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

<span class="kw">let </span><font color="Maroon" title="c1">e</font> be   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font> being   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>  st (  for <font color="Olive" title="b2">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/><font color="Olive" title="b1">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="power.html#K6" title="POWER:func.6">log</a> (2,<span class="p1">(<span class="default"><font color="Olive" title="b2">n</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">e</font></span>)</span>) ) holds <br/>( <font color="Olive" title="b1">f</font> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p1">(<span class="default"><a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> <font color="Maroon" title="c1">e</font></span>)</span> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p2">(<span class="default"><a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> <font color="Maroon" title="c1">e</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">f</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   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>  <a href="power.html#K6" title="POWER:func.6">log</a> (2,<span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">e</font></span>)</span>) ) implies ( <font color="Maroon" title="c2">f</font> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p1">(<span class="default"><a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> <font color="Maroon" title="c1">e</font></span>)</span> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p2">(<span class="default"><a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> <font color="Maroon" title="c1">e</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:20"/><span class="lab"><font color="Green" title="E15">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>  <a href="power.html#K6" title="POWER:func.6">log</a> (2,<span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">e</font></span>)</span>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">f</font> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p1">(<span class="default"><a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> <font color="Maroon" title="c1">e</font></span>)</span> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p2">(<span class="default"><a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> <font color="Maroon" title="c1">e</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c3">g</font> =  <a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> <font color="Maroon" title="c1">e</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">h</font> = <font color="Maroon" title="c2">f</font> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p1">(<span class="default"><a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> <font color="Maroon" title="c1">e</font></span>)</span>;<br/>
<div><span class="lab"><font color="Green" title="E16">A2</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">p</font> be   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies  ex <font color="Olive" title="b1">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/> for <font color="Olive" title="b2">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">N</font> holds <br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p4">(<span class="default"><a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> <font color="Maroon" title="c1">e</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">p</font> )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c6">p1</font> = <font color="Maroon" title="c5">p</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c7">i0</font> = <span class="p1"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">7 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c6">p1</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">e</font></span>)</span></span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c8">i1</font> = <span class="p1"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p5">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">e</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c9">N</font> =  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</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"><span class="p3">(<span class="default"><span class="p4">(<span class="default">7 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c6">p1</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">e</font></span>)</span></span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>,<span class="p2"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p5">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p0">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">e</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>)</span>)</span>,2);<br/><a NAME="E2:20_1"/><span class="lab"><font color="Green" title="E17">A3</font></span>: 
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</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"><span class="p3">(<span class="default"><span class="p4">(<span class="default">7 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c6">p1</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">e</font></span>)</span></span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>,<span class="p2"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p5">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p0">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">e</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>)</span>)</span>,2) <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <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"><span class="p3">(<span class="default">7 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c6">p1</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">e</font></span>)</span></span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>,<span class="p1"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p5">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">e</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</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="E3:20_1"/><span class="lab"><font color="Green" title="E18">A4</font></span>: 
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</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"><span class="p3">(<span class="default"><span class="p4">(<span class="default">7 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c6">p1</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">e</font></span>)</span></span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>,<span class="p2"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p5">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p0">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">e</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>)</span>)</span>,2) is   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:20_1_1_1"/>
(  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</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"><span class="p3">(<span class="default"><span class="p4">(<span class="default">7 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c6">p1</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">e</font></span>)</span></span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>,<span class="p2"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p5">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p0">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">e</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>)</span>)</span>,2) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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"><span class="p3">(<span class="default">7 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c6">p1</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">e</font></span>)</span></span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>,<span class="p1"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p5">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">e</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>) or  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</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"><span class="p3">(<span class="default"><span class="p4">(<span class="default">7 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c6">p1</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">e</font></span>)</span></span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>,<span class="p2"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p5">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p0">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">e</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>)</span>)</span>,2) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 </span><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/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:20_1_1_1_1"/>
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</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"><span class="p3">(<span class="default"><span class="p4">(<span class="default">7 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c6">p1</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">e</font></span>)</span></span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>,<span class="p2"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p5">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p0">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">e</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>)</span>)</span>,2) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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"><span class="p3">(<span class="default">7 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c6">p1</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">e</font></span>)</span></span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>,<span class="p1"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p5">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">e</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</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"><span class="p3">(<span class="default"><span class="p4">(<span class="default">7 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c6">p1</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">e</font></span>)</span></span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>,<span class="p2"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p5">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p0">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">e</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>)</span>)</span>,2) is   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:20_1_1_1_1"/>
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</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"><span class="p3">(<span class="default"><span class="p4">(<span class="default">7 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c6">p1</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">e</font></span>)</span></span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>,<span class="p2"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p5">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p0">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">e</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>)</span>)</span>,2) 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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:20_1_1_1_2"/>
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</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"><span class="p3">(<span class="default"><span class="p4">(<span class="default">7 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c6">p1</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">e</font></span>)</span></span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>,<span class="p2"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p5">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p0">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">e</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>)</span>)</span>,2) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</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"><span class="p3">(<span class="default"><span class="p4">(<span class="default">7 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c6">p1</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">e</font></span>)</span></span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>,<span class="p2"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p5">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p0">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">e</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>)</span>)</span>,2) is   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:20_1_1_1_2"/>
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</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"><span class="p3">(<span class="default"><span class="p4">(<span class="default">7 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c6">p1</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">e</font></span>)</span></span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>,<span class="p2"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p5">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p0">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">e</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>)</span>)</span>,2) is   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><a NAME="E4:20_1"/><span class="lab"><font color="Green" title="E19">A5</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">e</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">e</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</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#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E5:20_1"/>
<span class="p1"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p5">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">e</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">e</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#D7" title="INT_1:def.7">INT_1:def 7</a></span>;<br/><a NAME="E6:20_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A6</font></span>: 
<span class="p1"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p5">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">e</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c5">p</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">e</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:20_1"><span class="lab"><font color="Green" title="E19">A5</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">assume </span><a NAME="E7:20_1"/><span class="lab"><font color="Green" title="E21">A7</font></span>: 
<font color="Maroon" title="c5">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <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 <br/> for <font color="Olive" title="b2">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">N</font> holds <br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p4">(<span class="default"><a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> <font color="Maroon" title="c1">e</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">p</font></span><br/><a NAME="E8:20_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A8</font></span>: 
<font color="Maroon" title="c6">p1</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T34" title="POWER:th.34">POWER:34</a></span>;<br/><a NAME="E9:20_1"/>
 <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"><span class="p3">(<span class="default">7 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c6">p1</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">e</font></span>)</span></span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>,<span class="p1"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p5">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">e</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>) <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p5">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">e</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</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="E10:20_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A9</font></span>: 
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</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"><span class="p3">(<span class="default"><span class="p4">(<span class="default">7 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c6">p1</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">e</font></span>)</span></span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>,<span class="p2"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p5">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p0">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">e</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>)</span>)</span>,2) <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p5">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">e</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20_1"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E11:20_1"/><span class="lab"><font color="Green" title="E24">A10</font></span>: 
<span class="p1"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">7 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c6">p1</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">e</font></span>)</span></span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default">7 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">e</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#D7" title="INT_1:def.7">INT_1:def 7</a></span>;<br/><a NAME="E12:20_1"/>
 <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"><span class="p3">(<span class="default">7 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c6">p1</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">e</font></span>)</span></span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>,<span class="p1"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p5">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">e</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>) <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">7 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c6">p1</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">e</font></span>)</span></span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></span>;<br/><a NAME="E13:20_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A11</font></span>: 
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</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"><span class="p3">(<span class="default"><span class="p4">(<span class="default">7 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c6">p1</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">e</font></span>)</span></span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>,<span class="p2"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p5">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p0">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">e</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>)</span>)</span>,2) <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">7 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c6">p1</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">e</font></span>)</span></span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20_1"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E14:20_1"/><span class="lab"><font color="Green" title="E26">A12</font></span>: 
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</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"><span class="p3">(<span class="default"><span class="p4">(<span class="default">7 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c6">p1</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">e</font></span>)</span></span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>,<span class="p2"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p5">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p0">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">e</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>)</span>)</span>,2) <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 <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="E15:20_1"/><span class="lab"><font color="Green" title="E27">A13</font></span>: 
<font color="Maroon" title="c6">p1</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">e</font></span>)</span></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="txt" href="#E7:20_1"><span class="lab"><font color="Green" title="E21">A7</font></span></a>, <a class="ref" href="power.html#T34" title="POWER:th.34">POWER:34</a></span>;<br/><a NAME="E16:20_1"/><span class="lab"><font color="Green" title="E28">A14</font></span>: 
7 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 7 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:20_1"><span class="lab"><font color="Green" title="E21">A7</font></span></a>, <a class="ref" href="xreal_1.html#T68" title="XREAL_1:th.68">XREAL_1:68</a></span>;<br/><a NAME="E17:20_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A15</font></span>: 
<span class="p1">(<span class="default">7 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c6">p1</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">e</font></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#T34" title="POWER:th.34">POWER:34</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c10">N</font> =  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</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"><span class="p3">(<span class="default"><span class="p4">(<span class="default">7 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c6">p1</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">e</font></span>)</span></span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>,<span class="p2"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p5">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p0">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">e</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>)</span>)</span>,2) 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="#E14:20_1"><span class="lab"><font color="Green" title="E26">A12</font></span></a>, <a class="txt" href="#E3:20_1"><span class="lab"><font color="Green" title="E18">A4</font></span></a>, <a class="ref" href="int_1.html#T3" title="INT_1:th.3">INT_1:3</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c11">N</font> = <font color="Maroon" title="c10">N</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   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="c11">N</font> holds <br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p4">(<span class="default"><a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> <font color="Maroon" title="c1">e</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">p</font></span><br/><span class="kw">let </span><font color="Maroon" title="c12">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c11">N</font> implies  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p4">(<span class="default"><a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> <font color="Maroon" title="c1">e</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">p</font> )</span><br/><span class="kw">set </span><font color="Maroon" title="c13">c</font> = <font color="Maroon" title="c6">p1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">e</font></span>)</span>;<br/><span class="kw">assume </span><a NAME="E19:20_1"/><span class="lab"><font color="Green" title="E30">A16</font></span>: 
<font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c11">N</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p4">(<span class="default"><a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> <font color="Maroon" title="c1">e</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">p</font></span><br/><a NAME="E20:20_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">n</font> <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"><span class="p3">(<span class="default">7 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c6">p1</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">e</font></span>)</span></span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:20_1"><span class="lab"><font color="Green" title="E25">A11</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E21:20_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default">7 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">e</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:20_1"><span class="lab"><font color="Green" title="E24">A10</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E22:20_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">n</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">e</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">7 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">e</font></span>)</span></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">e</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:20_1"><span class="lab"><font color="Green" title="E29">A15</font></span></a>, <a class="txt" href="asympt_1.html#E13"><span class="lab"><font color="Green" title="E10">Lm6</font></span></a></span>;<br/><a NAME="E23:20_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">n</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">e</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default">7 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c6">p1</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">e</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">e</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:20_1"><span class="lab"><font color="Green" title="E28">A14</font></span></a>, <a class="ref" href="power.html#T33" title="POWER:th.33">POWER:33</a></span>;<br/><a NAME="E24:20_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">n</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">e</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default">7 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T106" title="XCMPLX_1:th.106">XCMPLX_1:106</a></span>;<br/><a NAME="E25:20_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">n</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">e</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 7 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c6">p1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T25" title="POWER:th.25">POWER:25</a></span>;<br/><a NAME="E26:20_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">p</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">e</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default">7 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c5">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:20_1"><span class="lab"><font color="Green" title="E21">A7</font></span></a>, <a class="ref" href="xreal_1.html#T64" title="XREAL_1:th.64">XREAL_1:64</a></span>;<br/><a NAME="E27:20_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">p</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">e</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 7
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:20_1"><span class="lab"><font color="Green" title="E21">A7</font></span></a>, <a class="ref" href="xcmplx_1.html#T87" title="XCMPLX_1:th.87">XCMPLX_1:87</a></span>;<br/><a NAME="E28:20_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">p</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">e</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 6
 <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="E29:20_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A17</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">e</font></span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">e</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="asympt_1.html#E16"><span class="lab"><font color="Green" title="E13">Lm9</font></span></a></span>;<br/><a NAME="E30:20_1"/>
<font color="Maroon" title="c12">n</font> <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"><span class="p3">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p5">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">e</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:20_1"><span class="lab"><font color="Green" title="E23">A9</font></span></a>, <a class="txt" href="#E19:20_1"><span class="lab"><font color="Green" title="E30">A16</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E31:20_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c5">p</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">e</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:20_1"><span class="lab"><font color="Green" title="E20">A6</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E32:20_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">n</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">e</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">e</font></span>)</span></span>)</span></span>)</span> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c1">e</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:20_1"><span class="lab"><font color="Green" title="E27">A13</font></span></a>, <a class="ref" href="power.html#T37" title="POWER:th.37">POWER:37</a></span>;<br/><a NAME="E33:20_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">n</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">e</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c6">p1</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">e</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c1">e</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:20_1"><span class="lab"><font color="Green" title="E21">A7</font></span></a>, <a class="ref" href="power.html#T33" title="POWER:th.33">POWER:33</a></span>;<br/><a NAME="E34:20_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">n</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">e</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c5">p</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">e</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c1">e</font></span>)</span></span>)</span>
 ;<br/><a NAME="E35:20_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">n</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">e</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c5">p</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T87" title="XCMPLX_1:th.87">XCMPLX_1:87</a></span>;<br/><a NAME="E36:20_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> 2</span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">e</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> 2</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 2</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:20_1"><span class="lab"><font color="Green" title="E22">A8</font></span></a>, <a class="ref" href="xreal_1.html#T68" title="XREAL_1:th.68">XREAL_1:68</a></span>;<br/><a NAME="E37:20_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> 2</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">e</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c6">p1</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 2</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:20_1"><span class="lab"><font color="Green" title="E21">A7</font></span></a>, <a class="ref" href="power.html#T27" title="POWER:th.27">POWER:27</a></span>;<br/><a NAME="E38:20_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> 2</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">e</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="ref" href="power.html#T24" title="POWER:th.24">POWER:24</a></span>;<br/><a NAME="E39:20_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">e</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="ref" href="power.html#T46" title="POWER:th.46">POWER:46</a></span>;<br/><a NAME="E40:20_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A18</font></span>: 
1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">e</font></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> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T88" title="XREAL_1:th.88">XREAL_1:88</a></span>;<br/><a NAME="E41:20_1"/>
2 <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">e</font></span>)</span></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#T34" title="POWER:th.34">POWER:34</a></span>;<br/><a NAME="E42:20_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A19</font></span>: 
<span class="p1">(<span class="default">2 <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">e</font></span>)</span></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="c6">p1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">e</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">e</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:20_1"><span class="lab"><font color="Green" title="E32">A18</font></span></a>, <a class="ref" href="xreal_1.html#T68" title="XREAL_1:th.68">XREAL_1:68</a></span>;<br/><a NAME="E43:20_1"/><span class="lab"><font color="Green" title="E34">A20</font></span>: 
<font color="Maroon" title="c12">n</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">e</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:20_1"><span class="lab"><font color="Green" title="E26">A12</font></span></a>, <a class="txt" href="#E19:20_1"><span class="lab"><font color="Green" title="E30">A16</font></span></a>, <a class="ref" href="power.html#T34" title="POWER:th.34">POWER:34</a></span>;<br/><a NAME="E44:20_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">p</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">e</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c5">p</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:20_1"><span class="lab"><font color="Green" title="E21">A7</font></span></a>, <a class="ref" href="xreal_1.html#T68" title="XREAL_1:th.68">XREAL_1:68</a></span>;<br/><a NAME="E45:20_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">p1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">e</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">e</font></span>)</span></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">p1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">e</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:20_1"><span class="lab"><font color="Green" title="E31">A17</font></span></a>, <a class="ref" href="xreal_1.html#T81" title="XREAL_1:th.81">XREAL_1:81</a></span>;<br/><a NAME="E46:20_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">n</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">e</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">e</font></span>)</span></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">p1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">e</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c5">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:20_1"><span class="lab"><font color="Green" title="E21">A7</font></span></a>, <a class="ref" href="xreal_1.html#T81" title="XREAL_1:th.81">XREAL_1:81</a></span>;<br/><a NAME="E47:20_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">n</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">e</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">e</font></span>)</span></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="c6">p1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">e</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c5">p</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T78" title="XCMPLX_1:th.78">XCMPLX_1:78</a></span>;<br/><a NAME="E48:20_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">n</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">e</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">e</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:20_1"><span class="lab"><font color="Green" title="E33">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="E49:20_1"/><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">n</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">e</font></span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="power.html#K6" title="POWER:func.6">log</a> (2,<span class="p1">(<span class="default">2 <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">e</font></span>)</span></span>)</span></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:20_1"><span class="lab"><font color="Green" title="E34">A20</font></span></a>, <a class="ref" href="power.html#T57" title="POWER:th.57">POWER:57</a></span>;<br/><a NAME="E50:20_1"/><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">n</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">e</font></span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">e</font></span>)</span></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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T55" title="POWER:th.55">POWER:55</a></span>;<br/><a NAME="E51:20_1"/><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">n</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">e</font></span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">e</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T52" title="POWER:th.52">POWER:52</a></span>;<br/><a NAME="E52:20_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A21</font></span>: 
<span class="p1">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> (2,<span class="p2">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">e</font></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="c12">n</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">e</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:20_1"><span class="lab"><font color="Green" title="E26">A12</font></span></a>, <a class="txt" href="#E19:20_1"><span class="lab"><font color="Green" title="E30">A16</font></span></a>, <a class="ref" href="power.html#T34" title="POWER:th.34">POWER:34</a>, <a class="ref" href="xreal_1.html#T83" title="XREAL_1:th.83">XREAL_1:83</a></span>;<br/><a NAME="E53:20_1"/>
<font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:20_1"><span class="lab"><font color="Green" title="E26">A12</font></span></a>, <a class="txt" href="#E19:20_1"><span class="lab"><font color="Green" title="E30">A16</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E54:20_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">n</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="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E55:20_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">n</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">e</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c12">n</font> <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T39" title="POWER:th.39">POWER:39</a></span>;<br/><a NAME="E56:20_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">n</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">e</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="ref" href="power.html#T24" title="POWER:th.24">POWER:24</a></span>;<br/><a NAME="E57:20_1"/><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">n</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">e</font></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> (2,1)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T57" title="POWER:th.57">POWER:57</a></span>;<br/><a NAME="E58:20_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A22</font></span>: 
 <a href="power.html#K6" title="POWER:func.6">log</a> (2,<span class="p1">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">e</font></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#T51" title="POWER:th.51">POWER:51</a></span>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p2">(<span class="default"><a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> <font color="Maroon" title="c1">e</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">n</font></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"><a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> <font color="Maroon" title="c1">e</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="asympt_1.html#E10"><span class="lab"><font color="Green" title="E7">Lm4</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> (2,<span class="p2">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">e</font></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"><a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> <font color="Maroon" title="c1">e</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="txt" href="#E14:20_1"><span class="lab"><font color="Green" title="E26">A12</font></span></a>, <a class="txt" href="#E19:20_1"><span class="lab"><font color="Green" title="E30">A16</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> (2,<span class="p2">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">e</font></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="c12">n</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">e</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:20_1"><span class="lab"><font color="Green" title="E26">A12</font></span></a>, <a class="txt" href="#E19:20_1"><span class="lab"><font color="Green" title="E30">A16</font></span></a>, <a class="ref" href="asympt_1.html#D3" target="_self" title="ASYMPT_1:def.3">Def3</a></span>
;<br/><span class="kw">hence </span><a NAME="E60:20_1"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p4">(<span class="default"><a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> <font color="Maroon" title="c1">e</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:20_1"><span class="lab"><font color="Green" title="E34">A20</font></span></a>, <a class="txt" href="#E52:20_1"><span class="lab"><font color="Green" title="E35">A21</font></span></a>, <a class="txt" href="#E58:20_1"><span class="lab"><font color="Green" title="E36">A22</font></span></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:20"/>
<font color="Maroon" title="c2">f</font> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p1">(<span class="default"><a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> <font color="Maroon" title="c1">e</font></span>)</span> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="seq_2.html#D6" title="SEQ_2:def.6">SEQ_2:def 6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p2">(<span class="default"><a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> <font color="Maroon" title="c1">e</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/>

<span class="kw">hence </span><a NAME="E4:20"/>
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p2">(<span class="default"><a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> <font color="Maroon" title="c1">e</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20"><span class="lab"><font color="Green" title="E16">A2</font></span></a>, <a class="ref" href="seq_2.html#D7" title="SEQ_2:def.7">SEQ_2:def 7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
