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

<span class="kw">let </span><font color="Maroon" title="c1">g</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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>  holds <br/> ( ( <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_1.html#K7" title="ASYMPT_1:func.7">POWEROF2SET</a>  implies <font color="Maroon" title="c1">g</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> ) &amp; (  not <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_1.html#K7" title="ASYMPT_1:func.7">POWEROF2SET</a>  implies <font color="Maroon" title="c1">g</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="series_1.html#K4" title="SERIES_1:func.4">to_power</a> 2 ) ) ) implies  ex <font color="Olive" title="b1">s</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="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c1">g</font> &amp;  <a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_0.html#K15" title="ASYMPT_0:func.15">Big_Theta</a> (<font color="Olive" title="b1">s</font>,<a href="asympt_1.html#K7" title="ASYMPT_1:func.7">POWEROF2SET</a>) &amp;  not  <a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_0.html#K12" title="ASYMPT_0:func.12">Big_Theta</a> <font color="Olive" title="b1">s</font> &amp; <font color="Olive" title="b1">s</font> <a href="asympt_0.html#K16" title="ASYMPT_0:func.16">taken_every</a> 2 <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="Olive" title="b1">s</font> &amp;  <a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> 1 is  <a href="asympt_0.html#V6" title="ASYMPT_0:attr.6">eventually-nondecreasing</a>  &amp;  not <font color="Olive" title="b1">s</font> is  <a href="asympt_0.html#V6" title="ASYMPT_0:attr.6">eventually-nondecreasing</a>  ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:99"/><span class="lab"><font color="Green" title="E57">A1</font></span>: 
 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>  holds <br/> ( ( <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_1.html#K7" title="ASYMPT_1:func.7">POWEROF2SET</a>  implies <font color="Maroon" title="c1">g</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> ) &amp; (  not <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_1.html#K7" title="ASYMPT_1:func.7">POWEROF2SET</a>  implies <font color="Maroon" title="c1">g</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="series_1.html#K4" title="SERIES_1:func.4">to_power</a> 2 ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">s</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="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c1">g</font> &amp;  <a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_0.html#K15" title="ASYMPT_0:func.15">Big_Theta</a> (<font color="Olive" title="b1">s</font>,<a href="asympt_1.html#K7" title="ASYMPT_1:func.7">POWEROF2SET</a>) &amp;  not  <a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_0.html#K12" title="ASYMPT_0:func.12">Big_Theta</a> <font color="Olive" title="b1">s</font> &amp; <font color="Olive" title="b1">s</font> <a href="asympt_0.html#K16" title="ASYMPT_0:func.16">taken_every</a> 2 <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="Olive" title="b1">s</font> &amp;  <a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> 1 is  <a href="asympt_0.html#V6" title="ASYMPT_0:attr.6">eventually-nondecreasing</a>  &amp;  not <font color="Olive" title="b1">s</font> is  <a href="asympt_0.html#V6" title="ASYMPT_0:attr.6">eventually-nondecreasing</a>  )</span><br/>

<a NAME="E2:99"/><span class="lab"><font color="Green" title="E58">A2</font></span>: 
<font color="Maroon" title="c1">g</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="99_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">take </span>
1
; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="asympt_0.html#D4" title="ASYMPT_0:def.4">ASYMPT_0:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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">g</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/>

<span class="kw">let </span><font color="Maroon" title="c2">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font> or  not <font color="Maroon" title="c1">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">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/>

<span class="kw">assume </span><a NAME="E1:99_1"/><span class="lab"><font color="Green" title="E59">A3</font></span>: 
<font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c1">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">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/>

<span class="kw">thus </span><a NAME="E2:99_1"/>
<font color="Maroon" title="c1">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">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> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="99_1_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="99_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:99_1_1_1"/>
( <font color="Maroon" title="c2">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_1.html#K7" title="ASYMPT_1:func.7">POWEROF2SET</a>  or  not <font color="Maroon" title="c2">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_1.html#K7" title="ASYMPT_1:func.7">POWEROF2SET</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="99_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:99_1_1_1_1"/>
<font color="Maroon" title="c2">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_1.html#K7" title="ASYMPT_1:func.7">POWEROF2SET</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">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> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:99_1_1_1_1"/>
<font color="Maroon" title="c1">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:99"><span class="lab"><font color="Green" title="E57">A1</font></span></a>, <a class="txt" href="#E1:99_1"><span class="lab"><font color="Green" title="E59">A3</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="99_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:99_1_1_1_2"/>
 not <font color="Maroon" title="c2">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_1.html#K7" title="ASYMPT_1:func.7">POWEROF2SET</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">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> </span><br/><div class="add"><a NAME="E2:99_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c1">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:99"><span class="lab"><font color="Green" title="E57">A1</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:99_1_1_1_2"/>
<font color="Maroon" title="c1">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:99_1"><span class="lab"><font color="Green" title="E59">A3</font></span></a>, <a class="ref" href="power.html#T34" title="POWER:th.34">POWER:34</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>


</div><span class="kw">end;</span></div>
<span class="kw">set </span><font color="Maroon" title="c2">h</font> = <font color="Maroon" title="c1">g</font> <a href="asympt_0.html#K16" title="ASYMPT_0:func.16">taken_every</a> 2;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c3">g</font> = <font color="Maroon" title="c1">g</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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:99"><span class="lab"><font color="Green" title="E58">A2</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c3">g</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">g</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c1">g</font> &amp;  <a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_0.html#K15" title="ASYMPT_0:func.15">Big_Theta</a> (<font color="Maroon" title="c3">g</font>,<a href="asympt_1.html#K7" title="ASYMPT_1:func.7">POWEROF2SET</a>) &amp;  not  <a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_0.html#K12" title="ASYMPT_0:func.12">Big_Theta</a> <font color="Maroon" title="c3">g</font> &amp; <font color="Maroon" title="c3">g</font> <a href="asympt_0.html#K16" title="ASYMPT_0:func.16">taken_every</a> 2 <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="c3">g</font> &amp;  <a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> 1 is  <a href="asympt_0.html#V6" title="ASYMPT_0:attr.6">eventually-nondecreasing</a>  &amp;  not <font color="Maroon" title="c3">g</font> is  <a href="asympt_0.html#V6" title="ASYMPT_0:attr.6">eventually-nondecreasing</a>  )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c4">X</font> =  <a href="asympt_1.html#K7" title="ASYMPT_1:func.7">POWEROF2SET</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">f</font> =  <a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> 1;<br/>
<a NAME="E4:99"/><span class="lab"><font color="Green" title="E59">A4</font></span>: 
<font color="Maroon" title="c1">g</font> <a href="asympt_0.html#K16" title="ASYMPT_0:func.16">taken_every</a> 2 is    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T8" title="FUNCT_2:th.8">FUNCT_2:8</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E60">A5</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="99_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies ( <span class="p1">(<span class="default"><font color="Maroon" title="c1">g</font> <a href="asympt_0.html#K16" title="ASYMPT_0:func.16">taken_every</a> 2</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 4 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">g</font> <a href="asympt_0.html#K16" title="ASYMPT_0:func.16">taken_every</a> 2</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) )</span><br/><span class="kw">assume </span><a NAME="E1:99_2"/>
<font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c1">g</font> <a href="asympt_0.html#K16" title="ASYMPT_0:func.16">taken_every</a> 2</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 4 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">g</font> <a href="asympt_0.html#K16" title="ASYMPT_0:func.16">taken_every</a> 2</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/><a NAME="E2:99_2"/><span class="lab"><font color="Green" title="E61">A6</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">g</font> <a href="asympt_0.html#K16" title="ASYMPT_0:func.16">taken_every</a> 2</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c6">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="asympt_0.html#D15" title="ASYMPT_0:def.15">ASYMPT_0:def 15</a></span>;<br/><span class="kw">thus </span><a NAME="E3:99_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">g</font> <a href="asympt_0.html#K16" title="ASYMPT_0:func.16">taken_every</a> 2</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 4 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font></span>)</span>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">g</font> <a href="asympt_0.html#K16" title="ASYMPT_0:func.16">taken_every</a> 2</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="99_2_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="99_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:99_2_1_1"/>
( <font color="Maroon" title="c6">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_1.html#K7" title="ASYMPT_1:func.7">POWEROF2SET</a>  or  not <font color="Maroon" title="c6">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_1.html#K7" title="ASYMPT_1:func.7">POWEROF2SET</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="99_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:99_2_1_1_1"/><span class="lab"><font color="Green" title="E62">A7</font></span>: 
<font color="Maroon" title="c6">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_1.html#K7" title="ASYMPT_1:func.7">POWEROF2SET</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">g</font> <a href="asympt_0.html#K16" title="ASYMPT_0:func.16">taken_every</a> 2</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 4 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font></span>)</span></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">m</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> <span class="kw"> such that </span><br/><a NAME="E3:99_2_1_1_1"/><span class="lab"><font color="Green" title="E63">A8</font></span>: 
<font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <font color="Maroon" title="c7">m</font>
 ;<br/>2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c6">n</font> = 
<span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> 1</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <font color="Maroon" title="c7">m</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:99_2_1_1_1"><span class="lab"><font color="Green" title="E63">A8</font></span></a>, <a class="ref" href="power.html#T25" title="POWER:th.25">POWER:25</a></span>
<br/>.= 
2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T27" title="POWER:th.27">POWER:27</a></span>
;<br/><a NAME="E5:99_2_1_1_1"/><span class="kw">then </span>
2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_1.html#K7" title="ASYMPT_1:func.7">POWEROF2SET</a> 
 ;<br/><a NAME="E6:99_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A9</font></span>: 
<font color="Maroon" title="c3">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c6">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:99"><span class="lab"><font color="Green" title="E57">A1</font></span></a></span>;<br/><a NAME="E7:99_2_1_1_1"/>
<font color="Maroon" title="c3">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:99"><span class="lab"><font color="Green" title="E57">A1</font></span></a>, <a class="txt" href="#E1:99_2_1_1_1"><span class="lab"><font color="Green" title="E62">A7</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E8:99_2_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">g</font> <a href="asympt_0.html#K16" title="ASYMPT_0:func.16">taken_every</a> 2</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 4 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:99_2"><span class="lab"><font color="Green" title="E61">A6</font></span></a>, <a class="txt" href="#E6:99_2_1_1_1"><span class="lab"><font color="Green" title="E64">A9</font></span></a>, <a class="ref" href="xreal_1.html#T64" title="XREAL_1:th.64">XREAL_1:64</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="99_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:99_2_1_1_2"/><span class="lab"><font color="Green" title="E62">A10</font></span>: 
 not <font color="Maroon" title="c6">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_1.html#K7" title="ASYMPT_1:func.7">POWEROF2SET</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">g</font> <a href="asympt_0.html#K16" title="ASYMPT_0:func.16">taken_every</a> 2</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 4 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font></span>)</span></span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="99_2_1_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:99_2_1_1_2_1"/>
2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_1.html#K7" title="ASYMPT_1:func.7">POWEROF2SET</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">m</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> <span class="kw"> such that </span><br/><a NAME="E3:99_2_1_1_2_1"/><span class="lab"><font color="Green" title="E63">A11</font></span>: 
2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <font color="Maroon" title="c7">m</font>
 ;<br/><span class="kw">thus </span><a NAME="E4:99_2_1_1_2_1"/>
contradiction
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="99_2_1_1_2_1_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="99_2_1_1_2_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:99_2_1_1_2_1_1_1"/>
( <font color="Maroon" title="c7">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c7">m</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>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="99_2_1_1_2_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:99_2_1_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E64">A12</font></span>: 
<font color="Maroon" title="c7">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><div><span class="lab"><font color="Green" title="E65">A13</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="99_2_1_1_2_1_1_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:99_2_1_1_2_1_1_1_1_1"/>
1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 is    <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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:99_2_1_1_2_1_1_1_1_1"/><span class="kw">then </span>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <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> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">hence </span><a NAME="E3:99_2_1_1_2_1_1_1_1_1"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E3:99_2_1_1_2_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 2</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K2" title="REAL_1:func.2">"</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K2" title="REAL_1:func.2">"</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:99_2_1_1_2_1"><span class="lab"><font color="Green" title="E63">A11</font></span></a>, <a class="txt" href="#E1:99_2_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E64">A12</font></span></a>, <a class="ref" href="power.html#T24" title="POWER:th.24">POWER:24</a></span>;<br/><span class="kw">hence </span><a NAME="E4:99_2_1_1_2_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:99_2_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E65">A13</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="99_2_1_1_2_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:99_2_1_1_2_1_1_1_2"/>
<font color="Maroon" title="c7">m</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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:99_2_1_1_2_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E3:99_2_1_1_2_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T19" title="XREAL_1:th.19">XREAL_1:19</a></span>;<br/><a NAME="E4:99_2_1_1_2_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A14</font></span>: 
<font color="Maroon" title="c7">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 is    <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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T3" title="INT_1:th.3">INT_1:3</a></span>;<br/>2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c6">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"><font color="Maroon" title="c7">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:99_2_1_1_2_1"><span class="lab"><font color="Green" title="E63">A11</font></span></a></span>
<br/>.= 
<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"><font color="Maroon" title="c7">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T27" title="POWER:th.27">POWER:27</a></span>
<br/>.= 
<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"><font color="Maroon" title="c7">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> 2
<span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T25" title="POWER:th.25">POWER:25</a></span>
;<br/><span class="kw">hence </span><a NAME="E6:99_2_1_1_2_1_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:99_2_1_1_2"><span class="lab"><font color="Green" title="E62">A10</font></span></a>, <a class="txt" href="#E4:99_2_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E64">A14</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A15</font></span>: <font color="Maroon" title="c3">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c6">n</font></span>)</span> = 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> 2
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:99"><span class="lab"><font color="Green" title="E57">A1</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T46" title="POWER:th.46">POWER:46</a></span>
<br/>.= 
4 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a></span>)</span>

;<br/><font color="Maroon" title="c3">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font> = 
<font color="Maroon" title="c6">n</font> <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> 2
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:99"><span class="lab"><font color="Green" title="E57">A1</font></span></a>, <a class="txt" href="#E1:99_2_1_1_2"><span class="lab"><font color="Green" title="E62">A10</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c6">n</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T46" title="POWER:th.46">POWER:46</a></span>
;<br/><span class="kw">hence </span><a NAME="E5:99_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">g</font> <a href="asympt_0.html#K16" title="ASYMPT_0:func.16">taken_every</a> 2</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 4 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:99_2_1_1_2"><span class="lab"><font color="Green" title="E63">A15</font></span></a>, <a class="ref" href="asympt_0.html#D15" title="ASYMPT_0:def.15">ASYMPT_0:def 15</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><span class="kw">thus </span><a NAME="E4:99_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">g</font> <a href="asympt_0.html#K16" title="ASYMPT_0:func.16">taken_every</a> 2</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="99_2_2"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="99_2_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:99_2_2_1"/>
( 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_1.html#K7" title="ASYMPT_1:func.7">POWEROF2SET</a>  or  not 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_1.html#K7" title="ASYMPT_1:func.7">POWEROF2SET</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="99_2_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:99_2_2_1_1"/>
2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_1.html#K7" title="ASYMPT_1:func.7">POWEROF2SET</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">g</font> <a href="asympt_0.html#K16" title="ASYMPT_0:func.16">taken_every</a> 2</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:99_2_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">g</font> <a href="asympt_0.html#K16" title="ASYMPT_0:func.16">taken_every</a> 2</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:99"><span class="lab"><font color="Green" title="E57">A1</font></span></a>, <a class="txt" href="#E2:99_2"><span class="lab"><font color="Green" title="E61">A6</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="99_2_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:99_2_2_1_2"/>
 not 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_1.html#K7" title="ASYMPT_1:func.7">POWEROF2SET</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">g</font> <a href="asympt_0.html#K16" title="ASYMPT_0:func.16">taken_every</a> 2</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/><div class="add"><a NAME="E2:99_2_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:99"><span class="lab"><font color="Green" title="E57">A1</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:99_2_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">g</font> <a href="asympt_0.html#K16" title="ASYMPT_0:func.16">taken_every</a> 2</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="asympt_0.html#D15" title="ASYMPT_0:def.15">ASYMPT_0:def 15</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<a NAME="E6:99"/><span class="lab"><font color="Green" title="E61">A16</font></span>: 
 <a href="asympt_0.html#K12" title="ASYMPT_0:func.12">Big_Theta</a> <font color="Maroon" title="c3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">t</font> where <font color="Olive" title="b1">t</font> is    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>) :  ex <font color="Olive" title="b2">c</font>, <font color="Olive" title="b3">d</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ex <font color="Olive" title="b4">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 <br/>( <font color="Olive" title="b2">c</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>  &amp; <font color="Olive" title="b3">d</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>  &amp; (  for <font color="Olive" title="b5">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="b5">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b4">N</font> holds <br/>( <font color="Olive" title="b3">d</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b5">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">t</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b5">n</font> &amp; <font color="Olive" title="b1">t</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b5">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">c</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b5">n</font></span>)</span> ) ) ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="asympt_0.html#T27" title="ASYMPT_0:th.27">ASYMPT_0:27</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E62">A17</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="99_3"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:99_3"/>
 <a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_0.html#K12" title="ASYMPT_0:func.12">Big_Theta</a> <font color="Maroon" title="c3">g</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">t</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>)<span class="kw"> such that </span><br/><a NAME="E3:99_3"/><span class="lab"><font color="Green" title="E63">A18</font></span>: 
<font color="Maroon" title="c6">t</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> 1
 <span class="kw">and </span><br/><a NAME="E4:99_3"/><span class="lab"><font color="Green" title="E64">A19</font></span>: 
 ex <font color="Olive" title="b1">c</font>, <font color="Olive" title="b2">d</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ex <font color="Olive" title="b3">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 <br/>( <font color="Olive" title="b1">c</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>  &amp; <font color="Olive" title="b2">d</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>  &amp; (  for <font color="Olive" title="b4">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="b4">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b3">N</font> holds <br/>( <font color="Olive" title="b2">d</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b4">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">t</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b4">n</font> &amp; <font color="Maroon" title="c6">t</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">c</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b4">n</font></span>)</span> ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:99"><span class="lab"><font color="Green" title="E61">A16</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c7">c</font>, <font color="Maroon" title="c8">d</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>, <font color="Maroon" title="c9">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> <span class="kw"> such that </span><br/><a NAME="E6:99_3"/>
<font color="Maroon" title="c7">c</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> 
 <span class="kw">and </span><br/><a NAME="E7:99_3"/><span class="lab"><font color="Green" title="E65">A20</font></span>: 
<font color="Maroon" title="c8">d</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> 
 <span class="kw">and </span><br/><a NAME="E8:99_3"/><span class="lab"><font color="Green" title="E66">A21</font></span>: 
 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="c9">N</font> holds <br/>( <font color="Maroon" title="c8">d</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">t</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> &amp; <font color="Maroon" title="c6">t</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">c</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:99_3"><span class="lab"><font color="Green" title="E64">A19</font></span></a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c10">N0</font> =  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> (<font color="Maroon" title="c9">N</font>,2)</span>)</span>,<span class="p1"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c8">d</font></span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>);<br/><a NAME="E9:99_3"/><span class="lab"><font color="Green" title="E67">A22</font></span>: 
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> (<font color="Maroon" title="c9">N</font>,2)</span>)</span>,<span class="p1"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c8">d</font></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="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> (<font color="Maroon" title="c9">N</font>,2)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></span>;<br/><a NAME="E10:99_3"/>
 <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> (<font color="Maroon" title="c9">N</font>,2) <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c9">N</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></span>;<br/><a NAME="E11:99_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A23</font></span>: 
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> (<font color="Maroon" title="c9">N</font>,2)</span>)</span>,<span class="p1"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c8">d</font></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> <font color="Maroon" title="c9">N</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:99_3"><span class="lab"><font color="Green" title="E67">A22</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E12:99_3"/><span class="lab"><font color="Green" title="E69">A24</font></span>: 
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> (<font color="Maroon" title="c9">N</font>,2)</span>)</span>,<span class="p1"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c8">d</font></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> <span class="p1"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c8">d</font></span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></span>;<br/><a NAME="E13:99_3"/><span class="lab"><font color="Green" title="E70">A25</font></span>: 
 <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> (<font color="Maroon" title="c9">N</font>,2) <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></span>;<br/><a NAME="E14:99_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A26</font></span>: 
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> (<font color="Maroon" title="c9">N</font>,2)</span>)</span>,<span class="p1"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c8">d</font></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> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:99_3"><span class="lab"><font color="Green" title="E67">A22</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E15:99_3"/>
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> (<font color="Maroon" title="c9">N</font>,2)</span>)</span>,<span class="p1"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c8">d</font></span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>) is   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T16" title="XXREAL_0:th.16">XXREAL_0:16</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">N0</font> =  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> (<font color="Maroon" title="c9">N</font>,2)</span>)</span>,<span class="p1"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c8">d</font></span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:99_3"><span class="lab"><font color="Green" title="E67">A22</font></span></a>, <a class="ref" href="int_1.html#T3" title="INT_1:th.3">INT_1:3</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c12">N1</font> = <span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <font color="Maroon" title="c11">N0</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1;<br/><a NAME="E17:99_3"/>
2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <font color="Maroon" title="c11">N0</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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T34" title="POWER:th.34">POWER:34</a></span>;<br/><a NAME="E18:99_3"/><span class="kw">then </span>
2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <font color="Maroon" title="c11">N0</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E19:99_3"/><span class="kw">then </span>
<span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <font color="Maroon" title="c11">N0</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T19" title="XREAL_1:th.19">XREAL_1:19</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">N1</font> = <span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <font color="Maroon" title="c11">N0</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T3" title="INT_1:th.3">INT_1:3</a></span>;<br/><a NAME="E21:99_3"/><span class="lab"><font color="Green" title="E72">A27</font></span>: 
<span class="p1"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c8">d</font></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> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c8">d</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#D7" title="INT_1:def.7">INT_1:def 7</a></span>;<br/><a NAME="E22:99_3"/>
 not <font color="Maroon" title="c13">N1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_1.html#K7" title="ASYMPT_1:func.7">POWEROF2SET</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:99_3"><span class="lab"><font color="Green" title="E67">A22</font></span></a>, <a class="txt" href="#E13:99_3"><span class="lab"><font color="Green" title="E70">A25</font></span></a>, <a class="txt" href="asympt_1.html#E90"><span class="lab"><font color="Green" title="E56">Lm49</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A28</font></span>: <font color="Maroon" title="c3">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">N1</font> = 
<font color="Maroon" title="c13">N1</font> <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> 2
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:99"><span class="lab"><font color="Green" title="E57">A1</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c13">N1</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T46" title="POWER:th.46">POWER:46</a></span>
;<br/><a NAME="E24:99_3"/>
2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <font color="Maroon" title="c11">N0</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c11">N0</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:99_3"><span class="lab"><font color="Green" title="E71">A26</font></span></a>, <a class="txt" href="asympt_1.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a></span>;<br/><a NAME="E25:99_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A29</font></span>: 
<font color="Maroon" title="c13">N1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c11">N0</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T20" title="XREAL_1:th.20">XREAL_1:20</a></span>;<br/><a NAME="E26:99_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A30</font></span>: 
<font color="Maroon" title="c13">N1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c9">N</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:99_3"><span class="lab"><font color="Green" title="E68">A23</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E27:99_3"/>
<font color="Maroon" title="c13">N1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&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">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c8">d</font></span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:99_3"><span class="lab"><font color="Green" title="E69">A24</font></span></a>, <a class="txt" href="#E25:99_3"><span class="lab"><font color="Green" title="E74">A29</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E28:99_3"/><span class="kw">then </span>
<font color="Maroon" title="c13">N1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c8">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:99_3"><span class="lab"><font color="Green" title="E72">A27</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E29:99_3"/><span class="kw">then </span>
<font color="Maroon" title="c13">N1</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c13">N1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c13">N1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c8">d</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:99_3"><span class="lab"><font color="Green" title="E74">A29</font></span></a>, <a class="ref" href="xreal_1.html#T68" title="XREAL_1:th.68">XREAL_1:68</a></span>;<br/><a NAME="E30:99_3"/><span class="kw">then </span>
<font color="Maroon" title="c8">d</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">N1</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">N1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c8">d</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c8">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:99_3"><span class="lab"><font color="Green" title="E65">A20</font></span></a>, <a class="ref" href="xreal_1.html#T68" title="XREAL_1:th.68">XREAL_1:68</a></span>;<br/><a NAME="E31:99_3"/><span class="kw">then </span>
<font color="Maroon" title="c8">d</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">N1</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c13">N1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c8">d</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c8">d</font></span>)</span>
 ;<br/><a NAME="E32:99_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A31</font></span>: 
<font color="Maroon" title="c8">d</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">N1</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c13">N1</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:99_3"><span class="lab"><font color="Green" title="E65">A20</font></span></a>, <a class="ref" href="xcmplx_1.html#T87" title="XCMPLX_1:th.87">XCMPLX_1:87</a></span>;<br/><font color="Maroon" title="c6">t</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">N1</font> = 
<font color="Maroon" title="c13">N1</font> <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:99_3"><span class="lab"><font color="Green" title="E63">A18</font></span></a>, <a class="txt" href="#E25:99_3"><span class="lab"><font color="Green" title="E74">A29</font></span></a>, <a class="ref" href="asympt_1.html#D3" target="_self" title="ASYMPT_1:def.3">Def3</a></span>
<br/>.= 
<font color="Maroon" title="c13">N1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T25" title="POWER:th.25">POWER:25</a></span>
;<br/><span class="kw">hence </span><a NAME="E34:99_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:99_3"><span class="lab"><font color="Green" title="E66">A21</font></span></a>, <a class="txt" href="#E26:99_3"><span class="lab"><font color="Green" title="E75">A30</font></span></a>, <a class="txt" href="#E32:99_3"><span class="lab"><font color="Green" title="E76">A31</font></span></a>, <a class="txt" href="#E23:99_3"><span class="lab"><font color="Green" title="E73">A28</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E8:99"/><span class="lab"><font color="Green" title="E63">A32</font></span>: 
3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 
;<br/>
<div><span class="lab"><font color="Green" title="E64">A33</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="99_4"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:99_4"/>
<font color="Maroon" title="c3">g</font> is  <a href="asympt_0.html#V6" title="ASYMPT_0:attr.6">eventually-nondecreasing</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E3:99_4"/><span class="lab"><font color="Green" title="E65">A34</font></span>: 
 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="Maroon" title="c3">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="asympt_0.html#D6" title="ASYMPT_0:def.6">ASYMPT_0:def 6</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c7">N0</font> =  <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> (<font color="Maroon" title="c6">N</font>,1);<br/><span class="kw">set </span><font color="Maroon" title="c8">N1</font> = <span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> (<font color="Maroon" title="c6">N</font>,1)</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1;<br/><a NAME="E4:99_4"/><span class="lab"><font color="Green" title="E66">A35</font></span>: 
 <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> (<font color="Maroon" title="c6">N</font>,1) <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">N</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></span>;<br/><a NAME="E5:99_4"/>
2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> (<font color="Maroon" title="c6">N</font>,1)</span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_ff.html#T8" title="PRE_FF:th.8">PRE_FF:8</a></span>;<br/><a NAME="E6:99_4"/><span class="kw">then </span>
2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> (<font color="Maroon" title="c6">N</font>,1)</span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T24" title="POWER:th.24">POWER:24</a></span>;<br/><a NAME="E7:99_4"/><span class="kw">then </span>
<span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> (<font color="Maroon" title="c6">N</font>,1)</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">N1</font> = <span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> (<font color="Maroon" title="c6">N</font>,1)</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T3" title="INT_1:th.3">INT_1:3</a></span>;<br/><a NAME="E9:99_4"/><span class="lab"><font color="Green" title="E67">A36</font></span>: 
2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> (<font color="Maroon" title="c6">N</font>,1)</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T64" title="XREAL_1:th.64">XREAL_1:64</a>, <a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></span>;<br/><a NAME="E10:99_4"/><span class="kw">then </span>
2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> (<font color="Maroon" title="c6">N</font>,1)</span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> (<font color="Maroon" title="c6">N</font>,1)</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="asympt_1.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a></span>;<br/><a NAME="E11:99_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A37</font></span>: 
<font color="Maroon" title="c9">N1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> (<font color="Maroon" title="c6">N</font>,1)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T20" title="XREAL_1:th.20">XREAL_1:20</a></span>;<br/><a NAME="E12:99_4"/>
2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> (<font color="Maroon" title="c6">N</font>,1)</span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:99_4"><span class="lab"><font color="Green" title="E67">A36</font></span></a>, <a class="ref" href="pre_ff.html#T8" title="PRE_FF:th.8">PRE_FF:8</a></span>;<br/><a NAME="E13:99_4"/><span class="kw">then </span>
2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> (<font color="Maroon" title="c6">N</font>,1)</span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T46" title="POWER:th.46">POWER:46</a></span>;<br/><a NAME="E14:99_4"/><span class="kw">then </span>
<font color="Maroon" title="c9">N1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 3
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:99"><span class="lab"><font color="Green" title="E63">A32</font></span></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/><a NAME="E15:99_4"/><span class="kw">then </span>
<font color="Maroon" title="c9">N1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E16:99_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A38</font></span>: 
<font color="Maroon" title="c9">N1</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c9">N1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="asympt_1.html#E88"><span class="lab"><font color="Green" title="E54">Lm47</font></span></a></span>;<br/><a NAME="E17:99_4"/>
2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> (<font color="Maroon" title="c6">N</font>,1)</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T64" title="XREAL_1:th.64">XREAL_1:64</a>, <a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></span>;<br/><a NAME="E18:99_4"/><span class="kw">then </span>
 not <font color="Maroon" title="c9">N1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_1.html#K7" title="ASYMPT_1:func.7">POWEROF2SET</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="asympt_1.html#E90"><span class="lab"><font color="Green" title="E56">Lm49</font></span></a></span>;<br/><a NAME="E19:99_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A39</font></span>: 
<font color="Maroon" title="c3">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">N1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">N1</font> <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:99"><span class="lab"><font color="Green" title="E57">A1</font></span></a></span>;<br/><a NAME="E20:99_4"/>
2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> (<font color="Maroon" title="c6">N</font>,1)</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> (<font color="Maroon" title="c6">N</font>,1)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T64" title="XREAL_1:th.64">XREAL_1:64</a></span>;<br/><a NAME="E21:99_4"/><span class="kw">then </span>
<font color="Maroon" title="c9">N1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> (<font color="Maroon" title="c6">N</font>,1)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:99_4"><span class="lab"><font color="Green" title="E68">A37</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E22:99_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A40</font></span>: 
<font color="Maroon" title="c9">N1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">N</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:99_4"><span class="lab"><font color="Green" title="E66">A35</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E23:99_4"/>
<font color="Maroon" title="c9">N1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_1.html#K7" title="ASYMPT_1:func.7">POWEROF2SET</a> 
 ;<br/><a NAME="E24:99_4"/><span class="kw">then </span>
<font color="Maroon" title="c3">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">N1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">N1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:99"><span class="lab"><font color="Green" title="E57">A1</font></span></a></span>;<br/><a NAME="E25:99_4"/><span class="kw">then </span>
<font color="Maroon" title="c3">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">N1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">N1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:99_4"><span class="lab"><font color="Green" title="E70">A39</font></span></a>, <a class="txt" href="#E16:99_4"><span class="lab"><font color="Green" title="E69">A38</font></span></a>, <a class="ref" href="power.html#T46" title="POWER:th.46">POWER:46</a></span>;<br/><span class="kw">hence </span><a NAME="E26:99_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:99_4"><span class="lab"><font color="Green" title="E65">A34</font></span></a>, <a class="txt" href="#E22:99_4"><span class="lab"><font color="Green" title="E71">A40</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E65">A41</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="99_5"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <span class="p1">(<span class="default"><a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> 1</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> 1</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E1:99_5"/>
<font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> 1</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> 1</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><br/><a NAME="E2:99_5"/><span class="lab"><font color="Green" title="E66">A42</font></span>: 
<font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E3:99_5"/><span class="lab"><font color="Green" title="E67">A43</font></span>: 
<span class="p1">(<span class="default"><a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> 1</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">n</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="99_5_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="99_5_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:99_5_1_1"/>
( <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c6">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>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="99_5_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:99_5_1_1_1"/>
<font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> 1</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">n</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:99_5_1_1_1"/>
<span class="p1">(<span class="default"><a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> 1</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="asympt_1.html#D3" target="_self" title="ASYMPT_1:def.3">Def3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="99_5_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:99_5_1_1_2"/>
<font color="Maroon" title="c6">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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> 1</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">n</font></span><br/><div class="add"><span class="kw">hence </span><span class="p1">(<span class="default"><a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> 1</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font> = 
<font color="Maroon" title="c6">n</font> <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="asympt_1.html#D3" target="_self" title="ASYMPT_1:def.3">Def3</a></span>
<br/>.= 
<font color="Maroon" title="c6">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T25" title="POWER:th.25">POWER:25</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><span class="p1">(<span class="default"><a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> 1</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="asympt_1.html#D3" target="_self" title="ASYMPT_1:def.3">Def3</a></span>
<br/>.= 
<font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T25" title="POWER:th.25">POWER:25</a></span>
;<br/><span class="kw">hence </span><a NAME="E5:99_5"/>
<span class="p1">(<span class="default"><a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> 1</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> 1</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:99_5"><span class="lab"><font color="Green" title="E67">A43</font></span></a>, <a class="txt" href="#E2:99_5"><span class="lab"><font color="Green" title="E66">A42</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E66">A44</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="99_6"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon" title="c6">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_1.html#K7" title="ASYMPT_1:func.7">POWEROF2SET</a>  implies ( 1 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> 1</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font> &amp; <span class="p1">(<span class="default"><a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> 1</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font></span>)</span> ) )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:99_6"/><span class="lab"><font color="Green" title="E67">A45</font></span>: 
<font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <span class="kw">and </span><br/><a NAME="E2:99_6"/><span class="lab"><font color="Green" title="E68">A46</font></span>: 
<font color="Maroon" title="c6">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_1.html#K7" title="ASYMPT_1:func.7">POWEROF2SET</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> 1</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font> &amp; <span class="p1">(<span class="default"><a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> 1</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font></span>)</span> )</span><br/><span class="lab"><font color="Green" title="E69">A47</font></span>: <span class="p1">(<span class="default"><a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> 1</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font> = 
<font color="Maroon" title="c6">n</font> <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:99_6"><span class="lab"><font color="Green" title="E67">A45</font></span></a>, <a class="ref" href="asympt_1.html#D3" target="_self" title="ASYMPT_1:def.3">Def3</a></span>
<br/>.= 
<font color="Maroon" title="c6">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T25" title="POWER:th.25">POWER:25</a></span>
;<br/><span class="kw">hence </span><a NAME="E4:99_6"/>
1 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> 1</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:99"><span class="lab"><font color="Green" title="E57">A1</font></span></a>, <a class="txt" href="#E2:99_6"><span class="lab"><font color="Green" title="E68">A46</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> 1</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font></span>)</span></span><br/><span class="kw">thus </span><a NAME="E5:99_6"/>
<span class="p1">(<span class="default"><a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> 1</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:99"><span class="lab"><font color="Green" title="E57">A1</font></span></a>, <a class="txt" href="#E2:99_6"><span class="lab"><font color="Green" title="E68">A46</font></span></a>, <a class="txt" href="#E3:99_6"><span class="lab"><font color="Green" title="E69">A47</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E12:99"/>
 <a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> 1 is    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T8" title="FUNCT_2:th.8">FUNCT_2:8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:99"/>
( <font color="Maroon" title="c3">g</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c1">g</font> &amp;  <a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_0.html#K15" title="ASYMPT_0:func.15">Big_Theta</a> (<font color="Maroon" title="c3">g</font>,<a href="asympt_1.html#K7" title="ASYMPT_1:func.7">POWEROF2SET</a>) &amp;  not  <a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_0.html#K12" title="ASYMPT_0:func.12">Big_Theta</a> <font color="Maroon" title="c3">g</font> &amp; <font color="Maroon" title="c3">g</font> <a href="asympt_0.html#K16" title="ASYMPT_0:func.16">taken_every</a> 2 <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="c3">g</font> &amp;  <a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> 1 is  <a href="asympt_0.html#V6" title="ASYMPT_0:attr.6">eventually-nondecreasing</a>  &amp;  not <font color="Maroon" title="c3">g</font> is  <a href="asympt_0.html#V6" title="ASYMPT_0:attr.6">eventually-nondecreasing</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:99"><span class="lab"><font color="Green" title="E66">A44</font></span></a>, <a class="txt" href="#E7:99"><span class="lab"><font color="Green" title="E62">A17</font></span></a>, <a class="txt" href="#E4:99"><span class="lab"><font color="Green" title="E59">A4</font></span></a>, <a class="txt" href="#E5:99"><span class="lab"><font color="Green" title="E60">A5</font></span></a>, <a class="txt" href="#E10:99"><span class="lab"><font color="Green" title="E65">A41</font></span></a>, <a class="txt" href="#E9:99"><span class="lab"><font color="Green" title="E64">A33</font></span></a>, <a class="ref" href="asympt_0.html#D6" title="ASYMPT_0:def.6">ASYMPT_0:def 6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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