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

<b>let </b><font color="Maroon">f</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>

<b>assume </b><a NAME="E1:92"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a>  <a href="power.html#K6">log</a> 2,<span class="p1">(<span class="default"><font color="Olive">n</font> <a href="newton.html#K12">!</a> </span>)</span>
 ;<br/>

<b>set </b><font color="Maroon">g</font> =  <a href="asympt_1.html#K2" target="_self">seq_logn</a> ;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="series_1.html#K4">Sum</a> <a href="asympt_1.html#K2" target="_self">seq_logn</a> ,<font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E2:92"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1"><b>proof </b></a><div class="add">

<i><font color="Green">E51</font></i>: <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> 0 = 
 <a href="power.html#K6">log</a> 2,1
<b>by </b><i>, <a class="ref" href="newton.html#T18">NEWTON:18</a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="power.html#T59">POWER:59</a></i>
;<br/>
 <a href="series_1.html#K4">Sum</a> <a href="asympt_1.html#K2" target="_self">seq_logn</a> ,0 = 
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <a href="asympt_1.html#K2" target="_self">seq_logn</a> </span>)</span> <a href="seq_1.html#K2">.</a> 0
<b>by </b><i><a class="ref" href="series_1.html#D6">SERIES_1:def 6</a></i>
<br/>.= 
<a href="asympt_1.html#K2" target="_self">seq_logn</a>  <a href="seq_1.html#K2">.</a> 0
<b>by </b><i><a class="ref" href="series_1.html#D1">SERIES_1:def 1</a></i>
<br/>.= 
0
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E3:92_1"/>
<font color="Maroon">S<sub>1</sub></font>[0]
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E3:92"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:92_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="series_1.html#K4">Sum</a> <a href="asympt_1.html#K2" target="_self">seq_logn</a> ,<font color="Maroon">k</font>
 ;<br/>

<a NAME="E2:92_2"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">k</font> <a href="newton.html#K12">!</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="newton.html#T23">NEWTON:23</a></i>;<br/>
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
 <a href="power.html#K6">log</a> 2,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="newton.html#K12">!</a> </span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="power.html#K6">log</a> 2,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="newton.html#K12">!</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T21">NEWTON:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="newton.html#K12">!</a> </span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="power.html#T61">POWER:61</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="series_1.html#K4">Sum</a> <a href="asympt_1.html#K2" target="_self">seq_logn</a> ,<font color="Maroon">k</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="asympt_1.html#K2" target="_self">seq_logn</a>  <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="series_1.html#K4">Sum</a> <a href="asympt_1.html#K2" target="_self">seq_logn</a> ,<font color="Maroon">k</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="asympt_1.html#K2" target="_self">seq_logn</a>  <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <a href="asympt_1.html#K2" target="_self">seq_logn</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i><a class="ref" href="series_1.html#D6">SERIES_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <a href="asympt_1.html#K2" target="_self">seq_logn</a> </span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="series_1.html#D1">SERIES_1:def 1</a></i>
<br/>.= 
 <a href="series_1.html#K4">Sum</a> <a href="asympt_1.html#K2" target="_self">seq_logn</a> ,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="series_1.html#D6">SERIES_1:def 6</a></i>
;<br/>
<b>hence </b><a NAME="E4:92_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:92"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<b>hence </b><a NAME="E5:92"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a>  <a href="series_1.html#K4">Sum</a> <a href="asympt_1.html#K2" target="_self">seq_logn</a> ,<font color="Olive">n</font>
 ;<br/>


</div>
