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

<b>let </b><font color="Maroon" title="c1">f</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; ( for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <font color="Olive" title="b1">n</font></span>)</span> ) implies  ex <font color="Olive" title="b1">s</font>, <font color="Olive" title="b2">s1</font> being <a href="asympt_0.html#V4" title="ASYMPT_0:attr.4">eventually-positive</a> <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a> st <br/>( <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> &amp; <font color="Olive" title="b2">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="asympt_1.html#K1" title="ASYMPT_1:func.1">seq_a^</a> 2,1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="asympt_0.html#K10" title="ASYMPT_0:func.10">Big_Oh</a> <font color="Olive" title="b1">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="asympt_0.html#K10" title="ASYMPT_0:func.10">Big_Oh</a> <font color="Olive" title="b2">s1</font> &amp; not  <a href="asympt_0.html#K10" title="ASYMPT_0:func.10">Big_Oh</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="asympt_0.html#K10" title="ASYMPT_0:func.10">Big_Oh</a> <font color="Olive" title="b2">s1</font> ) )</span><br/>

<b>assume </b><a NAME="E1:84"/><i><font color="Green" title="E56">A1</font></i>: 
( <font color="Maroon" title="c1">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; ( for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <font color="Olive" title="b1">n</font></span>)</span> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">s</font>, <font color="Olive" title="b2">s1</font> being <a href="asympt_0.html#V4" title="ASYMPT_0:attr.4">eventually-positive</a> <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a> st <br/>( <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> &amp; <font color="Olive" title="b2">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="asympt_1.html#K1" title="ASYMPT_1:func.1">seq_a^</a> 2,1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="asympt_0.html#K10" title="ASYMPT_0:func.10">Big_Oh</a> <font color="Olive" title="b1">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="asympt_0.html#K10" title="ASYMPT_0:func.10">Big_Oh</a> <font color="Olive" title="b2">s1</font> &amp; not  <a href="asympt_0.html#K10" title="ASYMPT_0:func.10">Big_Oh</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="asympt_0.html#K10" title="ASYMPT_0:func.10">Big_Oh</a> <font color="Olive" title="b2">s1</font> )</span><br/>

<b>set </b><font color="Maroon" title="c2">g</font> =  <a href="asympt_1.html#K1" title="ASYMPT_1:func.1">seq_a^</a> 2,1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ;<br/>
<b>set </b><font color="Maroon" title="c3">h</font> = <font color="Maroon" title="c1">f</font> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p1">(<span class="default"><a href="asympt_1.html#K1" title="ASYMPT_1:func.1">seq_a^</a> 2,1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>;<br/>
<a NAME="E2:84"/>
<font color="Maroon" title="c1">f</font> is <a href="asympt_0.html#V4" title="ASYMPT_0:attr.4">eventually-positive</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_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" title="ASYMPT_0:def.6">ASYMPT_0:def 6</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   holds <br/>(  not 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> or  not <font color="Maroon" title="c1">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/>

<b>let </b><font color="Maroon" title="c4">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> or  not <font color="Maroon" title="c1">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/>

<b>assume </b><a NAME="E1:84_1"/><i><font color="Green" title="E57">A2</font></i>: 
<font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not <font color="Maroon" title="c1">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/>

<a NAME="E2:84_1"/><b>then </b>
<font color="Maroon" title="c1">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <font color="Maroon" title="c4">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="asympt_1.html#E1:84"><i><font color="Green" title="E56">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:84_1"/>
<font color="Maroon" title="c1">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="asympt_1.html#E1:84_1"><i><font color="Green" title="E57">A2</font></i></a>, <a class="ref" href="power.html#T39" title="POWER:th.39">POWER:39</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c4">f</font> = <font color="Maroon" title="c1">f</font> as  <a href="asympt_0.html#V4" title="ASYMPT_0:attr.4">eventually-positive</a> <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a> ;<br/>
<b>reconsider </b><font color="Maroon" title="c5">g</font> =  <a href="asympt_1.html#K1" title="ASYMPT_1:func.1">seq_a^</a> 2,1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  as  <a href="asympt_0.html#V4" title="ASYMPT_0:attr.4">eventually-positive</a> <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a> ;<br/>
<b>take </b>
<font color="Maroon" title="c4">f</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">s1</font> being <a href="asympt_0.html#V4" title="ASYMPT_0:attr.4">eventually-positive</a> <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a> st <br/>( <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> &amp; <font color="Olive" title="b1">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="asympt_1.html#K1" title="ASYMPT_1:func.1">seq_a^</a> 2,1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="asympt_0.html#K10" title="ASYMPT_0:func.10">Big_Oh</a> <font color="Maroon" title="c4">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="asympt_0.html#K10" title="ASYMPT_0:func.10">Big_Oh</a> <font color="Olive" title="b1">s1</font> &amp; not  <a href="asympt_0.html#K10" title="ASYMPT_0:func.10">Big_Oh</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="asympt_0.html#K10" title="ASYMPT_0:func.10">Big_Oh</a> <font color="Olive" title="b1">s1</font> )</span><br/>

<b>take </b>
<font color="Maroon" title="c5">g</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> &amp; <font color="Maroon" title="c5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="asympt_1.html#K1" title="ASYMPT_1:func.1">seq_a^</a> 2,1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="asympt_0.html#K10" title="ASYMPT_0:func.10">Big_Oh</a> <font color="Maroon" title="c4">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="asympt_0.html#K10" title="ASYMPT_0:func.10">Big_Oh</a> <font color="Maroon" title="c5">g</font> &amp; not  <a href="asympt_0.html#K10" title="ASYMPT_0:func.10">Big_Oh</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="asympt_0.html#K10" title="ASYMPT_0:func.10">Big_Oh</a> <font color="Maroon" title="c5">g</font> )</span><br/>

<b>consider </b><font color="Maroon" title="c6">N</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E6:84"/><i><font color="Green" title="E57">A3</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">N</font> holds <br/><font color="Olive" title="b1">n</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Olive" title="b1">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b1">n</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 <b>by </b><i><a class="txt" href="asympt_1.html#E76"><i><font color="Green" title="E53">Lm46</font></i></a></i>;<br/>
<div><i><font color="Green" title="E58">A4</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">p</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies  ex <font color="Olive" title="b1">N1</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>for <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">N1</font> holds <br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p4">(<span class="default"><a href="asympt_1.html#K1" title="ASYMPT_1:func.1">seq_a^</a> 2,1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c7">p</font> )</span><br/><b>assume </b><a NAME="E1:84_2"/>
<font color="Maroon" title="c7">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">N1</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>for <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">N1</font> holds <br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p4">(<span class="default"><a href="asympt_1.html#K1" title="ASYMPT_1:func.1">seq_a^</a> 2,1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c7">p</font></span><br/><a NAME="E2:84_2"/><b>then </b>
<font color="Maroon" title="c7">p</font> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="xreal_1.html#T124" title="XREAL_1:th.124">XREAL_1:124</a></i>;<br/><a NAME="E3:84_2"/><b>then </b><i><font color="Green" title="E59">A5</font></i>: 
1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c7">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T217" title="XCMPLX_1:th.217">XCMPLX_1:217</a></i>;<br/><b>set </b><font color="Maroon" title="c8">N1</font> =  <a href="asympt_0.html#K2" title="ASYMPT_0:func.2">max</a> <font color="Maroon" title="c6">N</font>,<span class="p1">(<span class="default"><a href="asympt_0.html#K2" title="ASYMPT_0:func.2">max</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><span class="p4">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<span class="p5">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c7">p</font></span>)</span></span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span></span>)</span>,2</span>)</span>;<br/><a NAME="E4:84_2"/><i><font color="Green" title="E60">A6</font></i>: 
(  <a href="asympt_0.html#K2" title="ASYMPT_0:func.2">max</a> <font color="Maroon" title="c6">N</font>,<span class="p1">(<span class="default"><a href="asympt_0.html#K2" title="ASYMPT_0:func.2">max</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><span class="p4">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<span class="p5">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c7">p</font></span>)</span></span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span></span>)</span>,2</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">N</font> &amp;  <a href="asympt_0.html#K2" title="ASYMPT_0:func.2">max</a> <font color="Maroon" title="c6">N</font>,<span class="p1">(<span class="default"><a href="asympt_0.html#K2" title="ASYMPT_0:func.2">max</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><span class="p4">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<span class="p5">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c7">p</font></span>)</span></span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span></span>)</span>,2</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="asympt_0.html#K2" title="ASYMPT_0:func.2">max</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><span class="p3">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<span class="p4">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c7">p</font></span>)</span></span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span></span>)</span>,2 )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></i>;<br/><a NAME="E5:84_2"/><i><font color="Green" title="E61">A7</font></i>: 
(  <a href="asympt_0.html#K2" title="ASYMPT_0:func.2">max</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><span class="p3">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<span class="p4">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c7">p</font></span>)</span></span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span></span>)</span>,2 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><span class="p2">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c7">p</font></span>)</span></span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span> &amp;  <a href="asympt_0.html#K2" title="ASYMPT_0:func.2">max</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><span class="p3">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<span class="p4">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c7">p</font></span>)</span></span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span></span>)</span>,2 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></i>;<br/><a NAME="E6:84_2"/><b>then </b><i><font color="Green" title="E62">A8</font></i>: 
(  <a href="asympt_0.html#K2" title="ASYMPT_0:func.2">max</a> <font color="Maroon" title="c6">N</font>,<span class="p1">(<span class="default"><a href="asympt_0.html#K2" title="ASYMPT_0:func.2">max</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><span class="p4">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<span class="p5">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c7">p</font></span>)</span></span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span></span>)</span>,2</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><span class="p2">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c7">p</font></span>)</span></span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span> &amp;  <a href="asympt_0.html#K2" title="ASYMPT_0:func.2">max</a> <font color="Maroon" title="c6">N</font>,<span class="p1">(<span class="default"><a href="asympt_0.html#K2" title="ASYMPT_0:func.2">max</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><span class="p4">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<span class="p5">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c7">p</font></span>)</span></span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span></span>)</span>,2</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 )
 <b>by </b><i><a class="txt" href="asympt_1.html#E4:84_2"><i><font color="Green" title="E60">A6</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E7:84_2"/>
 <a href="asympt_0.html#K2" title="ASYMPT_0:func.2">max</a> <font color="Maroon" title="c6">N</font>,<span class="p1">(<span class="default"><a href="asympt_0.html#K2" title="ASYMPT_0:func.2">max</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><span class="p4">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<span class="p5">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c7">p</font></span>)</span></span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span></span>)</span>,2</span>)</span> is   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:84_2_1_1"/>
(  <a href="asympt_0.html#K2" title="ASYMPT_0:func.2">max</a> <font color="Maroon" title="c6">N</font>,<span class="p1">(<span class="default"><a href="asympt_0.html#K2" title="ASYMPT_0:func.2">max</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><span class="p4">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<span class="p5">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c7">p</font></span>)</span></span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span></span>)</span>,2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">N</font> or  <a href="asympt_0.html#K2" title="ASYMPT_0:func.2">max</a> <font color="Maroon" title="c6">N</font>,<span class="p1">(<span class="default"><a href="asympt_0.html#K2" title="ASYMPT_0:func.2">max</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><span class="p4">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<span class="p5">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c7">p</font></span>)</span></span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span></span>)</span>,2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="asympt_0.html#K2" title="ASYMPT_0:func.2">max</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><span class="p3">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<span class="p4">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c7">p</font></span>)</span></span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span></span>)</span>,2 )
 </span><b>by </b><i><a class="ref" href="xxreal_0.html#T16" title="XXREAL_0:th.16">XXREAL_0:16</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_2_1_1_1"><b>suppose </b></a><a NAME="E1:84_2_1_1_1"/>
 <a href="asympt_0.html#K2" title="ASYMPT_0:func.2">max</a> <font color="Maroon" title="c6">N</font>,<span class="p1">(<span class="default"><a href="asympt_0.html#K2" title="ASYMPT_0:func.2">max</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><span class="p4">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<span class="p5">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c7">p</font></span>)</span></span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span></span>)</span>,2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">N</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="asympt_0.html#K2" title="ASYMPT_0:func.2">max</a> <font color="Maroon" title="c6">N</font>,<span class="p1">(<span class="default"><a href="asympt_0.html#K2" title="ASYMPT_0:func.2">max</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><span class="p4">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<span class="p5">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c7">p</font></span>)</span></span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span></span>)</span>,2</span>)</span> is   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a></span><br/><div class="add"><b>hence </b><a NAME="E2:84_2_1_1_1"/>
 <a href="asympt_0.html#K2" title="ASYMPT_0:func.2">max</a> <font color="Maroon" title="c6">N</font>,<span class="p1">(<span class="default"><a href="asympt_0.html#K2" title="ASYMPT_0:func.2">max</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><span class="p4">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<span class="p5">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c7">p</font></span>)</span></span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span></span>)</span>,2</span>)</span> is   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_2_1_1_2"><b>suppose </b></a><a NAME="E1:84_2_1_1_2"/>
 <a href="asympt_0.html#K2" title="ASYMPT_0:func.2">max</a> <font color="Maroon" title="c6">N</font>,<span class="p1">(<span class="default"><a href="asympt_0.html#K2" title="ASYMPT_0:func.2">max</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><span class="p4">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<span class="p5">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c7">p</font></span>)</span></span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span></span>)</span>,2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="asympt_0.html#K2" title="ASYMPT_0:func.2">max</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><span class="p3">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<span class="p4">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c7">p</font></span>)</span></span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span></span>)</span>,2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="asympt_0.html#K2" title="ASYMPT_0:func.2">max</a> <font color="Maroon" title="c6">N</font>,<span class="p1">(<span class="default"><a href="asympt_0.html#K2" title="ASYMPT_0:func.2">max</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><span class="p4">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<span class="p5">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c7">p</font></span>)</span></span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span></span>)</span>,2</span>)</span> is   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a></span><br/><div class="add"><b>hence </b><a NAME="E2:84_2_1_1_2"/>
 <a href="asympt_0.html#K2" title="ASYMPT_0:func.2">max</a> <font color="Maroon" title="c6">N</font>,<span class="p1">(<span class="default"><a href="asympt_0.html#K2" title="ASYMPT_0:func.2">max</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><span class="p4">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<span class="p5">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c7">p</font></span>)</span></span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span></span>)</span>,2</span>)</span> is   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T16" title="XXREAL_0:th.16">XXREAL_0:16</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c9">N1</font> =  <a href="asympt_0.html#K2" title="ASYMPT_0:func.2">max</a> <font color="Maroon" title="c6">N</font>,<span class="p1">(<span class="default"><a href="asympt_0.html#K2" title="ASYMPT_0:func.2">max</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><span class="p4">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<span class="p5">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c7">p</font></span>)</span></span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span></span>)</span>,2</span>)</span> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="asympt_1.html#E4:84_2"><i><font color="Green" title="E60">A6</font></i></a>, <a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/><b>take </b><font color="Maroon" title="c10">N1</font> = <font color="Maroon" title="c9">N1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c10">N1</font> holds <br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p4">(<span class="default"><a href="asympt_1.html#K1" title="ASYMPT_1:func.1">seq_a^</a> 2,1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c7">p</font></span><br/><b>let </b><font color="Maroon" title="c11">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c11">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c10">N1</font> implies  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p4">(<span class="default"><a href="asympt_1.html#K1" title="ASYMPT_1:func.1">seq_a^</a> 2,1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c7">p</font> )</span><br/><b>assume </b><a NAME="E9:84_2"/><i><font color="Green" title="E63">A9</font></i>: 
<font color="Maroon" title="c11">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c10">N1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p4">(<span class="default"><a href="asympt_1.html#K1" title="ASYMPT_1:func.1">seq_a^</a> 2,1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c7">p</font></span><br/><a NAME="E10:84_2"/><i><font color="Green" title="E64">A10</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p2">(<span class="default"><a href="asympt_1.html#K1" title="ASYMPT_1:func.1">seq_a^</a> 2,1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E10"><i><font color="Green" title="E7">Lm4</font></i></a></i>;<br/><i><font color="Green" title="E65">A11</font></i>: <font color="Maroon" title="c4">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">n</font> = 
<font color="Maroon" title="c11">n</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <font color="Maroon" title="c11">n</font></span>)</span>
<b>by </b><i><a class="txt" href="asympt_1.html#E1:84"><i><font color="Green" title="E56">A1</font></i></a>, <a class="txt" href="asympt_1.html#E4:84_2"><i><font color="Green" title="E60">A6</font></i></a>, <a class="txt" href="asympt_1.html#E5:84_2"><i><font color="Green" title="E61">A7</font></i></a>, <a class="txt" href="asympt_1.html#E9:84_2"><i><font color="Green" title="E63">A9</font></i></a></i>
<br/>.= 
2 <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c11">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="asympt_1.html#E4:84_2"><i><font color="Green" title="E60">A6</font></i></a>, <a class="txt" href="asympt_1.html#E5:84_2"><i><font color="Green" title="E61">A7</font></i></a>, <a class="txt" href="asympt_1.html#E9:84_2"><i><font color="Green" title="E63">A9</font></i></a>, <a class="txt" href="asympt_1.html#E6"><i><font color="Green" title="E4">Lm3</font></i></a></i>
;<br/><font color="Maroon" title="c5">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">n</font> = 
2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
<b>by </b><i><a class="ref" href="asympt_1.html#D1" target="_self" title="ASYMPT_1:def.1">Def1</a></i>
<br/>.= 
2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c11">n</font>

;<br/><b>then </b><i><font color="Green" title="E66">A12</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p2">(<span class="default"><a href="asympt_1.html#K1" title="ASYMPT_1:func.1">seq_a^</a> 2,1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">n</font> = 
2 <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c11">n</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c11">n</font></span>)</span>
<b>by </b><i><a class="txt" href="asympt_1.html#E10:84_2"><i><font color="Green" title="E64">A10</font></i></a>, <a class="txt" href="asympt_1.html#E11:84_2"><i><font color="Green" title="E65">A11</font></i></a>, <a class="ref" href="power.html#T34" title="POWER:th.34">POWER:34</a></i>
<br/>.= 
2 <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">n</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c11">n</font></span>)</span></span>)</span></span>)</span></span>)</span>

;<br/><a NAME="E14:84_2"/>
<font color="Maroon" title="c11">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">N</font>
 <b>by </b><i><a class="txt" href="asympt_1.html#E4:84_2"><i><font color="Green" title="E60">A6</font></i></a>, <a class="txt" href="asympt_1.html#E9:84_2"><i><font color="Green" title="E63">A9</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E15:84_2"/><b>then </b>
<font color="Maroon" title="c11">n</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c11">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c11">n</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 <b>by </b><i><a class="txt" href="asympt_1.html#E6:84"><i><font color="Green" title="E57">A3</font></i></a></i>;<br/><a NAME="E16:84_2"/><b>then </b>
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">n</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c11">n</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">n</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T26" title="XREAL_1:th.26">XREAL_1:26</a></i>;<br/><a NAME="E17:84_2"/><b>then </b><i><font color="Green" title="E67">A13</font></i>: 
2 <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">n</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c11">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 2 <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">n</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="power.html#T44" title="POWER:th.44">POWER:44</a></i>;<br/><a NAME="E18:84_2"/><i><font color="Green" title="E68">A14</font></i>: 
<span class="p1"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><span class="p2">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c7">p</font></span>)</span></span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="power.html#K6" title="POWER:func.6">log</a> 2,<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c7">p</font></span>)</span>
 <b>by </b><i><a class="ref" href="int_1.html#D5" title="INT_1:def.5">INT_1:def 5</a></i>;<br/><a NAME="E19:84_2"/>
<font color="Maroon" title="c11">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><span class="p2">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c7">p</font></span>)</span></span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E6:84_2"><i><font color="Green" title="E62">A8</font></i></a>, <a class="txt" href="asympt_1.html#E9:84_2"><i><font color="Green" title="E63">A9</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E20:84_2"/><b>then </b>
<font color="Maroon" title="c11">n</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><span class="p2">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c7">p</font></span>)</span></span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>
 <b>by </b><i><a class="ref" href="real_2.html#T177" title="REAL_2:th.177">REAL_2:177</a></i>;<br/><a NAME="E21:84_2"/><b>then </b>
<font color="Maroon" title="c11">n</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="power.html#K6" title="POWER:func.6">log</a> 2,<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c7">p</font></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E18:84_2"><i><font color="Green" title="E68">A14</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E22:84_2"/><b>then </b>
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">n</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c7">p</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T26" title="XREAL_1:th.26">XREAL_1:26</a></i>;<br/><a NAME="E23:84_2"/><b>then </b>
2 <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">n</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c7">p</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="pre_ff.html#T10" title="PRE_FF:th.10">PRE_FF:10</a></i>;<br/><a NAME="E24:84_2"/><b>then </b>
2 <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">n</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p2">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c7">p</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="power.html#T33" title="POWER:th.33">POWER:33</a></i>;<br/><a NAME="E25:84_2"/><b>then </b>
2 <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">n</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c7">p</font></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E3:84_2"><i><font color="Green" title="E59">A5</font></i></a>, <a class="ref" href="power.html#D3" title="POWER:def.3">POWER:def 3</a></i>;<br/><a NAME="E26:84_2"/><b>then </b>
2 <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">n</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">p</font>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T56" title="XCMPLX_1:th.56">XCMPLX_1:56</a></i>;<br/><a NAME="E27:84_2"/><b>then </b><i><font color="Green" title="E69">A15</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p2">(<span class="default"><a href="asympt_1.html#K1" title="ASYMPT_1:func.1">seq_a^</a> 2,1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c7">p</font>
 <b>by </b><i><a class="txt" href="asympt_1.html#E13:84_2"><i><font color="Green" title="E66">A12</font></i></a>, <a class="txt" href="asympt_1.html#E17:84_2"><i><font color="Green" title="E67">A13</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E28:84_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p2">(<span class="default"><a href="asympt_1.html#K1" title="ASYMPT_1:func.1">seq_a^</a> 2,1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="asympt_1.html#E13:84_2"><i><font color="Green" title="E66">A12</font></i></a>, <a class="ref" href="power.html#T39" title="POWER:th.39">POWER:39</a></i>;<br/><b>hence </b><a NAME="E29:84_2"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p4">(<span class="default"><a href="asympt_1.html#K1" title="ASYMPT_1:func.1">seq_a^</a> 2,1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c7">p</font>
 <b>by </b><i><a class="txt" href="asympt_1.html#E27:84_2"><i><font color="Green" title="E69">A15</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E8:84"/><b>then </b><i><font color="Green" title="E59">A16</font></i>: 
<font color="Maroon" title="c1">f</font> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p1">(<span class="default"><a href="asympt_1.html#K1" title="ASYMPT_1:func.1">seq_a^</a> 2,1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 
<b>by </b><i><a class="ref" href="seq_2.html#D6" title="SEQ_2:def.6">SEQ_2:def 6</a></i>;<br/>
<a NAME="E9:84"/><b>then </b>
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p2">(<span class="default"><a href="asympt_1.html#K1" title="ASYMPT_1:func.1">seq_a^</a> 2,1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="asympt_1.html#E7:84"><i><font color="Green" title="E58">A4</font></i></a>, <a class="ref" href="seq_2.html#D7" title="SEQ_2:def.7">SEQ_2:def 7</a></i>;<br/>
<a NAME="E10:84"/><b>then </b>
( <font color="Maroon" title="c4">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_0.html#K10" title="ASYMPT_0:func.10">Big_Oh</a> <font color="Maroon" title="c5">g</font> &amp; not <font color="Maroon" title="c5">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_0.html#K10" title="ASYMPT_0:func.10">Big_Oh</a> <font color="Maroon" title="c4">f</font> )
 
<b>by </b><i><a class="txt" href="asympt_1.html#E8:84"><i><font color="Green" title="E59">A16</font></i></a>, <a class="ref" href="asympt_0.html#T16" title="ASYMPT_0:th.16">ASYMPT_0:16</a></i>;<br/>
<a NAME="E11:84"/><b>then </b>
( <font color="Maroon" title="c4">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_0.html#K10" title="ASYMPT_0:func.10">Big_Oh</a> <font color="Maroon" title="c5">g</font> &amp; not <font color="Maroon" title="c4">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_0.html#K11" title="ASYMPT_0:func.11">Big_Omega</a> <font color="Maroon" title="c5">g</font> )
 
<b>by </b><i><a class="ref" href="asympt_0.html#T19" title="ASYMPT_0:th.19">ASYMPT_0:19</a></i>;<br/>
<b>hence </b><a NAME="E12:84"/>
( <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> &amp; <font color="Maroon" title="c5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="asympt_1.html#K1" title="ASYMPT_1:func.1">seq_a^</a> 2,1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="asympt_0.html#K10" title="ASYMPT_0:func.10">Big_Oh</a> <font color="Maroon" title="c4">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="asympt_0.html#K10" title="ASYMPT_0:func.10">Big_Oh</a> <font color="Maroon" title="c5">g</font> &amp; not  <a href="asympt_0.html#K10" title="ASYMPT_0:func.10">Big_Oh</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="asympt_0.html#K10" title="ASYMPT_0:func.10">Big_Oh</a> <font color="Maroon" title="c5">g</font> )
 <b>by </b><i><a class="ref" href="asympt_1.html#T4" target="_self" title="ASYMPT_1:th.4">Th4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
