="Green">E80</font></i></a>, <a class="ref" href="power.html#T32">POWER:32</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="ref" href="power.html#T30">POWER:30</a></i>
;<br/><a NAME="E23:108_6"/>
<span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="int_1.html#K2">\]</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="int_1.html#D5">INT_1:def 5</a></i>;<br/><a NAME="E24:108_6"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="asympt_1.html#E9:108_6"><i><font color="Green">E79</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E25:108_6"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="series_1.html#K3">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E22:108_6"><i><font color="Green">E86</font></i></a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E26:108_6"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="asympt_0.html#K15">taken_every</a> 2</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E21:108_6"><i><font color="Green">E85</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E27:108_6"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="asympt_0.html#K15">taken_every</a> 2</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E2:108"><i><font color="Green">E68</font></i></a>, <a class="txt" href="asympt_1.html#E9:108_6"><i><font color="Green">E79</font></i></a>, <a class="txt" href="asympt_1.html#E10:108_6"><i><font color="Green">E80</font></i></a></i>;<br/><b>hence </b><a NAME="E28:108_6"/>
contradiction
 <b>by </b><i><a class="txt" href="asympt_1.html#E4:108_6"><i><font color="Green">E76</font></i></a>, <a class="txt" href="asympt_1.html#E8:108_6"><i><font color="Green">E78</font></i></a>, <a class="txt" href="asympt_1.html#E11:108_6"><i><font color="Green">E81</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E12:108"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="asympt_0.html#K14">Big_Theta</a> <font color="Maroon">c<sub>5</sub></font>,<a href="asympt_1.html#K9" target="_self">POWEROF2SET</a>  &amp; not <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="asympt_0.html#K11">Big_Theta</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="asympt_0.html#V6">eventually-nondecreasing</a> &amp; <font color="Maroon">c<sub>5</sub></font> is <a href="asympt_0.html#V6">eventually-nondecreasing</a> &amp; not <font color="Maroon">c<sub>5</sub></font> <a href="asympt_0.html#R2">is_smooth_wrt</a> 2 )
 <b>by </b><i><a class="txt" href="asympt_1.html#E6:108"><i><font color="Green">E70</font></i></a>, <a class="txt" href="asympt_1.html#E7:108"><i><font color="Green">E71</font></i></a>, <a class="txt" href="asympt_1.html#E8:108"><i><font color="Green">E72</font></i></a>, <a class="txt" href="asympt_1.html#E9:108"><i><font color="Green">E73</font></i></a>, <a class="txt" href="asympt_1.html#E10:108"><i><font color="Green">E74</font></i></a>, <a class="ref" href="asympt_0.html#D8">ASYMPT_0:def 8</a></i>;<br/>


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