<?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>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="E1:36"/><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="series_1.html#K4">Sum</a> <span class="p1">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> <font color="Maroon">k</font></span>)</span>,<font color="Olive">n</font>
 ;<br/>

<b>set </b><font color="Maroon">g</font> =  <a href="asympt_1.html#K3" target="_self">seq_n^</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/>
<a NAME="E2:36"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">f</font> is    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <a href="numbers.html#K5">NAT</a> ,<a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/>
<a NAME="E3:36"/>
2 <a href="series_1.html#K3">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="power.html#T39">POWER:39</a></i>;<br/>
<a NAME="E4:36"/><b>then </b><i><font color="Green">E51</font></i>: 
<span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</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#K2">"</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T124">XREAL_1:124</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:36_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 ;<br/><b>set </b><font color="Maroon">p</font> =  <a href="asympt_1.html#K3" target="_self">seq_n^</a> <font color="Maroon">k</font>;<br/><a NAME="E2:36_1"/>
 ex <font color="Olive">s</font> being  <a href="seq_1.html#NM1">Real_Sequence</a> st <br/>( <font color="Olive">s</font> <a href="seq_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> 0 &amp; ( for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 holds <br/><font color="Olive">s</font> <a href="seq_1.html#K2">.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> <font color="Maroon">k</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1"><b>proof </b></a><div class="add">

<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> ,  <a href="real_1.html#NM1">Real</a>] <b>means </b>( ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 ) &amp; ( <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> <font color="Maroon">k</font> ) );<br/>
<a NAME="E1:36_1_1"/><i><font color="Green">E59</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   ex <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1"><b>proof </b></a><div class="add">

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

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_1_1_1_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:36_1_1_1_1_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:36_1_1_1_1_1"/>
 ex <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Olive">y</font>]
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:36_1_1_1_1_2"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:36_1_1_1_1_2"/>
 ex <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Olive">y</font>]
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>consider </b><font color="Maroon">h</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E3:36_1_1"/><i><font color="Green">E60</font></i>: 
for <font color="Olive">x</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">x</font>,<font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>();<br/></i>
<b>take </b>
<font color="Maroon">h</font>
;<br/>

<b>thus </b><a NAME="E4:36_1_1"/>
<font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> 0
 <b>by </b><i/>;<br/>

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

<b>thus </b><a NAME="E5:36_1_1"/>
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 implies <font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> <font color="Maroon">k</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon">q</font> being   <a href="seq_1.html#NM1">Real_Sequence</a><b> such that </b><br/><a NAME="E4:36_1"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">q</font> <a href="seq_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> 0 &amp; ( for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 holds <br/><font color="Maroon">q</font> <a href="seq_1.html#K2">.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> <font color="Maroon">k</font> ) )
 ;<br/><a NAME="E5:36_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="series_1.html#K4">Sum</a> <span class="p1">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> <font color="Maroon">k</font></span>)</span>,<font color="Maroon">n</font>
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:36_1_2"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_1_2_1"/>
( <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_2_1_1"><b>suppose </b></a><a NAME="E1:36_1_2_1_1"/>
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:36_1_2_1_1"/>
<span class="p1">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_2_1_2"><b>suppose </b></a><a NAME="E1:36_1_2_1_2"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:36_1_2_1_2"/><b>then </b><i><font color="Green">E67</font></i>: 
<span class="p1">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="series_1.html#K3">to_power</a> <font color="Maroon">k</font>
 <b>by </b><i/>;<br/><a NAME="E3:36_1_2_1_2"/>
<font color="Maroon">q</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> <font color="Maroon">k</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E4:36_1_2_1_2"/>
<span class="p1">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font>
 <b>by </b><i>, , , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E7:36_1"/><b>then </b><i><font color="Green">E69</font></i>: 
 <a href="series_1.html#K4">Sum</a> <span class="p1">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> <font color="Maroon">k</font></span>)</span>,<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="series_1.html#K4">Sum</a> <font color="Maroon">q</font>,<font color="Maroon">n</font>
 <b>by </b><i/>;<br/> <a href="series_1.html#K4">Sum</a> <font color="Maroon">q</font>,<font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="power.html#T30">POWER:30</a></i>
<br/>.= 
<font color="Maroon">n</font> <a href="series_1.html#K3">to_power</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="power.html#T32">POWER:32</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</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="seq_1.html#K2">.</a> <font color="Maroon">n</font>
<b>by </b><i>, </i>
;<br/><b>hence </b><a NAME="E9:36_1"/>
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:36_1_4"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_1_4_1"/>
( <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_4_1_1"><b>suppose </b></a><a NAME="E1:36_1_4_1_1"/>
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:36_1_4_1_1"/>
<span class="p1">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_4_1_2"><b>suppose </b></a><a NAME="E1:36_1_4_1_2"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:36_1_4_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="series_1.html#K3">to_power</a> <font color="Maroon">k</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:36_1_4_1_2"/>
<span class="p1">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E11:36_1"/>
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<a NAME="E6:36"/><b>then </b><i><font color="Green">E70</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="asympt_0.html#K9">Big_Oh</a> <span class="p1">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:36_2"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 ;<br/><b>set </b><font color="Maroon">p</font> =  <a href="asympt_1.html#K3" target="_self">seq_n^</a> <font color="Maroon">k</font>;<br/><a NAME="E2:36_2"/>
 ex <font color="Olive">s</font> being  <a href="seq_1.html#NM1">Real_Sequence</a> st <br/>( <font color="Olive">s</font> <a href="seq_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> 0 &amp; ( for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 holds <br/><font color="Olive">s</font> <a href="seq_1.html#K2">.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">k</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_2_1"><b>proof </b></a><div class="add">

<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> ,  <a href="real_1.html#NM1">Real</a>] <b>means </b>( ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 ) &amp; ( <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">k</font> ) );<br/>
<a NAME="E1:36_2_1"/><i><font color="Green">E72</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   ex <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_2_1_1"><b>proof </b></a><div class="add">

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

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_2_1_1_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:36_2_1_1_1_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:36_2_1_1_1_1"/>
 ex <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Olive">y</font>]
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:36_2_1_1_1_2"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:36_2_1_1_1_2"/>
 ex <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Olive">y</font>]
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>consider </b><font color="Maroon">h</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E3:36_2_1"/><i><font color="Green">E73</font></i>: 
for <font color="Olive">x</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">x</font>,<font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>();<br/></i>
<b>take </b>
<font color="Maroon">h</font>
;<br/>

<b>thus </b><a NAME="E4:36_2_1"/>
<font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> 0
 <b>by </b><i/>;<br/>

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

<b>thus </b><a NAME="E5:36_2_1"/>
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 implies <font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">k</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon">q</font> being   <a href="seq_1.html#NM1">Real_Sequence</a><b> such that </b><br/><a NAME="E4:36_2"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">q</font> <a href="seq_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> 0 &amp; ( for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 holds <br/><font color="Maroon">q</font> <a href="seq_1.html#K2">.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">k</font> ) )
 ;<br/><a NAME="E5:36_2"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="series_1.html#K4">Sum</a> <span class="p1">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> <font color="Maroon">k</font></span>)</span>,<font color="Maroon">n</font>
 <b>by </b><i/>;<br/><a NAME="E6:36_2"/><i><font color="Green">E77</font></i>: 
<span class="p1">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</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="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="series_1.html#K3">to_power</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>, </i>;<br/><b>set </b><font color="Maroon">n1</font> = <span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="int_1.html#K2">\]</a></span>;<br/><a NAME="E7:36_2"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">n</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E8:36_2"/><b>then </b><i><font color="Green">E79</font></i>: 
<font color="Maroon">n</font> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><a NAME="E9:36_2"/><i><font color="Green">E80</font></i>: 
<span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="int_1.html#K2">\]</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n</font> <a href="real_1.html#K6">/</a> 2
 <b>by </b><i><a class="ref" href="int_1.html#D5">INT_1:def 5</a></i>;<br/><b>then reconsider </b><font color="Maroon">n1</font> = <span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="int_1.html#K2">\]</a></span> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/><b>set </b><font color="Maroon">n2</font> = <font color="Maroon">n1</font> <a href="real_1.html#K5">-</a> 1;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:36_2_2"/>
<font color="Maroon">n1</font> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;</a> 0
 ;<br/><a NAME="E2:36_2_2"/><b>then </b>
<font color="Maroon">n1</font> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> 1
 <b>by </b><i><a class="ref" href="int_1.html#T21">INT_1:21</a></i>;<br/><a NAME="E3:36_2_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K3">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="real_1.html#K3">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><b>hence </b><a NAME="E4:36_2_2"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">n2</font> = <font color="Maroon">n1</font> <a href="real_1.html#K5">-</a> 1 as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/><div><i><font color="Green">E81</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="36_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:36_2_3"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_2_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_2_3_1"/>
( <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_2_3_1_1"><b>suppose </b></a><a NAME="E1:36_2_3_1_1"/>
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:36_2_3_1_1"/>
<span class="p1">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_2_3_1_2"><b>suppose </b></a><a NAME="E1:36_2_3_1_2"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:36_2_3_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="series_1.html#K3">to_power</a> <font color="Maroon">k</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:36_2_3_1_2"/>
<span class="p1">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E14:36_2"/><b>then </b><i><font color="Green">E82</font></i>: 
 <a href="series_1.html#K4">Sum</a> <span class="p1">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> <font color="Maroon">k</font></span>)</span>,<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i/>;<br/><a NAME="E15:36_2"/><i><font color="Green">E83</font></i>: 
 <a href="series_1.html#K5">Sum</a> <span class="p1">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> <font color="Maroon">k</font></span>)</span>,<font color="Maroon">n</font>,<font color="Maroon">n2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K4">Sum</a> <span class="p2">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> <font color="Maroon">k</font></span>)</span>,<font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="series_1.html#K4">Sum</a> <span class="p2">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> <font color="Maroon">k</font></span>)</span>,<font color="Maroon">n2</font></span>)</span>
 <b>by </b><i><a class="ref" href="series_1.html#D7">SERIES_1:def 7</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_2_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:36_2_4"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n2</font>
 ;<br/><a NAME="E2:36_2_4"/>
<font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E3:36_2_4"/><b>then </b>
<font color="Maroon">n2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n1</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/><a NAME="E4:36_2_4"/><b>then </b><i><font color="Green">E85</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n1</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E5:36_2_4"/>
<font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/><a NAME="E6:36_2_4"/><b>then </b>
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E7:36_2_4"/>
<span class="p1">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i/>;<br/></div><b>end;</b></div><a NAME="E17:36_2"/><b>then </b>
 <a href="series_1.html#K4">Sum</a> <span class="p1">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> <font color="Maroon">k</font></span>)</span>,<font color="Maroon">n2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i/>;<br/><a NAME="E18:36_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="series_1.html#K4">Sum</a> <span class="p2">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> <font color="Maroon">k</font></span>)</span>,<font color="Maroon">n</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="series_1.html#K4">Sum</a> <span class="p2">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> <font color="Maroon">k</font></span>)</span>,<font color="Maroon">n2</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="series_1.html#K4">Sum</a> <span class="p2">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> <font color="Maroon">k</font></span>)</span>,<font color="Maroon">n</font></span>)</span> <a href="real_1.html#K3">+</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E19:36_2"/><b>then </b><i><font color="Green">E86</font></i>: 
 <a href="series_1.html#K4">Sum</a> <span class="p1">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> <font color="Maroon">k</font></span>)</span>,<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="series_1.html#K5">Sum</a> <span class="p1">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> <font color="Maroon">k</font></span>)</span>,<font color="Maroon">n</font>,<font color="Maroon">n2</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/><a NAME="E20:36_2"/><i><font color="Green">E87</font></i>: 
for <font color="Olive">N0</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">N0</font> &amp; <font color="Olive">N0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> holds <br/><font color="Maroon">q</font> <a href="seq_1.html#K2">.</a> <font color="Olive">N0</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">N0</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_2_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">N0</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:36_2_5"/><i><font color="Green">E88</font></i>: 
( <font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">N0</font> &amp; <font color="Maroon">N0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> )
 ;<br/>

<a NAME="E2:36_2_5"/><b>then </b><i><font color="Green">E89</font></i>: 
<font color="Maroon">N0</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:36_2_5"/><b>then </b><i><font color="Green">E90</font></i>: 
<font color="Maroon">q</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">N0</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">k</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:36_2_5"/><i><font color="Green">E91</font></i>: 
<span class="p1">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">N0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N0</font> <a href="series_1.html#K3">to_power</a> <font color="Maroon">k</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:36_2_5"/>
<font color="Maroon">N0</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n</font> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E6:36_2_5"/>
<font color="Maroon">q</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">N0</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">N0</font>
 <b>by </b><i>, <a class="txt" href="asympt_1.html#E1"><i><font color="Green">Lemma47</font></i></a>, , </i>;<br/>


</div><b>end;</b></div><a NAME="E21:36_2"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n1</font>
 <b>by </b><i/>;<br/><a NAME="E22:36_2"/><b>then </b>
<font color="Maroon">n</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n1</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E23:36_2"/><b>then </b>
<font color="Maroon">n</font> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n1</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>
 ;<br/><a NAME="E24:36_2"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E25:36_2"/><b>then </b><i><font color="Green">E92</font></i>: 
 <a href="series_1.html#K5">Sum</a> <span class="p1">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> <font color="Maroon">k</font></span>)</span>,<font color="Maroon">n</font>,<font color="Maroon">n2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="series_1.html#K5">Sum</a> <font color="Maroon">q</font>,<font color="Maroon">n</font>,<font color="Maroon">n2</font>
 <b>by </b><i>, </i>;<br/><a NAME="E26:36_2"/><i><font color="Green">E93</font></i>: 
 <a href="series_1.html#K5">Sum</a> <font color="Maroon">q</font>,<font color="Maroon">n</font>,<font color="Maroon">n2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K5">-</a> <font color="Maroon">n2</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i>, </i>;<br/><div><i><font color="Green">E94</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="36_2_6"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:36_2_6"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">n</font> <a href="real_1.html#K5">-</a> <font color="Maroon">n2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font> <a href="real_1.html#K6">/</a> 2
 ;<br/><a NAME="E2:36_2_6"/>
<span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="int_1.html#K2">\]</a></span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K3">+</a> 1
 <b>by </b><i><a class="ref" href="int_1.html#D5">INT_1:def 5</a></i>;<br/><a NAME="E3:36_2_6"/><b>then </b>
<font color="Maroon">n2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font> <a href="real_1.html#K6">/</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E4:36_2_6"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">n2</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K6">/</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><b>hence </b><a NAME="E5:36_2_6"/>
contradiction
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/></div><b>end;</b></div><a NAME="E28:36_2"/>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, <a class="ref" href="power.html#T39">POWER:39</a></i>;<br/><a NAME="E29:36_2"/><b>then </b>
 <a href="series_1.html#K5">Sum</a> <font color="Maroon">q</font>,<font color="Maroon">n</font>,<font color="Maroon">n2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="asympt_1.html#E1"><i><font color="Green">Lemma47</font></i></a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E30:36_2"/><b>then </b>
 <a href="series_1.html#K5">Sum</a> <font color="Maroon">q</font>,<font color="Maroon">n</font>,<font color="Maroon">n2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="power.html#K4">to_power</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="power.html#T30">POWER:30</a></i>;<br/><a NAME="E31:36_2"/><b>then </b>
 <a href="series_1.html#K5">Sum</a> <font color="Maroon">q</font>,<font color="Maroon">n</font>,<font color="Maroon">n2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="power.html#K4">to_power</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="power.html#T32">POWER:32</a></i>;<br/><a NAME="E32:36_2"/><b>then </b>
 <a href="series_1.html#K5">Sum</a> <font color="Maroon">q</font>,<font color="Maroon">n</font>,<font color="Maroon">n2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="series_1.html#K3">to_power</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#K6">/</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="power.html#T36">POWER:36</a></i>;<br/><a NAME="E33:36_2"/><b>then </b>
 <a href="series_1.html#K5">Sum</a> <font color="Maroon">q</font>,<font color="Maroon">n</font>,<font color="Maroon">n2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="series_1.html#K3">to_power</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#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p3">(<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#K2">"</a> </span>)</span>
 <b>by </b><i><a class="ref" href="xcmplx_0.html#D9">XCMPLX_0:def 9</a></i>;<br/><a NAME="E34:36_2"/><b>then </b>
 <a href="series_1.html#K5">Sum</a> <span class="p1">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> <font color="Maroon">k</font></span>)</span>,<font color="Maroon">n</font>,<font color="Maroon">n2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="series_1.html#K3">to_power</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#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p3">(<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#K2">"</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E35:36_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p3">(<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#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, , , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>thus </b><a NAME="E36:36_2"/>
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<a NAME="E8:36"/><b>then </b>
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="asympt_0.html#K10">Big_Omega</a> <span class="p1">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E9:36"/>
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="asympt_0.html#K11">Big_Theta</a> <span class="p1">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div>
