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

<b>let </b><font color="Maroon">h</font> be  <a href="asympt_0.html#V2">eventually-nonnegative</a> <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>

<b>reconsider </b><font color="Maroon">F</font> = <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> 1</span>)</span></span><a href="domain_1.html#K6">}</a></span> as    <a href="fraenkel.html#M1">FUNCTION_DOMAIN</a> of  <a href="numbers.html#K5">NAT</a> , <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="fraenkel.html#T10">FRAENKEL:10</a></i>;<br/>
<b>take </b>
<font color="Maroon">F</font>
;<br/>

<b>thus </b><a NAME="E2:41"/>
<font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> 1</span>)</span></span><a href="domain_1.html#K6">}</a></span>
 ;<br/>

<b>set </b><font color="Maroon">G</font> =  <a href="asympt_0.html#K9">Big_Oh</a> <span class="p1">(<span class="default"><a href="asympt_1.html#K4" target="_self">seq_const</a> 1</span>)</span>;<br/>
<b>set </b><font color="Maroon">p</font> =  <a href="asympt_1.html#K4" target="_self">seq_const</a> 1;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><b>now </b></a><div class="add"><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:41_1_1"/>
<font color="Maroon">h</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="asympt_0.html#K18">to_power</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K9">Big_Oh</a> <span class="p2">(<span class="default"><a href="asympt_1.html#K4" target="_self">seq_const</a> 1</span>)</span></span>)</span>
 ;<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="E3:41_1_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font>
 <b>and </b><br/><a NAME="E4:41_1_1"/><i><font color="Green">E50</font></i>: 
 ex <font color="Olive">f</font>, <font color="Olive">g</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>  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">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Olive">g</font> <a href="hidden.html#R2">in</a>  <a href="asympt_0.html#K9">Big_Oh</a> <span class="p1">(<span class="default"><a href="asympt_1.html#K4" target="_self">seq_const</a> 1</span>)</span> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">N</font> holds <br/><font color="Maroon">t</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><font color="Olive">g</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> ) )
 ;<br/><b>consider </b><font color="Maroon">f</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> , <font color="Maroon">g</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> , <font color="Maroon">N0</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="E6:41_1_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>and </b><br/><a NAME="E7:41_1_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a>  <a href="asympt_0.html#K9">Big_Oh</a> <span class="p1">(<span class="default"><a href="asympt_1.html#K4" target="_self">seq_const</a> 1</span>)</span>
 <b>and </b><br/><a NAME="E8:41_1_1"/><i><font color="Green">E59</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">N0</font> holds <br/><font color="Maroon">t</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> <a href="power.html#K4">to_power</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/>;<br/><a NAME="E9:41_1_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="asympt_1.html#K3" target="_self">seq_n^</a> 1
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>consider </b><font color="Maroon">g'</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="E11:41_1_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g'</font>
 <b>and </b><br/><a NAME="E12:41_1_1"/><i><font color="Green">E62</font></i>: 
 ex <font color="Olive">c</font> being  <a href="real_1.html#NM1">Real</a> ex <font color="Olive">N</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">c</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">N</font> holds <br/>( <font color="Maroon">g'</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">c</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="asympt_1.html#K4" target="_self">seq_const</a> 1</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> &amp; <font color="Maroon">g'</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) ) )
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">c</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">N1</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="E14:41_1_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">c</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>and </b><br/><a NAME="E15:41_1_1"/><i><font color="Green">E64</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">N1</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">c</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="asympt_1.html#K4" target="_self">seq_const</a> 1</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> &amp; <font color="Maroon">g'</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 <b>by </b><i/>;<br/><b>set </b><font color="Maroon">N</font> =  <a href="asympt_0.html#K4">max</a> 2,<span class="p1">(<span class="default"><a href="asympt_0.html#K4">max</a> <font color="Maroon">N0</font>,<font color="Maroon">N1</font></span>)</span>;<br/><a NAME="E16:41_1_1"/>
(  <a href="asympt_0.html#K4">max</a> <font color="Maroon">N0</font>,<font color="Maroon">N1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">N0</font> &amp;  <a href="asympt_0.html#K4">max</a> <font color="Maroon">N0</font>,<font color="Maroon">N1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">N1</font> &amp;  <a href="asympt_0.html#K4">max</a> 2,<span class="p1">(<span class="default"><a href="asympt_0.html#K4">max</a> <font color="Maroon">N0</font>,<font color="Maroon">N1</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="asympt_0.html#K4">max</a> <font color="Maroon">N0</font>,<font color="Maroon">N1</font> )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/><a NAME="E17:41_1_1"/><b>then </b><i><font color="Green">E67</font></i>: 
(  <a href="asympt_0.html#K4">max</a> 2,<span class="p1">(<span class="default"><a href="asympt_0.html#K4">max</a> <font color="Maroon">N0</font>,<font color="Maroon">N1</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">N0</font> &amp;  <a href="asympt_0.html#K4">max</a> 2,<span class="p1">(<span class="default"><a href="asympt_0.html#K4">max</a> <font color="Maroon">N0</font>,<font color="Maroon">N1</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">N1</font> &amp;  <a href="asympt_0.html#K4">max</a> 2,<span class="p1">(<span class="default"><a href="asympt_0.html#K4">max</a> <font color="Maroon">N0</font>,<font color="Maroon">N1</font></span>)</span> <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>, <a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/><b>set </b><font color="Maroon">k</font> = <span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><font color="Maroon">c</font></span><a href="int_1.html#K2">\]</a></span>;<br/><a NAME="E18:41_1_1"/><i><font color="Green">E69</font></i>: 
<span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><font color="Maroon">c</font></span><a href="int_1.html#K2">\]</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c</font>
 <b>by </b><i><a class="ref" href="int_1.html#D5">INT_1:def 5</a></i>;<br/><a NAME="E19:41_1_1"/>
<span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><font color="Maroon">c</font></span><a href="int_1.html#K2">\]</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, <a class="ref" href="int_1.html#D5">INT_1:def 5</a></i>;<br/><b>then reconsider </b><font color="Maroon">k</font> = <span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><font color="Maroon">c</font></span><a href="int_1.html#K2">\]</a></span> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/><b>reconsider </b><font color="Maroon">i</font> = 1 as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><b>take </b><font color="Maroon">N</font> =  <a href="asympt_0.html#K4">max</a> 2,<span class="p1">(<span class="default"><a href="asympt_0.html#K4">max</a> <font color="Maroon">N0</font>,<font color="Maroon">N1</font></span>)</span>;<br/><b>take </b><font color="Maroon">i</font> = <font color="Maroon">i</font>;<br/><b>take </b><font color="Maroon">k</font> = <font color="Maroon">k</font>;<br/><b>thus </b><a NAME="E22:41_1_1"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<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="E23:41_1_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">N</font>
 ;<br/><a NAME="E24:41_1_1"/><b>then </b><i><font color="Green">E71</font></i>: 
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">N0</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">N1</font> &amp; <font color="Maroon">n</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="E25:41_1_1"/><b>then </b><i><font color="Green">E72</font></i>: 
( <font color="Maroon">g'</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="asympt_1.html#K4" target="_self">seq_const</a> 1</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">g'</font> <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/>;<br/><a NAME="E26:41_1_1"/><i><font color="Green">E73</font></i>: 
<span class="p1">(<span class="default"><a href="asympt_1.html#K4" target="_self">seq_const</a> 1</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/><a NAME="E27:41_1_1"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">n</font> <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/><font color="Maroon">f</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> 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/><a NAME="E29:41_1_1"/><b>then </b><i><font color="Green">E75</font></i>: 
<font color="Maroon">h</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="power.html#K4">to_power</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/><a NAME="E30:41_1_1"/>
<font color="Maroon">n</font> <a href="power.html#K4">to_power</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#NR2" target="_self">&gt;=</a> <font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> 0
 <b>by </b><i>, , <a class="txt" href="asympt_1.html#E1"><i><font color="Green">Lemma47</font></i></a>, <a class="ref" href="pre_ff.html#T10">PRE_FF:10</a></i>;<br/><b>hence </b><a NAME="E31:41_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="power.html#T29">POWER:29</a></i>;<br/><a NAME="E32:41_1_1"/>
<font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="asympt_1.html#K4" target="_self">seq_const</a> 1</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, , </i>;<br/><a NAME="E33:41_1_1"/><b>then </b><i><font color="Green">E77</font></i>: 
<font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="real_1.html#K4">*</a> 1</span>)</span>
 <b>by </b><i>, <a class="txt" href="asympt_1.html#E1"><i><font color="Green">Lemma47</font></i></a>, , <a class="ref" href="pre_ff.html#T10">PRE_FF:10</a></i>;<br/><a NAME="E34:41_1_1"/>
<font color="Maroon">n</font> <a href="power.html#K4">to_power</a> <font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="txt" href="asympt_1.html#E1"><i><font color="Green">Lemma47</font></i></a>, <a class="ref" href="pre_ff.html#T10">PRE_FF:10</a></i>;<br/><a NAME="E35:41_1_1"/><b>then </b>
<font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="txt" href="asympt_1.html#E3:41_1_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E36:41_1_1"/>
<font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> <font color="Maroon">k</font></span>)</span> <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><b>given </b><font color="Maroon">N0</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">c</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><a NAME="E2:41_1"/>
<font color="Maroon">c</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>and </b><br/><a NAME="E3:41_1"/><i><font color="Green">E78</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">N0</font> holds <br/>( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> &amp; <font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> )
 ;<br/><b>reconsider </b><font color="Maroon">t</font> = <font color="Maroon">h</font> as    <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/><b>set </b><font color="Maroon">N</font> =  <a href="asympt_0.html#K4">max</a> <font color="Maroon">N0</font>,2;<br/><a NAME="E5:41_1"/><i><font color="Green">E79</font></i>: 
(  <a href="asympt_0.html#K4">max</a> <font color="Maroon">N0</font>,2 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">N0</font> &amp;  <a href="asympt_0.html#K4">max</a> <font color="Maroon">N0</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="E6:41_1"/><b>then </b><i><font color="Green">E80</font></i>: 
 <a href="asympt_0.html#K4">max</a> <font color="Maroon">N0</font>,2 <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="E7:41_1"/><i><font color="Green">E81</font></i>: 
(  <a href="asympt_0.html#K4">max</a> <font color="Maroon">N0</font>,2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="asympt_0.html#K4">max</a> <font color="Maroon">N0</font>,2 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/><b>set </b><font color="Maroon">c1</font> =  <a href="asympt_0.html#K2">max</a> <font color="Maroon">c</font>,2;<br/><a NAME="E8:41_1"/><i><font color="Green">E82</font></i>: 
(  <a href="asympt_0.html#K2">max</a> <font color="Maroon">c</font>,2 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c</font> &amp;  <a href="asympt_0.html#K2">max</a> <font color="Maroon">c</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="E9:41_1"/><b>then </b><i><font color="Green">E83</font></i>: 
(  <a href="asympt_0.html#K2">max</a> <font color="Maroon">c</font>,2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="asympt_0.html#K2">max</a> <font color="Maroon">c</font>,2 <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/><b>set </b><font color="Maroon">a</font> =  <a href="power.html#K6">log</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K4">max</a> <font color="Maroon">N0</font>,2</span>)</span>,<span class="p1">(<span class="default"><a href="asympt_0.html#K2">max</a> <font color="Maroon">c</font>,2</span>)</span>;<br/><b>set </b><font color="Maroon">b</font> = <font color="Maroon">k</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="power.html#K6">log</a> <span class="p2">(<span class="default"><a href="asympt_0.html#K4">max</a> <font color="Maroon">N0</font>,2</span>)</span>,<span class="p2">(<span class="default"><a href="asympt_0.html#K2">max</a> <font color="Maroon">c</font>,2</span>)</span></span>)</span>;<br/><b>reconsider </b><font color="Maroon">f</font> =  <a href="asympt_1.html#K3" target="_self">seq_n^</a> 1 as    <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/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ,  <a href="real_1.html#NM1">Real</a>] <b>means </b>( ( <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="asympt_0.html#K4">max</a> <font color="Maroon">N0</font>,2 implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> 1 ) &amp; ( <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="asympt_0.html#K4">max</a> <font color="Maroon">N0</font>,2 implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="power.html#K6">log</a> <font color="Maroon">a<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> ) );<br/><a NAME="E11:41_1"/><i><font color="Green">E84</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   ex <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_2"><b>proof </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/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:41_1_2_1"/>
( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="asympt_0.html#K4">max</a> <font color="Maroon">N0</font>,2 or <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="asympt_0.html#K4">max</a> <font color="Maroon">N0</font>,2 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_2_1_1"><b>suppose </b></a><a NAME="E1:41_1_2_1_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="asympt_0.html#K4">max</a> <font color="Maroon">N0</font>,2
 ;<br/><div class="add"><b>hence </b><a NAME="E2:41_1_2_1_1"/>
 ex <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Olive">y</font>]
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_2_1_2"><b>suppose </b></a><a NAME="E1:41_1_2_1_2"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="asympt_0.html#K4">max</a> <font color="Maroon">N0</font>,2
 ;<br/><div class="add"><b>hence </b><a NAME="E2:41_1_2_1_2"/>
 ex <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Olive">y</font>]
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>consider </b><font color="Maroon">g</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E13:41_1"/><i><font color="Green">E85</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>();<br/></i><a NAME="E14:41_1"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">g</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">E87</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_3"><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="E1:41_1_3"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="asympt_0.html#K4">max</a> <font color="Maroon">N0</font>,2
 ;<br/><a NAME="E2:41_1_3"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">N0</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E3:41_1_3"/><b>then </b><i><font color="Green">E89</font></i>: 
<font color="Maroon">t</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i/>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="power.html#K4">to_power</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> = 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> 1</span>)</span> <a href="power.html#K4">to_power</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/>.= 
<font color="Maroon">n</font> <a href="power.html#K4">to_power</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="power.html#T30">POWER:30</a></i>
<br/>.= 
<font color="Maroon">n</font> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="power.html#K6">log</a> <font color="Maroon">n</font>,<span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">t</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
<b>by </b><i>, , , <a class="ref" href="power.html#D3">POWER:def 3</a></i>
;<br/></div><b>end;</b></div><a NAME="E16:41_1"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_4"><b>now </b></a><div class="add"><a NAME="E1:41_1_4"/>
 <a href="power.html#K6">log</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K4">max</a> <font color="Maroon">N0</font>,2</span>)</span>,1 <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="ref" href="power.html#T59">POWER:59</a></i>;<br/><a NAME="E2:41_1_4"/><b>then </b>
 <a href="power.html#K6">log</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K4">max</a> <font color="Maroon">N0</font>,2</span>)</span>,<span class="p1">(<span class="default"><a href="asympt_0.html#K2">max</a> <font color="Maroon">c</font>,2</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, , <a class="ref" href="power.html#T65">POWER:65</a></i>;<br/><a NAME="E3:41_1_4"/><b>then </b>
<font color="Maroon">k</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="power.html#K6">log</a> <span class="p2">(<span class="default"><a href="asympt_0.html#K4">max</a> <font color="Maroon">N0</font>,2</span>)</span>,<span class="p2">(<span class="default"><a href="asympt_0.html#K2">max</a> <font color="Maroon">c</font>,2</span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><b>hence </b><a NAME="E4:41_1_4"/>
<font color="Maroon">k</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="power.html#K6">log</a> <span class="p2">(<span class="default"><a href="asympt_0.html#K4">max</a> <font color="Maroon">N0</font>,2</span>)</span>,<span class="p2">(<span class="default"><a href="asympt_0.html#K2">max</a> <font color="Maroon">c</font>,2</span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<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="E5:41_1_4"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="asympt_0.html#K4">max</a> <font color="Maroon">N0</font>,2
 ;<br/><a NAME="E6:41_1_4"/><b>then </b><i><font color="Green">E92</font></i>: 
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">N0</font> &amp; <font color="Maroon">n</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="E7:41_1_4"/><b>then </b><i><font color="Green">E93</font></i>: 
<font color="Maroon">n</font> <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="E8:41_1_4"/><i><font color="Green">E94</font></i>: 
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 )
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E9:41_1_4"/><i><font color="Green">E95</font></i>: 
 <a href="power.html#K6">log</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K4">max</a> <font color="Maroon">N0</font>,2</span>)</span>,<span class="p1">(<span class="default"><a href="asympt_0.html#K2">max</a> <font color="Maroon">c</font>,2</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="power.html#K6">log</a> <font color="Maroon">n</font>,<span class="p1">(<span class="default"><a href="asympt_0.html#K2">max</a> <font color="Maroon">c</font>,2</span>)</span>
 <b>by </b><i>, , , </i>;<br/><a NAME="E10:41_1_4"/><i><font color="Green">E96</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> &amp; <font color="Maroon">t</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> )
 <b>by </b><i>, </i>;<br/><a NAME="E11:41_1_4"/><i><font color="Green">E97</font></i>: 
<span class="p1">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> <font color="Maroon">k</font></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> <a href="series_1.html#K3">to_power</a> <font color="Maroon">k</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E12:41_1_4"/><b>then </b><i><font color="Green">E98</font></i>: 
<span class="p1">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> <font color="Maroon">k</font></span>)</span> <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/><a NAME="E13:41_1_4"/>
<font color="Maroon">c</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> <font color="Maroon">k</font></span>)</span> <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_0.html#K2">max</a> <font color="Maroon">c</font>,2</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> <font color="Maroon">k</font></span>)</span> <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/><a NAME="E14:41_1_4"/><b>then </b>
<font color="Maroon">t</font> <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_0.html#K2">max</a> <font color="Maroon">c</font>,2</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E1"><i><font color="Green">Lemma47</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E15:41_1_4"/><b>then </b><i><font color="Green">E99</font></i>: 
 <a href="power.html#K6">log</a> <font color="Maroon">n</font>,<span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="power.html#K6">log</a> <font color="Maroon">n</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="asympt_0.html#K2">max</a> <font color="Maroon">c</font>,2</span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="txt" href="asympt_1.html#E1"><i><font color="Green">Lemma47</font></i></a>, <a class="ref" href="pre_ff.html#T12">PRE_FF:12</a></i>;<br/><font color="Maroon">t</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="power.html#K4">to_power</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/>.= 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> 1</span>)</span> <a href="power.html#K4">to_power</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/>.= 
<font color="Maroon">n</font> <a href="power.html#K4">to_power</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="power.html#T30">POWER:30</a></i>
;<br/><b>then </b><i><font color="Green">E100</font></i>:  <a href="power.html#K6">log</a> <font color="Maroon">n</font>,<span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> = 
<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="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="power.html#K6">log</a> <font color="Maroon">n</font>,<font color="Maroon">n</font></span>)</span>
<b>by </b><i>, <a class="ref" href="power.html#T63">POWER:63</a></i>
<br/>.= 
<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="real_1.html#K4">*</a> 1
<b>by </b><i>, <a class="ref" href="power.html#T60">POWER:60</a></i>
;<br/><i><font color="Green">E101</font></i>:  <a href="power.html#K6">log</a> <font color="Maroon">n</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="asympt_0.html#K2">max</a> <font color="Maroon">c</font>,2</span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="power.html#K6">log</a> <font color="Maroon">n</font>,<span class="p2">(<span class="default"><a href="asympt_0.html#K2">max</a> <font color="Maroon">c</font>,2</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="power.html#K6">log</a> <font color="Maroon">n</font>,<span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> <font color="Maroon">k</font></span>)</span></span>)</span>
<b>by </b><i>, , , , <a class="ref" href="power.html#T61">POWER:61</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="power.html#K6">log</a> <font color="Maroon">n</font>,<span class="p2">(<span class="default"><a href="asympt_0.html#K2">max</a> <font color="Maroon">c</font>,2</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="power.html#K6">log</a> <font color="Maroon">n</font>,<font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="power.html#T63">POWER:63</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="power.html#K6">log</a> <font color="Maroon">n</font>,<span class="p2">(<span class="default"><a href="asympt_0.html#K2">max</a> <font color="Maroon">c</font>,2</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K2">*</a> 1</span>)</span>
<b>by </b><i>, <a class="ref" href="power.html#T60">POWER:60</a></i>
;<br/><a NAME="E19:41_1_4"/><i><font color="Green">E102</font></i>: 
<span class="p1">(<span class="default"><a href="power.html#K6">log</a> <font color="Maroon">n</font>,<span class="p2">(<span class="default"><a href="asympt_0.html#K2">max</a> <font color="Maroon">c</font>,2</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="power.html#K6">log</a> <span class="p2">(<span class="default"><a href="asympt_0.html#K4">max</a> <font color="Maroon">N0</font>,2</span>)</span>,<span class="p2">(<span class="default"><a href="asympt_0.html#K2">max</a> <font color="Maroon">c</font>,2</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E20:41_1_4"/>
<span class="p1">(<span class="default"><a href="asympt_1.html#K4" target="_self">seq_const</a> 1</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/><b>hence </b><a NAME="E21:41_1_4"/>
<font color="Maroon">g</font> <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"><font color="Maroon">k</font> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="power.html#K6">log</a> <span class="p3">(<span class="default"><a href="asympt_0.html#K4">max</a> <font color="Maroon">N0</font>,2</span>)</span>,<span class="p3">(<span class="default"><a href="asympt_0.html#K2">max</a> <font color="Maroon">c</font>,2</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="asympt_1.html#K4" target="_self">seq_const</a> 1</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, , , , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E22:41_1_4"/>
<font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="power.html#K6">log</a> <font color="Maroon">n</font>,<span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E23:41_1_4"/><b>then </b>
<font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="power.html#K6">log</a> <font color="Maroon">n</font>,1
 <b>by </b><i>, <a class="txt" href="asympt_1.html#E1"><i><font color="Green">Lemma47</font></i></a>, <a class="ref" href="pre_ff.html#T12">PRE_FF:12</a></i>;<br/><b>hence </b><a NAME="E24:41_1_4"/>
<font color="Maroon">g</font> <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="power.html#T59">POWER:59</a></i>;<br/></div><b>end;</b></div><a NAME="E18:41_1"/><b>then </b>
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a>  <a href="asympt_0.html#K9">Big_Oh</a> <span class="p1">(<span class="default"><a href="asympt_1.html#K4" target="_self">seq_const</a> 1</span>)</span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E19:41_1"/>
<font color="Maroon">h</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="asympt_0.html#K18">to_power</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K9">Big_Oh</a> <span class="p2">(<span class="default"><a href="asympt_1.html#K4" target="_self">seq_const</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:41"/>
( <font color="Maroon">h</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="asympt_0.html#K18">to_power</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K9">Big_Oh</a> <span class="p2">(<span class="default"><a href="asympt_1.html#K4" target="_self">seq_const</a> 1</span>)</span></span>)</span> iff  ex <font color="Olive">N</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ex <font color="Olive">c</font> being  <a href="real_1.html#NM1">Real</a> ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">c</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">N</font> holds <br/>( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> &amp; <font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">c</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> <font color="Olive">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> ) ) ) )
 ;<br/>


</div>
