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

<span class="kw">set </span><font color="Maroon" title="c1">X</font> =  <a href="asympt_1.html#K7" title="ASYMPT_1:func.7">POWEROF2SET</a> ;<br/>
<span class="kw">let </span><font color="Maroon" title="c2">f</font>, <font color="Maroon" title="c3">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="Red">::  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>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p1">(<span class="default">2 <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p2"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p3">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Olive" title="b1">n</font></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span></span>)</span> ) &amp; ( for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/><font color="Maroon" title="c3">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#K3" title="SERIES_1:func.3">to_power</a> <font color="Olive" title="b1">n</font> ) 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="c3">g</font> &amp; <font color="Maroon" title="c2">f</font> <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 <font color="Maroon" title="c2">f</font> <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="Maroon" title="c2">f</font> is  <a href="asympt_0.html#V6" title="ASYMPT_0:attr.6">eventually-nondecreasing</a>  &amp; <font color="Olive" title="b1">s</font> is  <a href="asympt_0.html#V6" title="ASYMPT_0:attr.6">eventually-nondecreasing</a>  &amp; not <font color="Olive" title="b1">s</font> <a href="asympt_0.html#R2" title="ASYMPT_0:pred.2">is_smooth_wrt</a> 2 ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><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>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p1">(<span class="default">2 <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p2"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p3">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Olive" title="b1">n</font></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:99"/><span class="lab"><font color="Green" title="E58">A2</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#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/><font color="Maroon" title="c3">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#K3" title="SERIES_1:func.3">to_power</a> <font color="Olive" title="b1">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="c3">g</font> &amp; <font color="Maroon" title="c2">f</font> <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 <font color="Maroon" title="c2">f</font> <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="Maroon" title="c2">f</font> is  <a href="asympt_0.html#V6" title="ASYMPT_0:attr.6">eventually-nondecreasing</a>  &amp; <font color="Olive" title="b1">s</font> is  <a href="asympt_0.html#V6" title="ASYMPT_0:attr.6">eventually-nondecreasing</a>  &amp; not <font color="Olive" title="b1">s</font> <a href="asympt_0.html#R2" title="ASYMPT_0:pred.2">is_smooth_wrt</a> 2 )</span><br/>

<a NAME="E3:99"/><span class="lab"><font color="Green" title="E59">A3</font></span>: 
<font color="Maroon" title="c3">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="Red">:: according to </font></span><a class="ref" href="asympt_0.html#D6" title="ASYMPT_0:def.6">ASYMPT_0:def 6</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="c3">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="c4">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="c4">n</font> or  not <font color="Maroon" title="c3">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:99_1"/><span class="lab"><font color="Green" title="E60">A4</font></span>: 
<font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> not <font color="Maroon" title="c3">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/>

<a NAME="E2:99_1"/><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="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font> <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c4">n</font>
 
<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">hence </span><a NAME="E3:99_1"/>
not <font color="Maroon" title="c3">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:99_1"><span class="lab"><font color="Green" title="E60">A4</font></span></a>, <a class="ref" href="power.html#T39" title="POWER:th.39">POWER:39</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">set </span><font color="Maroon" title="c4">h</font> = <font color="Maroon" title="c3">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="c5">g</font> = <font color="Maroon" title="c3">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="#E3:99"><span class="lab"><font color="Green" title="E59">A3</font></span></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="Red">::  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="c5">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> <font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font> &amp; <font color="Maroon" title="c2">f</font> <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="c5">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_2"/><span class="lab"><font color="Green" title="E61">A6</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_2"/><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="Red">::  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="c5">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> <font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font> &amp; <font color="Maroon" title="c2">f</font> <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="c5">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">consider </span><font color="Maroon" title="c7">n1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E4:99_2"/><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#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c7">n1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:99_2"><span class="lab"><font color="Green" title="E62">A7</font></span></a></span>;<br/><a NAME="E5:99_2"/><span class="lab"><font color="Green" title="E64">A9</font></span>: 
<font color="Maroon" title="c2">f</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> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p1">(<span class="default">2 <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p2"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p3">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c6">n</font></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span></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="#E1:99_2"><span class="lab"><font color="Green" title="E61">A6</font></span></a></span>;<br/> <a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c6">n</font> = 
<font color="Maroon" title="c7">n1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,2</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:99_2"><span class="lab"><font color="Green" title="E63">A8</font></span></a>, <a class="ref" href="power.html#T63" title="POWER:th.63">POWER:63</a></span>
<br/>.= 
<font color="Maroon" title="c7">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="ref" href="power.html#T60" title="POWER:th.60">POWER:60</a></span>
;<br/><a NAME="E7:99_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A10</font></span>: 
<font color="Maroon" title="c2">f</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> <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c6">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:99_2"><span class="lab"><font color="Green" title="E64">A9</font></span></a>, <a class="txt" href="#E4:99_2"><span class="lab"><font color="Green" title="E63">A8</font></span></a>, <a class="ref" href="int_1.html#T47" title="INT_1:th.47">INT_1:47</a></span>;<br/><span class="kw">hence </span><a NAME="E8:99_2"/>
1 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">f</font> <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="#E2:99"><span class="lab"><font color="Green" title="E58">A2</font></span></a>, <a class="txt" href="#E1:99_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="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">f</font> <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="c5">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="E9:99_2"/>
<font color="Maroon" title="c2">f</font> <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="c5">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"><span class="lab"><font color="Green" title="E58">A2</font></span></a>, <a class="txt" href="#E1:99_2"><span class="lab"><font color="Green" title="E61">A6</font></span></a>, <a class="txt" href="#E7:99_2"><span class="lab"><font color="Green" title="E65">A10</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E6:99"/><span class="lab"><font color="Green" title="E61">A11</font></span>: 
 <a href="asympt_0.html#K12" title="ASYMPT_0:func.12">Big_Theta</a> <font color="Maroon" title="c5">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="c5">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="c5">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">A12</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"/>
<font color="Maroon" title="c2">f</font> <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="c5">g</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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">A13</font></span>: 
<font color="Maroon" title="c6">t</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c2">f</font>
 <span class="kw">and </span><br/><a NAME="E4:99_3"/><span class="lab"><font color="Green" title="E64">A14</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="c5">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="c5">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">A11</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">A15</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">A16</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="c5">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="c5">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">A14</font></span></a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c10">N1</font> =  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</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="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>;<br/><a NAME="E9:99_3"/><span class="lab"><font color="Green" title="E67">A17</font></span>: 
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</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="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> <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="E10:99_3"/><span class="lab"><font color="Green" title="E68">A18</font></span>: 
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</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="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> 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/><a NAME="E11:99_3"/><span class="lab"><font color="Green" title="E69">A19</font></span>: 
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</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="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> <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="E12:99_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="E13:99_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A20</font></span>: 
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</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="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> <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="E69">A19</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E14:99_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> 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="E15:99_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A21</font></span>: 
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</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="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> <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="#E11:99_3"><span class="lab"><font color="Green" title="E69">A19</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">reconsider </span><font color="Maroon" title="c11">N1</font> =  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</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="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> 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="#E11:99_3"><span class="lab"><font color="Green" title="E69">A19</font></span></a>, <a class="txt" href="#E10:99_3"><span class="lab"><font color="Green" title="E68">A18</font></span></a>, <a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c12">N2</font> = 2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c11">N1</font> 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>  ;<br/><a NAME="E18:99_3"/><span class="lab"><font color="Green" title="E72">A22</font></span>: 
<font color="Maroon" title="c12">N2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c11">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="#E15:99_3"><span class="lab"><font color="Green" title="E71">A21</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="E19:99_3"/>
<font color="Maroon" title="c11">N1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:99_3"><span class="lab"><font color="Green" title="E71">A21</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E20:99_3"/><span class="kw">then </span>
<span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">N1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c11">N1</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="asympt_1.html#E91"><span class="lab"><font color="Green" title="E55">Lm48</font></span></a></span>;<br/><a NAME="E21:99_3"/><span class="kw">then </span>
2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">N1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c12">N2</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#T22" title="XREAL_1:th.22">XREAL_1:22</a></span>;<br/><a NAME="E22:99_3"/><span class="kw">then </span>
 <a href="power.html#K6" title="POWER:func.6">log</a> 2,<span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">N1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="power.html#K6" title="POWER:func.6">log</a> 2,<span class="p1">(<span class="default"><font color="Maroon" title="c12">N2</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#T65" title="POWER:th.65">POWER:65</a></span>;<br/><a NAME="E23:99_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">N1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,2</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="power.html#K6" title="POWER:func.6">log</a> 2,<span class="p1">(<span class="default"><font color="Maroon" title="c12">N2</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#T63" title="POWER:th.63">POWER:63</a></span>;<br/><a NAME="E24:99_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A23</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">N1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="power.html#K6" title="POWER:func.6">log</a> 2,<span class="p1">(<span class="default"><font color="Maroon" title="c12">N2</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#T60" title="POWER:th.60">POWER:60</a></span>;<br/><div><span class="lab"><font color="Green" title="E74">A24</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="99_3_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:99_3_1"/>
<span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<span class="p3">(<span class="default"><font color="Maroon" title="c12">N2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c11">N1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:99_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A25</font></span>: 
<span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<span class="p3">(<span class="default"><font color="Maroon" title="c12">N2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c11">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="ref" href="int_1.html#T20" title="INT_1:th.20">INT_1:20</a></span>;<br/><a NAME="E3:99_3_1"/>
 <a href="power.html#K6" title="POWER:func.6">log</a> 2,<span class="p1">(<span class="default"><font color="Maroon" title="c12">N2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<span class="p3">(<span class="default"><font color="Maroon" title="c12">N2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#D4" title="INT_1:def.4">INT_1:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E4:99_3_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:99_3"><span class="lab"><font color="Green" title="E73">A23</font></span></a>, <a class="txt" href="#E2:99_3_1"><span class="lab"><font color="Green" title="E75">A25</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E26:99_3"/><span class="lab"><font color="Green" title="E75">A26</font></span>: 
<font color="Maroon" title="c5">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">N2</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> <span class="p1">(<span class="default"><font color="Maroon" title="c12">N2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">N2</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="#E2:99"><span class="lab"><font color="Green" title="E58">A2</font></span></a></span>;<br/><a NAME="E27:99_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A27</font></span>: 
<font color="Maroon" title="c5">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">N2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <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#T39" title="POWER:th.39">POWER:39</a></span>;<br/><a NAME="E28:99_3"/>
<font color="Maroon" title="c11">N1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c11">N1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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#T10" title="XREAL_1:th.10">XREAL_1:10</a></span>;<br/><a NAME="E29:99_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A28</font></span>: 
<font color="Maroon" title="c12">N2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c11">N1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:99_3"><span class="lab"><font color="Green" title="E72">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="E30:99_3"/><span class="lab"><font color="Green" title="E78">A29</font></span>: 
<font color="Maroon" title="c12">N2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c12">N2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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#T10" title="XREAL_1:th.10">XREAL_1:10</a></span>;<br/><a NAME="E31:99_3"/><span class="kw">then </span>
<font color="Maroon" title="c12">N2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c11">N1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:99_3"><span class="lab"><font color="Green" title="E77">A28</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E32:99_3"/><span class="kw">then </span>
<font color="Maroon" title="c12">N2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c9">N</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:99_3"><span class="lab"><font color="Green" title="E70">A20</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E33:99_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A30</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="c5">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">N2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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> <span class="p1">(<span class="default"><font color="Maroon" title="c12">N2</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="#E8:99_3"><span class="lab"><font color="Green" title="E66">A16</font></span></a></span>;<br/><a NAME="E34:99_3"/>
<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#D5" title="INT_1:def.5">INT_1:def 5</a></span>;<br/><a NAME="E35:99_3"/><span class="kw">then </span>
<font color="Maroon" title="c11">N1</font> <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="txt" href="#E9:99_3"><span class="lab"><font color="Green" title="E67">A17</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E36:99_3"/><span class="kw">then </span>
<font color="Maroon" title="c12">N2</font> <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="txt" href="#E29:99_3"><span class="lab"><font color="Green" title="E77">A28</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E37:99_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A31</font></span>: 
<font color="Maroon" title="c12">N2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</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> <a href="real_1.html#K7" title="REAL_1:func.7">+</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#T10" title="XREAL_1:th.10">XREAL_1:10</a></span>;<br/> <a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c12">N2</font> = 
<font color="Maroon" title="c11">N1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,2</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T63" title="POWER:th.63">POWER:63</a></span>
<br/>.= 
<font color="Maroon" title="c11">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="ref" href="power.html#T60" title="POWER:th.60">POWER:60</a></span>
;<br/><a NAME="E39:99_3"/><span class="kw">then </span>
 <a href="power.html#K6" title="POWER:func.6">log</a> 2,<span class="p1">(<span class="default"><font color="Maroon" title="c12">N2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c11">N1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:99_3"><span class="lab"><font color="Green" title="E72">A22</font></span></a>, <a class="txt" href="#E30:99_3"><span class="lab"><font color="Green" title="E78">A29</font></span></a>, <a class="ref" href="power.html#T65" title="POWER:th.65">POWER:65</a></span>;<br/><a NAME="E40:99_3"/><span class="kw">then </span>
<span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<span class="p3">(<span class="default"><font color="Maroon" title="c12">N2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c11">N1</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_ff.html#T11" title="PRE_FF:th.11">PRE_FF:11</a></span>;<br/><a NAME="E41:99_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A32</font></span>: 
<span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<span class="p3">(<span class="default"><font color="Maroon" title="c12">N2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c11">N1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T47" title="INT_1:th.47">INT_1:47</a></span>;<br/><a NAME="E42:99_3"/>
<font color="Maroon" title="c6">t</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">N2</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> <span class="p1">(<span class="default"><font color="Maroon" title="c12">N2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p1">(<span class="default">2 <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p2"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p3">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<span class="p4">(<span class="default"><font color="Maroon" title="c12">N2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span></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="#E3:99_3"><span class="lab"><font color="Green" title="E63">A13</font></span></a></span>;<br/><span class="kw">then </span><span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">N2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">N2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">N2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">N2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">N2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c12">N2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:99_3"><span class="lab"><font color="Green" title="E75">A26</font></span></a>, <a class="txt" href="#E41:99_3"><span class="lab"><font color="Green" title="E81">A32</font></span></a>, <a class="txt" href="#E25:99_3"><span class="lab"><font color="Green" title="E74">A24</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">N2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <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="c12">N2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c12">N2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T34" title="POWER:th.34">POWER:34</a></span>
<br/>.= 
<font color="Maroon" title="c12">N2</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#T30" title="POWER:th.30">POWER:30</a></span>
;<br/><a NAME="E44:99_3"/><span class="kw">then </span>
1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">N2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">N2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c8">d</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:99_3"><span class="lab"><font color="Green" title="E65">A15</font></span></a>, <a class="txt" href="#E37:99_3"><span class="lab"><font color="Green" title="E80">A31</font></span></a>, <a class="ref" href="xreal_1.html#T90" title="XREAL_1:th.90">XREAL_1:90</a></span>;<br/><a NAME="E45:99_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">N2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">N2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">d</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T57" title="XCMPLX_1:th.57">XCMPLX_1:57</a></span>;<br/><a NAME="E46:99_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">N2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">N2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">N2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <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="c5">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">N2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:99_3"><span class="lab"><font color="Green" title="E76">A27</font></span></a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></span>;<br/><span class="kw">hence </span><a NAME="E47:99_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:99_3"><span class="lab"><font color="Green" title="E79">A30</font></span></a>, <a class="txt" href="#E27:99_3"><span class="lab"><font color="Green" title="E76">A27</font></span></a>, <a class="ref" href="xcmplx_1.html#T88" title="XCMPLX_1:th.88">XCMPLX_1:88</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="E63">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="c5">g</font> <a href="asympt_0.html#R2" title="ASYMPT_0:pred.2">is_smooth_wrt</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:99_4"/><span class="kw">then </span>
<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="c5">g</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="asympt_0.html#D19" title="ASYMPT_0:def.19">ASYMPT_0:def 19</a></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="E4:99_4"/><span class="lab"><font color="Green" title="E64">A34</font></span>: 
<font color="Maroon" title="c6">t</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">g</font> <a href="asympt_0.html#K16" title="ASYMPT_0:func.16">taken_every</a> 2
 <span class="kw">and </span><br/><a NAME="E5:99_4"/><span class="lab"><font color="Green" title="E65">A35</font></span>: 
 ex <font color="Olive" title="b1">c</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ex <font color="Olive" title="b2">N</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <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; ( for <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 <font color="Olive" title="b3">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b2">N</font> holds <br/>( <font color="Maroon" title="c6">t</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">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="c5">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">n</font></span>)</span> &amp; <font color="Maroon" title="c6">t</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">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>  ) ) )
 ;<br/><span class="kw">consider </span><font color="Maroon" title="c7">c</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>, <font color="Maroon" title="c8">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="E7:99_4"/>
<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="E8:99_4"/><span class="lab"><font color="Green" title="E66">A36</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="c8">N</font> holds <br/>( <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="c5">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> &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#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="#E5:99_4"><span class="lab"><font color="Green" title="E65">A35</font></span></a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c9">N0</font> =  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><font color="Maroon" title="c7">c</font></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c8">N</font>,2</span>)</span>;<br/><a NAME="E9:99_4"/><span class="lab"><font color="Green" title="E67">A37</font></span>: 
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><font color="Maroon" title="c7">c</font></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c8">N</font>,2</span>)</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"><font color="Maroon" title="c7">c</font></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="E10:99_4"/><span class="lab"><font color="Green" title="E68">A38</font></span>: 
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><font color="Maroon" title="c7">c</font></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c8">N</font>,2</span>)</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/><a NAME="E11:99_4"/><span class="lab"><font color="Green" title="E69">A39</font></span>: 
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><font color="Maroon" title="c7">c</font></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c8">N</font>,2</span>)</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="c8">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="E12:99_4"/>
 <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c8">N</font>,2 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c8">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="E13:99_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A40</font></span>: 
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><font color="Maroon" title="c7">c</font></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c8">N</font>,2</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c8">N</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:99_4"><span class="lab"><font color="Green" title="E69">A39</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E14:99_4"/><span class="lab"><font color="Green" title="E71">A41</font></span>: 
 <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c8">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="E15:99_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A42</font></span>: 
2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p2"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><font color="Maroon" title="c7">c</font></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>,<span class="p2">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c8">N</font>,2</span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p2"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><font color="Maroon" title="c7">c</font></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>,<span class="p2">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c8">N</font>,2</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:99_4"><span class="lab"><font color="Green" title="E69">A39</font></span></a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></span>;<br/><a NAME="E16:99_4"/><span class="lab"><font color="Green" title="E73">A43</font></span>: 
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><font color="Maroon" title="c7">c</font></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c8">N</font>,2</span>)</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="#E11:99_4"><span class="lab"><font color="Green" title="E69">A39</font></span></a>, <a class="txt" href="#E14:99_4"><span class="lab"><font color="Green" title="E71">A41</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E17:99_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A44</font></span>: 
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><font color="Maroon" title="c7">c</font></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c8">N</font>,2</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <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/><span class="kw">reconsider </span><font color="Maroon" title="c10">N0</font> =  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><font color="Maroon" title="c7">c</font></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c8">N</font>,2</span>)</span> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:99_4"><span class="lab"><font color="Green" title="E69">A39</font></span></a>, <a class="txt" href="#E10:99_4"><span class="lab"><font color="Green" title="E68">A38</font></span></a>, <a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></span>;<br/><a NAME="E19:99_4"/>
<span class="p1"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><font color="Maroon" title="c7">c</font></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="c7">c</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#D5" title="INT_1:def.5">INT_1:def 5</a></span>;<br/><a NAME="E20:99_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A45</font></span>: 
<font color="Maroon" title="c10">N0</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:99_4"><span class="lab"><font color="Green" title="E67">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="E21:99_4"/>
<font color="Maroon" title="c10">N0</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:99_4"><span class="lab"><font color="Green" title="E73">A43</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>
<font color="Maroon" title="c10">N0</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">N0</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c10">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="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E23:99_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A46</font></span>: 
<font color="Maroon" title="c10">N0</font> <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c10">N0</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c10">N0</font> <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">N0</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="#E17:99_4"><span class="lab"><font color="Green" title="E74">A44</font></span></a>, <a class="ref" href="pre_ff.html#T10" title="PRE_FF:th.10">PRE_FF:10</a></span>;<br/><font color="Maroon" title="c10">N0</font> <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">N0</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="c10">N0</font> <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c10">N0</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">N0</font> <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:99_4"><span class="lab"><font color="Green" title="E69">A39</font></span></a>, <a class="txt" href="#E14:99_4"><span class="lab"><font color="Green" title="E71">A41</font></span></a>, <a class="ref" href="power.html#T32" title="POWER:th.32">POWER:32</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">N0</font> <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c10">N0</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c10">N0</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T30" title="POWER:th.30">POWER:30</a></span>
;<br/><a NAME="E25:99_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A47</font></span>: 
<font color="Maroon" title="c10">N0</font> <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">N0</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</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="c10">N0</font> <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c10">N0</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:99_4"><span class="lab"><font color="Green" title="E75">A45</font></span></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></span>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c3">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="c10">N0</font> = 
<font color="Maroon" title="c5">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="c10">N0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="asympt_0.html#D18" title="ASYMPT_0:def.18">ASYMPT_0:def 18</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="c10">N0</font></span>)</span> <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c10">N0</font></span>)</span>
<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>, <a class="txt" href="#E15:99_4"><span class="lab"><font color="Green" title="E72">A42</font></span></a></span>
;<br/><a NAME="E27:99_4"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">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="c10">N0</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c10">N0</font> <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c10">N0</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:99_4"><span class="lab"><font color="Green" title="E69">A39</font></span></a>, <a class="txt" href="#E14:99_4"><span class="lab"><font color="Green" title="E71">A41</font></span></a>, <a class="txt" href="#E15:99_4"><span class="lab"><font color="Green" title="E72">A42</font></span></a>, <a class="ref" href="power.html#T42" title="POWER:th.42">POWER:42</a></span>;<br/><a NAME="E28:99_4"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">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="c10">N0</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c10">N0</font> <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">N0</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="#E23:99_4"><span class="lab"><font color="Green" title="E76">A46</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_4"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">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="c10">N0</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</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="c10">N0</font> <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c10">N0</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:99_4"><span class="lab"><font color="Green" title="E77">A47</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E30:99_4"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">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="c10">N0</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</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="c5">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">N0</font></span>)</span>
 <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>, <a class="txt" href="#E11:99_4"><span class="lab"><font color="Green" title="E69">A39</font></span></a>, <a class="txt" href="#E14:99_4"><span class="lab"><font color="Green" title="E71">A41</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E31:99_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:99_4"><span class="lab"><font color="Green" title="E64">A34</font></span></a>, <a class="txt" href="#E8:99_4"><span class="lab"><font color="Green" title="E66">A36</font></span></a>, <a class="txt" href="#E13:99_4"><span class="lab"><font color="Green" title="E70">A40</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="E64">A48</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="Red">::  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> 2 implies <font color="Maroon" title="c2">f</font> <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> <font color="Maroon" title="c2">f</font> <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="E1:99_5"/><span class="lab"><font color="Green" title="E65">A49</font></span>: 
<font color="Maroon" title="c2">f</font> <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> <a href="hidden.html#R1" title="HIDDEN:pred.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> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p1">(<span class="default">2 <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p2"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p3">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<span class="p4">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span></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></span>;<br/><span class="kw">assume </span><a NAME="E2:99_5"/><span class="lab"><font color="Green" title="E66">A50</font></span>: 
<font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">f</font> <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> <font color="Maroon" title="c2">f</font> <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="E3:99_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A51</font></span>: 
<font color="Maroon" title="c2">f</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> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p1">(<span class="default">2 <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p2"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p3">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c6">n</font></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span></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></span>;<br/><a NAME="E4:99_5"/><span class="lab"><font color="Green" title="E68">A52</font></span>: 
<font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></span>;<br/><a NAME="E5:99_5"/><span class="kw">then </span>
 <a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="power.html#K6" title="POWER:func.6">log</a> 2,<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="#E2:99_5"><span class="lab"><font color="Green" title="E66">A50</font></span></a>, <a class="ref" href="power.html#T65" title="POWER:th.65">POWER:65</a></span>;<br/><a NAME="E6:99_5"/><span class="kw">then </span>
<span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c6">n</font></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<span class="p3">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_ff.html#T11" title="PRE_FF:th.11">PRE_FF:11</a></span>;<br/><a NAME="E7:99_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A53</font></span>: 
2 <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c6">n</font></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<span class="p3">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_ff.html#T10" title="PRE_FF:th.10">PRE_FF:10</a></span>;<br/><a NAME="E8:99_5"/>
<font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <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 href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:99_5"><span class="lab"><font color="Green" title="E66">A50</font></span></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></span>;<br/><a NAME="E9:99_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A54</font></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="power.html#K3" title="POWER:func.3">to_power</a> <span class="p1">(<span class="default">2 <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p2"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p3">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c6">n</font></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p1">(<span class="default">2 <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p2"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p3">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<span class="p4">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:99_5"><span class="lab"><font color="Green" title="E69">A53</font></span></a>, <a class="ref" href="pre_ff.html#T10" title="PRE_FF:th.10">PRE_FF:10</a></span>;<br/><a NAME="E10:99_5"/>
 <a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="power.html#K6" title="POWER:func.6">log</a> 2,2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:99_5"><span class="lab"><font color="Green" title="E66">A50</font></span></a>, <a class="ref" href="pre_ff.html#T12" title="PRE_FF:th.12">PRE_FF:12</a></span>;<br/><a NAME="E11:99_5"/><span class="kw">then </span>
 <a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c6">n</font> <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#T60" title="POWER:th.60">POWER:60</a></span>;<br/><a NAME="E12:99_5"/><span class="kw">then </span>
<span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c6">n</font></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default">1</span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_ff.html#T11" title="PRE_FF:th.11">PRE_FF:11</a></span>;<br/><a NAME="E13:99_5"/><span class="kw">then </span>
<span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c6">n</font></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></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="int_1.html#T47" title="INT_1:th.47">INT_1:47</a></span>;<br/><a NAME="E14:99_5"/><span class="kw">then </span>
2 <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c6">n</font></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2 <a href="series_1.html#K3" title="SERIES_1:func.3">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="power.html#T44" title="POWER:th.44">POWER:44</a></span>;<br/><a NAME="E15:99_5"/><span class="kw">then </span>
<font color="Maroon" title="c6">n</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p1">(<span class="default">2 <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p2"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p3">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c6">n</font></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p1">(<span class="default">2 <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p2"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p3">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c6">n</font></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:99_5"><span class="lab"><font color="Green" title="E66">A50</font></span></a>, <a class="txt" href="#E4:99_5"><span class="lab"><font color="Green" title="E68">A52</font></span></a>, <a class="ref" href="power.html#T42" title="POWER:th.42">POWER:42</a></span>;<br/><span class="kw">hence </span><a NAME="E16:99_5"/>
<font color="Maroon" title="c2">f</font> <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> <font color="Maroon" title="c2">f</font> <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">A51</font></span></a>, <a class="txt" href="#E1:99_5"><span class="lab"><font color="Green" title="E65">A49</font></span></a>, <a class="txt" href="#E9:99_5"><span class="lab"><font color="Green" title="E70">A54</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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">A55</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="Red">::  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 implies <font color="Maroon" title="c5">g</font> <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> <font color="Maroon" title="c5">g</font> <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_6"/><span class="lab"><font color="Green" title="E66">A56</font></span>: 
<font color="Maroon" title="c6">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="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">g</font> <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> <font color="Maroon" title="c5">g</font> <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_6"/><span class="lab"><font color="Green" title="E67">A57</font></span>: 
<font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></span>;<br/><a NAME="E3:99_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A58</font></span>: 
<font color="Maroon" title="c6">n</font> <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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> <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c6">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:99_6"><span class="lab"><font color="Green" title="E66">A56</font></span></a>, <a class="ref" href="power.html#T42" title="POWER:th.42">POWER:42</a></span>;<br/><a NAME="E4:99_6"/>
<font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <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_6"><span class="lab"><font color="Green" title="E66">A56</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E5:99_6"/><span class="kw">then </span>
<font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <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="E6:99_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A59</font></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#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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> <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</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="#E2:99_6"><span class="lab"><font color="Green" title="E67">A57</font></span></a>, <a class="ref" href="power.html#T44" title="POWER:th.44">POWER:44</a></span>;<br/><a NAME="E7:99_6"/><span class="lab"><font color="Green" title="E70">A60</font></span>: 
<font color="Maroon" title="c5">g</font> <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> <a href="hidden.html#R1" title="HIDDEN:pred.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> <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</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="#E2:99"><span class="lab"><font color="Green" title="E58">A2</font></span></a></span>;<br/><a NAME="E8:99_6"/>
<font color="Maroon" title="c5">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> <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c6">n</font>
 <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>, <a class="txt" href="#E1:99_6"><span class="lab"><font color="Green" title="E66">A56</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E9:99_6"/>
<font color="Maroon" title="c5">g</font> <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> <font color="Maroon" title="c5">g</font> <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="#E7:99_6"><span class="lab"><font color="Green" title="E70">A60</font></span></a>, <a class="txt" href="#E6:99_6"><span class="lab"><font color="Green" title="E69">A59</font></span></a>, <a class="txt" href="#E3:99_6"><span class="lab"><font color="Green" title="E68">A58</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">take </span>
<font color="Maroon" title="c5">g</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">g</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">g</font> &amp; <font color="Maroon" title="c2">f</font> <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="c5">g</font>,<a href="asympt_1.html#K7" title="ASYMPT_1:func.7">POWEROF2SET</a>  &amp; not <font color="Maroon" title="c2">f</font> <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="c5">g</font> &amp; <font color="Maroon" title="c2">f</font> is  <a href="asympt_0.html#V6" title="ASYMPT_0:attr.6">eventually-nondecreasing</a>  &amp; <font color="Maroon" title="c5">g</font> is  <a href="asympt_0.html#V6" title="ASYMPT_0:attr.6">eventually-nondecreasing</a>  &amp; not <font color="Maroon" title="c5">g</font> <a href="asympt_0.html#R2" title="ASYMPT_0:pred.2">is_smooth_wrt</a> 2 )</span><br/>

<a NAME="E11:99"/>
<font color="Maroon" title="c2">f</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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T11" title="FUNCT_2:th.11">FUNCT_2:11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:99"/>
( <font color="Maroon" title="c5">g</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">g</font> &amp; <font color="Maroon" title="c2">f</font> <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="c5">g</font>,<a href="asympt_1.html#K7" title="ASYMPT_1:func.7">POWEROF2SET</a>  &amp; not <font color="Maroon" title="c2">f</font> <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="c5">g</font> &amp; <font color="Maroon" title="c2">f</font> is  <a href="asympt_0.html#V6" title="ASYMPT_0:attr.6">eventually-nondecreasing</a>  &amp; <font color="Maroon" title="c5">g</font> is  <a href="asympt_0.html#V6" title="ASYMPT_0:attr.6">eventually-nondecreasing</a>  &amp; not <font color="Maroon" title="c5">g</font> <a href="asympt_0.html#R2" title="ASYMPT_0:pred.2">is_smooth_wrt</a> 2 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:99"><span class="lab"><font color="Green" title="E60">A5</font></span></a>, <a class="txt" href="#E7:99"><span class="lab"><font color="Green" title="E62">A12</font></span></a>, <a class="txt" href="#E9:99"><span class="lab"><font color="Green" title="E64">A48</font></span></a>, <a class="txt" href="#E10:99"><span class="lab"><font color="Green" title="E65">A55</font></span></a>, <a class="txt" href="#E8:99"><span class="lab"><font color="Green" title="E63">A33</font></span></a>, <a class="ref" href="asympt_0.html#D8" title="ASYMPT_0:def.8">ASYMPT_0:def 8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
