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

<b>let </b><font color="Maroon">g</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>

<b>assume </b><a NAME="E1:109"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <br/> ( ( <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="asympt_1.html#K9" target="_self">POWEROF2SET</a>  implies <font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Olive">n</font> ) &amp; ( not <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="asympt_1.html#K9" target="_self">POWEROF2SET</a>  implies <font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Olive">n</font> <a href="series_1.html#K3">to_power</a> 2 ) )
 ;<br/>

<b>set </b><font color="Maroon">f</font> =  <a href="asympt_1.html#K3" target="_self">seq_n^</a> 1;<br/>
<b>set </b><font color="Maroon">h</font> = <font color="Maroon">g</font> <a href="asympt_0.html#K15">taken_every</a> 2;<br/>
<b>set </b><font color="Maroon">X</font> =  <a href="asympt_1.html#K9" target="_self">POWEROF2SET</a> ;<br/>
<a NAME="E2:109"/>
<font color="Maroon">g</font> is <a href="asympt_0.html#V4">eventually-positive</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_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">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:109_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 ;<br/>

<b>thus </b><a NAME="E2:109_1"/>
<font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="109_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:109_1_1_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="asympt_1.html#K9" target="_self">POWEROF2SET</a>  or  not <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="asympt_1.html#K9" target="_self">POWEROF2SET</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="109_1_1_1_1"><b>suppose </b></a><a NAME="E1:109_1_1_1_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="asympt_1.html#K9" target="_self">POWEROF2SET</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:109_1_1_1_1"/>
<font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="109_1_1_1_2"><b>suppose </b></a><a NAME="E1:109_1_1_1_2"/>
not <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="asympt_1.html#K9" target="_self">POWEROF2SET</a> 
 ;<br/><div class="add"><a NAME="E2:109_1_1_1_2"/><b>then </b>
<font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> 2
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:109_1_1_1_2"/>
<font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <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/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">g</font> = <font color="Maroon">g</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">g</font>
;<br/>

<a NAME="E4:109"/><i><font color="Green">E51</font></i>: 
3 <a href="hidden.html#R1">=</a> 4 <a href="real_1.html#K5">-</a> 1
 
;<br/>
<a NAME="E5:109"/><i><font color="Green">E52</font></i>: 
 <a href="asympt_1.html#K3" target="_self">seq_n^</a> 1 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">E59</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="109_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:109_2"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>and </b><br/><a NAME="E2:109_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="asympt_1.html#K9" target="_self">POWEROF2SET</a> 
 ;<br/><i><font color="Green">E62</font></i>: <span class="p1">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> 1</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> = 
<font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> 1
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="power.html#T30">POWER:30</a></i>
;<br/><b>hence </b><a NAME="E4:109_2"/>
1 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> 1</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, </i>;<br/><b>thus </b><a NAME="E5:109_2"/>
<span class="p1">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> 1</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</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">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div>
<a NAME="E7:109"/><i><font color="Green">E63</font></i>: 
 <a href="asympt_0.html#K11">Big_Theta</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">t</font> where <font color="Olive">t</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>  :  ex <font color="Olive">c</font>, <font color="Olive">d</font> being  <a href="real_1.html#NM1">Real</a> ex <font color="Olive">N</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">c</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Olive">d</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">N</font> holds <br/>( <font color="Olive">d</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">t</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> &amp; <font color="Olive">t</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">c</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> ) ) ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="asympt_0.html#T27">ASYMPT_0:27</a></i>;<br/>
<div><i><font color="Green">E64</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="109_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:109_3"/>
 <a href="asympt_1.html#K3" target="_self">seq_n^</a> 1 <a href="hidden.html#R2">in</a>  <a href="asympt_0.html#K11">Big_Theta</a> <font color="Maroon">g</font>
 ;<br/><b>then consider </b><font color="Maroon">t</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="E3:109_3"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a>  <a href="asympt_1.html#K3" target="_self">seq_n^</a> 1
 <b>and </b><br/><a NAME="E4:109_3"/><i><font color="Green">E69</font></i>: 
 ex <font color="Olive">c</font>, <font color="Olive">d</font> being  <a href="real_1.html#NM1">Real</a> ex <font color="Olive">N</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">c</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Olive">d</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">N</font> holds <br/>( <font color="Olive">d</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> &amp; <font color="Maroon">t</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">c</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> ) ) )
 <b>by </b><i><a class="txt" href="asympt_1.html#E8:109_3"><i><font color="Green">E71</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">d</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">N</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:109_3"/>
<font color="Maroon">c</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>and </b><br/><a NAME="E7:109_3"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">d</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>and </b><br/><a NAME="E8:109_3"/><i><font color="Green">E71</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">N</font> holds <br/>( <font color="Maroon">d</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> &amp; <font color="Maroon">t</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> )
 <b>by </b><i><a class="txt" href="asympt_1.html#E19:109_3"><i><font color="Green">E74</font></i></a></i>;<br/><b>set </b><font color="Maroon">N0</font> =  <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K4">max</a> <font color="Maroon">N</font>,2</span>)</span>,<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">d</font></span>)</span></span><a href="int_1.html#K2">\]</a></span>;<br/><a NAME="E9:109_3"/><i><font color="Green">E72</font></i>: 
(  <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K4">max</a> <font color="Maroon">N</font>,2</span>)</span>,<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">d</font></span>)</span></span><a href="int_1.html#K2">\]</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="asympt_0.html#K4">max</a> <font color="Maroon">N</font>,2 &amp;  <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K4">max</a> <font color="Maroon">N</font>,2</span>)</span>,<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">d</font></span>)</span></span><a href="int_1.html#K2">\]</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">d</font></span>)</span></span><a href="int_1.html#K2">\]</a></span> )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/><a NAME="E10:109_3"/>
(  <a href="asympt_0.html#K4">max</a> <font color="Maroon">N</font>,2 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">N</font> &amp;  <a href="asympt_0.html#K4">max</a> <font color="Maroon">N</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="E11:109_3"/><b>then </b><i><font color="Green">E73</font></i>: 
(  <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K4">max</a> <font color="Maroon">N</font>,2</span>)</span>,<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">d</font></span>)</span></span><a href="int_1.html#K2">\]</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">N</font> &amp;  <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K4">max</a> <font color="Maroon">N</font>,2</span>)</span>,<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">d</font></span>)</span></span><a href="int_1.html#K2">\]</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 )
 <b>by </b><i><a class="txt" href="asympt_1.html#E21:109_3"><i><font color="Green">E77</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E12:109_3"/>
 <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K4">max</a> <font color="Maroon">N</font>,2</span>)</span>,<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">d</font></span>)</span></span><a href="int_1.html#K2">\]</a></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">N0</font> =  <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K4">max</a> <font color="Maroon">N</font>,2</span>)</span>,<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">d</font></span>)</span></span><a href="int_1.html#K2">\]</a></span> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="asympt_1.html#E21:109_3"><i><font color="Green">E77</font></i></a>, <a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/><b>set </b><font color="Maroon">N1</font> = <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">N0</font></span>)</span> <a href="real_1.html#K5">-</a> 1;<br/><a NAME="E14:109_3"/>
2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">N0</font> <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="E15:109_3"/><b>then </b>
2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">N0</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E16:109_3"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">N0</font></span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><b>then reconsider </b><font color="Maroon">N1</font> = <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">N0</font></span>)</span> <a href="real_1.html#K5">-</a> 1 as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/><a NAME="E18:109_3"/>
2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">N0</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">N0</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="asympt_1.html#E27:109_3"><i><font color="Green">E78</font></i></a>, </i>;<br/><a NAME="E19:109_3"/><b>then </b><i><font color="Green">E74</font></i>: 
<font color="Maroon">N1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">N0</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/><a NAME="E20:109_3"/><b>then </b><i><font color="Green">E75</font></i>: 
<font color="Maroon">N1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">N</font>
 <b>by </b><i><a class="txt" href="asympt_1.html#E27:109_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="E21:109_3"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">N1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">d</font></span>)</span></span><a href="int_1.html#K2">\]</a></span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E21:109_3"><i><font color="Green">E77</font></i></a>, <a class="txt" href="asympt_1.html#E29:109_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="E22:109_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">d</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">d</font>
 <b>by </b><i><a class="ref" href="int_1.html#D5">INT_1:def 5</a></i>;<br/><a NAME="E23:109_3"/><b>then </b>
<font color="Maroon">N1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="real_1.html#K6">/</a> <font color="Maroon">d</font>
 <b>by </b><i><a class="txt" href="asympt_1.html#E10:109"><i><font color="Green">E81</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E24:109_3"/><b>then </b>
<font color="Maroon">N1</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">N1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">N1</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">d</font></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E29:109_3"><i><font color="Green">E79</font></i></a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E25:109_3"/><b>then </b>
<font color="Maroon">d</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">N1</font> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">N1</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">d</font>
 <b>by </b><i><a class="txt" href="asympt_1.html#E20:109_3"><i><font color="Green">E75</font></i></a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E26:109_3"/><b>then </b>
<font color="Maroon">d</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">N1</font> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">N1</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">d</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">d</font></span>)</span>
 ;<br/><a NAME="E27:109_3"/><b>then </b><i><font color="Green">E78</font></i>: 
<font color="Maroon">d</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">N1</font> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">N1</font> <a href="nat_1.html#K2">*</a> 1
 <b>by </b><i><a class="txt" href="asympt_1.html#E20:109_3"><i><font color="Green">E75</font></i></a>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/><a NAME="E28:109_3"/>
not <font color="Maroon">N1</font> <a href="hidden.html#R2">in</a>  <a href="asympt_1.html#K9" target="_self">POWEROF2SET</a> 
 <b>by </b><i><a class="txt" href="asympt_1.html#E27:109_3"><i><font color="Green">E78</font></i></a>, </i>;<br/><b>then </b><i><font color="Green">E79</font></i>: <font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">N1</font> = 
<font color="Maroon">N1</font> <a href="series_1.html#K3">to_power</a> 2
<b>by </b><i><a class="txt" href="asympt_1.html#E3:109_3"><i><font color="Green">E67</font></i></a></i>
<br/>.= 
<font color="Maroon">N1</font> <a href="square_1.html#K5">^2</a> 
<b>by </b><i><a class="ref" href="power.html#T53">POWER:53</a></i>
;<br/><font color="Maroon">t</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">N1</font> = 
<font color="Maroon">N1</font> <a href="series_1.html#K3">to_power</a> 1
<b>by </b><i><a class="txt" href="asympt_1.html#E11:109_3"><i><font color="Green">E73</font></i></a>, <a class="txt" href="asympt_1.html#E29:109_3"><i><font color="Green">E79</font></i></a>, </i>
<br/>.= 
<font color="Maroon">N1</font>
<b>by </b><i><a class="ref" href="power.html#T30">POWER:30</a></i>
;<br/><b>hence </b><a NAME="E31:109_3"/>
contradiction
 <b>by </b><i>, <a class="txt" href="asympt_1.html#E9:109"><i><font color="Green">E80</font></i></a>, , </i>;<br/></div><b>end;</b></div>
<a NAME="E9:109"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">g</font> <a href="asympt_0.html#K15">taken_every</a> 2 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">E81</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="109_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:109_4"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/><a NAME="E2:109_4"/><i><font color="Green">E82</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="asympt_0.html#K15">taken_every</a> 2</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</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">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="asympt_0.html#D18">ASYMPT_0:def 18</a></i>;<br/><b>thus </b><a NAME="E3:109_4"/>
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="asympt_0.html#K15">taken_every</a> 2</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 4 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_4_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="109_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:109_4_1_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="asympt_1.html#K9" target="_self">POWEROF2SET</a>  or  not <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="asympt_1.html#K9" target="_self">POWEROF2SET</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="109_4_1_1_1"><b>suppose </b></a><a NAME="E1:109_4_1_1_1"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="asympt_1.html#K9" target="_self">POWEROF2SET</a> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">m</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:109_4_1_1_1"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">m</font>
 ;<br/>2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> 1</span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">m</font></span>)</span>
<b>by </b><i>, <a class="ref" href="power.html#T30">POWER:30</a></i>
<br/>.= 
2 <a href="series_1.html#K3">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="power.html#T32">POWER:32</a></i>
;<br/><a NAME="E5:109_4_1_1_1"/><b>then </b>
2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="asympt_1.html#K9" target="_self">POWEROF2SET</a> 
 ;<br/><a NAME="E6:109_4_1_1_1"/><b>then </b><i><font color="Green">E85</font></i>: 
<font color="Maroon">g</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">n</font></span>)</span> <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/><a NAME="E7:109_4_1_1_1"/>
<font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E8:109_4_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="asympt_0.html#K15">taken_every</a> 2</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 4 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="109_4_1_1_2"><b>suppose </b></a><a NAME="E1:109_4_1_1_2"/><i><font color="Green">E86</font></i>: 
not <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="asympt_1.html#K9" target="_self">POWEROF2SET</a> 
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_4_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:109_4_1_1_2_1"/>
2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="asympt_1.html#K9" target="_self">POWEROF2SET</a> 
 ;<br/><b>then consider </b><font color="Maroon">m</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:109_4_1_1_2_1"/><i><font color="Green">E87</font></i>: 
2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">m</font>
 ;<br/><b>thus </b><a NAME="E4:109_4_1_1_2_1"/>
contradiction
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_4_1_1_2_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="109_4_1_1_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:109_4_1_1_2_1_1_1"/>
( <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="109_4_1_1_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:109_4_1_1_2_1_1_1_1"/>
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:109_4_1_1_2_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="power.html#T29">POWER:29</a></i>;<br/><a NAME="E3:109_4_1_1_2_1_1_1_1"/><b>then </b><i><font color="Green">E88</font></i>: 
<font color="Maroon">n</font> <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">2 <a href="real_1.html#K2">"</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_4_1_1_2_1_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:109_4_1_1_2_1_1_1_1_1"/>
1 <a href="real_1.html#K6">/</a> 2 is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 ;<br/><a NAME="E2:109_4_1_1_2_1_1_1_1_1"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K6">/</a> 2
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E3:109_4_1_1_2_1_1_1_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:109_4_1_1_2_1_1_1_1"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="109_4_1_1_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:109_4_1_1_2_1_1_1_2"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:109_4_1_1_2_1_1_1_2"/><b>then </b>
<font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:109_4_1_1_2_1_1_1_2"/><b>then </b><i><font color="Green">E89</font></i>: 
<font color="Maroon">m</font> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><i><font color="Green">E90</font></i>: 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> = 
2 <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="power.html#T32">POWER:32</a></i>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> 2
<b>by </b><i><a class="ref" href="power.html#T30">POWER:30</a></i>
;<br/><a NAME="E5:109_4_1_1_2_1_1_1_2"/>
<font color="Maroon">m</font> <a href="real_1.html#K5">-</a> 1 is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i>, <a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/><b>hence </b><a NAME="E6:109_4_1_1_2_1_1_1_2"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><b>then </b><i><font color="Green">E91</font></i>: <font color="Maroon">g</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">n</font></span>)</span> = 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="series_1.html#K3">to_power</a> 2
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></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/>.= 
4 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="square_1.html#K5">^2</a> </span>)</span>

;<br/><font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> = 
<font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> 2
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">n</font> <a href="square_1.html#K5">^2</a> 
<b>by </b><i><a class="ref" href="power.html#T53">POWER:53</a></i>
;<br/><b>hence </b><a NAME="E5:109_4_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="asympt_0.html#K15">taken_every</a> 2</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 4 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="asympt_0.html#D18">ASYMPT_0:def 18</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>thus </b><a NAME="E4:109_4"/>
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="asympt_0.html#K15">taken_every</a> 2</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_4_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="109_4_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:109_4_2_1"/>
( 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="asympt_1.html#K9" target="_self">POWEROF2SET</a>  or  not 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="asympt_1.html#K9" target="_self">POWEROF2SET</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="109_4_2_1_1"><b>suppose </b></a><a NAME="E1:109_4_2_1_1"/>
2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="asympt_1.html#K9" target="_self">POWEROF2SET</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:109_4_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="asympt_0.html#K15">taken_every</a> 2</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="109_4_2_1_2"><b>suppose </b></a><a NAME="E1:109_4_2_1_2"/>
not 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="asympt_1.html#K9" target="_self">POWEROF2SET</a> 
 ;<br/><div class="add"><a NAME="E2:109_4_2_1_2"/><b>then </b>
<font color="Maroon">g</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">n</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="series_1.html#K3">to_power</a> 2
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:109_4_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="asympt_0.html#K15">taken_every</a> 2</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="asympt_0.html#D18">ASYMPT_0:def 18</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div>
<div><i><font color="Green">E92</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="109_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:109_5"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/><a NAME="E2:109_5"/><i><font color="Green">E93</font></i>: 
<span class="p1">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> 1</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_5_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="109_5_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:109_5_1_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="109_5_1_1_1"><b>suppose </b></a><a NAME="E1:109_5_1_1_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:109_5_1_1_1"/>
<span class="p1">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> 1</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="109_5_1_1_2"><b>suppose </b></a><a NAME="E1:109_5_1_1_2"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> 1</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> = 
<font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> 1
<b>by </b><i/>
<br/>.= 
<font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="power.html#T30">POWER:30</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><i><font color="Green">E94</font></i>: <span class="p1">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> 1</span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="series_1.html#K3">to_power</a> 1
<b>by </b><i/>
<br/>.= 
<font color="Maroon">n</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="E4:109_5"/>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</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/><b>hence </b><a NAME="E5:109_5"/>
<span class="p1">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> 1</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> 1</span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_6"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:109_6"/>
<font color="Maroon">g</font> is <a href="asympt_0.html#V6">eventually-nondecreasing</a>
 ;<br/><b>then consider </b><font color="Maroon">N</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:109_6"/><i><font color="Green">E95</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">N</font> holds <br/><font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="asympt_0.html#D8">ASYMPT_0:def 8</a></i>;<br/><b>set </b><font color="Maroon">N0</font> =  <a href="asympt_0.html#K4">max</a> <font color="Maroon">N</font>,1;<br/><b>set </b><font color="Maroon">N1</font> = <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><a href="asympt_0.html#K4">max</a> <font color="Maroon">N</font>,1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1;<br/><a NAME="E4:109_6"/>
2 <a href="series_1.html#K3">to_power</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="asympt_0.html#K4">max</a> <font color="Maroon">N</font>,1</span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 <a href="series_1.html#K3">to_power</a> 0
 <b>by </b><i><a class="ref" href="pre_ff.html#T10">PRE_FF:10</a></i>;<br/><a NAME="E5:109_6"/><b>then </b>
2 <a href="series_1.html#K3">to_power</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="asympt_0.html#K4">max</a> <font color="Maroon">N</font>,1</span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="power.html#T29">POWER:29</a></i>;<br/><a NAME="E6:109_6"/><b>then </b><i><font color="Green">E96</font></i>: 
<span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><a href="asympt_0.html#K4">max</a> <font color="Maroon">N</font>,1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E7:109_6"/><i><font color="Green">E97</font></i>: 
 <a href="asympt_0.html#K4">max</a> <font color="Maroon">N</font>,1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/><a NAME="E8:109_6"/><b>then </b><i><font color="Green">E98</font></i>: 
2 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K4">max</a> <font color="Maroon">N</font>,1</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 <a href="nat_1.html#K2">*</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><b>reconsider </b><font color="Maroon">N1</font> = <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><a href="asympt_0.html#K4">max</a> <font color="Maroon">N</font>,1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1 as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, <a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/><a NAME="E10:109_6"/>
not <font color="Maroon">N1</font> <a href="hidden.html#R2">in</a>  <a href="asympt_1.html#K9" target="_self">POWEROF2SET</a> 
 <b>by </b><i>, </i>;<br/><a NAME="E11:109_6"/><b>then </b><i><font color="Green">E99</font></i>: 
<font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">N1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N1</font> <a href="series_1.html#K3">to_power</a> 2
 <b>by </b><i/>;<br/><a NAME="E12:109_6"/>
<font color="Maroon">N1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="asympt_1.html#K9" target="_self">POWEROF2SET</a> 
 ;<br/><a NAME="E13:109_6"/><b>then </b><i><font color="Green">E100</font></i>: 
<font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">N1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">N1</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i/>;<br/><a NAME="E14:109_6"/><i><font color="Green">E101</font></i>: 
2 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K4">max</a> <font color="Maroon">N</font>,1</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 <a href="nat_1.html#K2">*</a> 1
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E15:109_6"/><b>then </b>
2 <a href="series_1.html#K3">to_power</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="asympt_0.html#K4">max</a> <font color="Maroon">N</font>,1</span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 <a href="series_1.html#K3">to_power</a> 2
 <b>by </b><i><a class="ref" href="pre_ff.html#T10">PRE_FF:10</a></i>;<br/><a NAME="E16:109_6"/><b>then </b>
2 <a href="series_1.html#K3">to_power</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="asympt_0.html#K4">max</a> <font color="Maroon">N</font>,1</span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 <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="E17:109_6"/><b>then </b>
<font color="Maroon">N1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 3
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E18:109_6"/><b>then </b>
<font color="Maroon">N1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E19:109_6"/><b>then </b>
<font color="Maroon">N1</font> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">N1</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="asympt_1.html#E8:109"><i><font color="Green">E64</font></i></a></i>;<br/><a NAME="E20:109_6"/><b>then </b><i><font color="Green">E102</font></i>: 
<font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">N1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">N1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E9:109"><i><font color="Green">E80</font></i></a>, <a class="txt" href="asympt_1.html#E10:109"><i><font color="Green">E81</font></i></a>, <a class="ref" href="power.html#T53">POWER:53</a></i>;<br/><a NAME="E21:109_6"/><i><font color="Green">E103</font></i>: 
 <a href="asympt_0.html#K4">max</a> <font color="Maroon">N</font>,1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">N</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/><a NAME="E22:109_6"/>
2 <a href="series_1.html#K3">to_power</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="asympt_0.html#K4">max</a> <font color="Maroon">N</font>,1</span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="asympt_0.html#K4">max</a> <font color="Maroon">N</font>,1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, </i>;<br/><a NAME="E23:109_6"/><b>then </b><i><font color="Green">E104</font></i>: 
<font color="Maroon">N1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 2 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K4">max</a> <font color="Maroon">N</font>,1</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/><a NAME="E24:109_6"/>
2 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K4">max</a> <font color="Maroon">N</font>,1</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K4">max</a> <font color="Maroon">N</font>,1</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E25:109_6"/><b>then </b>
<font color="Maroon">N1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="asympt_0.html#K4">max</a> <font color="Maroon">N</font>,1
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E26:109_6"/><b>then </b>
<font color="Maroon">N1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">N</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E27:109_6"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E13:109"/>
( <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> &amp;  <a href="asympt_1.html#K3" target="_self">seq_n^</a> 1 <a href="hidden.html#R2">in</a>  <a href="asympt_0.html#K14">Big_Theta</a> <font color="Maroon">g</font>,<a href="asympt_1.html#K9" target="_self">POWEROF2SET</a>  &amp; not  <a href="asympt_1.html#K3" target="_self">seq_n^</a> 1 <a href="hidden.html#R2">in</a>  <a href="asympt_0.html#K11">Big_Theta</a> <font color="Maroon">g</font> &amp; <font color="Maroon">g</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">g</font> &amp;  <a href="asympt_1.html#K3" target="_self">seq_n^</a> 1 is <a href="asympt_0.html#V6">eventually-nondecreasing</a> &amp; not <font color="Maroon">g</font> is <a href="asympt_0.html#V6">eventually-nondecreasing</a> )
 <b>by </b><i>, , , , , , <a class="ref" href="asympt_0.html#D8">ASYMPT_0:def 8</a></i>;<br/>


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