<?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>, <font color="Maroon">g</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:2"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> 0 &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 holds <br/><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">12 <a href="nat_1.html#K2">*</a> <span class="p5">(<span class="default"><font color="Olive">n</font> <a href="series_1.html#K3">to_power</a> 3</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Olive">n</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">5 <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Olive">n</font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Olive">n</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 36 ) )
 <b>and </b><br/><a NAME="E2:2"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> 0 &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 holds <br/><font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="series_1.html#K3">to_power</a> 3</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Olive">n</font></span>)</span> ) )
 ;<br/>

<a NAME="E3:2"/>
<font color="Maroon">f</font> is <a href="asympt_0.html#V4">eventually-positive</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1"><b>proof </b></a><div class="add">

<b>take </b>
3
; <i><font color="Red">:: according to </font></i><a class="ref" href="asympt_0.html#D6">ASYMPT_0:def 6</a><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>assume </b><a NAME="E1:2_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 3
 ;<br/>

<a NAME="E2:2_1"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E3:2_1"/><b>then </b><i><font color="Green">E52</font></i>: 
<font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> 3 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> 2
 
<b>by </b><i><a class="ref" href="power.html#T44">POWER:44</a></i>;<br/>
<a NAME="E4:2_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> 2 <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="E5:2_1"/><i><font color="Green">E60</font></i>: 
 <a href="power.html#K6">log</a> 2,<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="power.html#K6">log</a> 2,3
 
<b>by </b><i>, <a class="ref" href="pre_ff.html#T12">PRE_FF:12</a></i>;<br/>
<a NAME="E6:2_1"/>
 <a href="power.html#K6">log</a> 2,3 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="power.html#K6">log</a> 2,2
 
<b>by </b><i><a class="ref" href="power.html#T65">POWER:65</a></i>;<br/>
<a NAME="E7:2_1"/><b>then </b>
 <a href="power.html#K6">log</a> 2,3 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 
<b>by </b><i><a class="ref" href="power.html#T60">POWER:60</a></i>;<br/>
<a NAME="E8:2_1"/><b>then </b>
 <a href="power.html#K6">log</a> 2,<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E9:2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> 3</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> 2</span>)</span> <a href="nat_1.html#K2">*</a> 1
 
<b>by </b><i>, , <a class="ref" href="real_2.html#T199">REAL_2:199</a></i>;<br/>
<a NAME="E10:2_1"/><b>then </b>
12 <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="series_1.html#K3">to_power</a> 3</span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 5 <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> 2</span>)</span>
 
<b>by </b><i>, <a class="ref" href="real_2.html#T199">REAL_2:199</a></i>;<br/>
<a NAME="E11:2_1"/><b>then </b>
<span class="p1">(<span class="default">12 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> 3</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default">5 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 0
 
<b>by </b><i><a class="ref" href="power.html#T53">POWER:53</a></i>;<br/>
<a NAME="E12:2_1"/><b>then </b><i><font color="Green">E61</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">12 <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> 3</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">5 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/>
<a NAME="E13:2_1"/>
<span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">n</font></span>)</span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T65">XREAL_1:65</a></i>;<br/>
<a NAME="E14:2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">12 <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> 3</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">5 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="square_1.html#K5">^2</a> </span>)</span></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="power.html#K6">log</a> 2,<font color="Maroon">n</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 <a href="nat_1.html#K1">+</a> 0
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E15:2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">12 <a href="nat_1.html#K2">*</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> 3</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">5 <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">n</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 36 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 <a href="nat_1.html#K1">+</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<b>hence </b><a NAME="E16:2_1"/>
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as  <a href="asympt_0.html#V4">eventually-positive</a> <a href="seq_1.html#NM1">Real_Sequence</a> ;<br/>
<a NAME="E5:2"/>
<font color="Maroon">g</font> is <a href="asympt_0.html#V4">eventually-positive</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2"><b>proof </b></a><div class="add">

<b>take </b>
2
; <i><font color="Red">:: according to </font></i><a class="ref" href="asympt_0.html#D6">ASYMPT_0:def 6</a><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>assume </b><a NAME="E1:2_2"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2
 ;<br/>

<a NAME="E2:2_2"/><b>then </b>
 <a href="power.html#K6">log</a> 2,<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="power.html#K6">log</a> 2,2
 
<b>by </b><i><a class="ref" href="pre_ff.html#T12">PRE_FF:12</a></i>;<br/>
<a NAME="E3:2_2"/><b>then </b><i><font color="Green">E63</font></i>: 
 <a href="power.html#K6">log</a> 2,<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="power.html#T60">POWER:60</a></i>;<br/>
<a NAME="E4:2_2"/>
<font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> 3 <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="E5:2_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> 3</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> 3</span>)</span> <a href="nat_1.html#K2">*</a> 0
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<b>hence </b><a NAME="E6:2_2"/>
<font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">g</font> = <font color="Maroon">g</font> as  <a href="asympt_0.html#V4">eventually-positive</a> <a href="seq_1.html#NM1">Real_Sequence</a> ;<br/>
<b>take </b>
<font color="Maroon">f</font>
;<br/>

<b>take </b>
<font color="Maroon">g</font>
;<br/>

<a NAME="E7:2"/>
 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">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</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">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">12 <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><font color="Olive">n</font> <a href="series_1.html#K3">to_power</a> 3</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Olive">n</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">5 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_3"><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"><span class="p2">(<span class="default">12 <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="series_1.html#K3">to_power</a> 3</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">5 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> ) );<br/>
<a NAME="E1:2_3"/><i><font color="Green">E64</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="2_3_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/>

<a NAME="E1:2_3_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 )
 
;<br/>
<b>hence </b><a NAME="E2:2_3_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>
<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="E3:2_3"/><i><font color="Green">E67</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="E4:2_3"/>
<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="E5:2_3"/>
( <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"><span class="p2">(<span class="default">12 <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> 3</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">5 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">p</font> being   <a href="seq_1.html#NM1">Real_Sequence</a><b> such that </b><br/><a NAME="E9:2"/><i><font color="Green">E69</font></i>: 
( <font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> 0 &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 holds <br/><font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">12 <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><font color="Olive">n</font> <a href="series_1.html#K3">to_power</a> 3</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Olive">n</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">5 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> ) )
 ;<br/>
<a NAME="E10:2"/>
<font color="Maroon">p</font> is <a href="asympt_0.html#V4">eventually-positive</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_4"><b>proof </b></a><div class="add">

<b>take </b>
3
; <i><font color="Red">:: according to </font></i><a class="ref" href="asympt_0.html#D6">ASYMPT_0:def 6</a><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>assume </b><a NAME="E1:2_4"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 3
 ;<br/>

<a NAME="E2:2_4"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E3:2_4"/><b>then </b><i><font color="Green">E71</font></i>: 
<font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> 3 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> 2
 
<b>by </b><i><a class="ref" href="power.html#T44">POWER:44</a></i>;<br/>
<a NAME="E4:2_4"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> 2 <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="E5:2_4"/><i><font color="Green">E73</font></i>: 
 <a href="power.html#K6">log</a> 2,<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="power.html#K6">log</a> 2,3
 
<b>by </b><i>, <a class="ref" href="pre_ff.html#T12">PRE_FF:12</a></i>;<br/>
<a NAME="E6:2_4"/>
 <a href="power.html#K6">log</a> 2,3 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="power.html#K6">log</a> 2,2
 
<b>by </b><i><a class="ref" href="power.html#T65">POWER:65</a></i>;<br/>
<a NAME="E7:2_4"/><b>then </b>
 <a href="power.html#K6">log</a> 2,3 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 
<b>by </b><i><a class="ref" href="power.html#T60">POWER:60</a></i>;<br/>
<a NAME="E8:2_4"/><b>then </b>
 <a href="power.html#K6">log</a> 2,<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E9:2_4"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> 3</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> 2</span>)</span> <a href="nat_1.html#K2">*</a> 1
 
<b>by </b><i>, , <a class="ref" href="real_2.html#T199">REAL_2:199</a></i>;<br/>
<a NAME="E10:2_4"/><b>then </b>
12 <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="series_1.html#K3">to_power</a> 3</span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 5 <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> 2</span>)</span>
 
<b>by </b><i>, <a class="ref" href="real_2.html#T199">REAL_2:199</a></i>;<br/>
<a NAME="E11:2_4"/><b>then </b>
<span class="p1">(<span class="default">12 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> 3</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default">5 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 0
 
<b>by </b><i><a class="ref" href="power.html#T53">POWER:53</a></i>;<br/>
<a NAME="E12:2_4"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">12 <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> 3</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">5 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/>
<b>hence </b><a NAME="E13:2_4"/>
<font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">p</font> = <font color="Maroon">p</font> as  <a href="asympt_0.html#V4">eventually-positive</a> <a href="seq_1.html#NM1">Real_Sequence</a> ;<br/>
<a NAME="E12:2"/>
 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">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</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">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Olive">n</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> 36 ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_5"><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"><span class="p2">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> 36 ) );<br/>
<a NAME="E1:2_5"/><i><font color="Green">E74</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="2_5_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/>

<a NAME="E1:2_5_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 )
 
;<br/>
<b>hence </b><a NAME="E2:2_5_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>
<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="E3:2_5"/><i><font color="Green">E75</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="E4:2_5"/>
( <font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> 0 &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 holds <br/><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Olive">n</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> 36 ) )
 <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="E14:2"/><i><font color="Green">E77</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">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</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">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Olive">n</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> 36 ) )
 ;<br/>
<a NAME="E15:2"/>
<font color="Maroon">q</font> is <a href="asympt_0.html#V4">eventually-positive</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_6"><b>proof </b></a><div class="add">

<b>take </b>
1
; <i><font color="Red">:: according to </font></i><a class="ref" href="asympt_0.html#D6">ASYMPT_0:def 6</a><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>assume </b><a NAME="E1:2_6"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 ;<br/>

<a NAME="E2:2_6"/>
<span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">n</font></span>)</span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T65">XREAL_1:65</a></i>;<br/>
<a NAME="E3:2_6"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">n</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> 36 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 <a href="nat_1.html#K1">+</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<b>hence </b><a NAME="E4:2_6"/>
<font color="Maroon">q</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">q</font> = <font color="Maroon">q</font> as  <a href="asympt_0.html#V4">eventually-positive</a> <a href="seq_1.html#NM1">Real_Sequence</a> ;<br/>
<b>set </b><font color="Maroon">t</font> =  <a href="asympt_0.html#K8">max</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>;<br/>
<a NAME="E17:2"/>
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> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_7"><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/>

<b>thus </b><a NAME="E1:2_7"/>
<font color="Maroon">f</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">p</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_7_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_7_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_7_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="2_7_1_1_1"><b>suppose </b></a><a NAME="E1:2_7_1_1_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:2_7_1_1_1"/>
<font color="Maroon">f</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">p</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_7_1_1_2"><b>suppose </b></a><a NAME="E1:2_7_1_1_2"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:2_7_1_1_2"/><b>then </b>
<font color="Maroon">p</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"><span class="p2">(<span class="default">12 <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> 3</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">5 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
 <b>by </b><i/>;<br/><b>then </b><span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">12 <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> 3</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">5 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">n</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> 36</span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">12 <a href="nat_1.html#K2">*</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> 3</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">5 <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">n</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 36

;<br/><b>hence </b><a NAME="E4:2_7_1_1_2"/>
<font color="Maroon">f</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">p</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>


</div><b>end;</b></div>
<b>then </b><i><font color="Green">E80</font></i>:  <a href="asympt_0.html#K9">Big_Oh</a> <font color="Maroon">f</font> = 
 <a href="asympt_0.html#K9">Big_Oh</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="asympt_0.html#K5">+</a> <font color="Maroon">q</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T11">SEQ_1:11</a></i>
<br/>.= 
 <a href="asympt_0.html#K9">Big_Oh</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K8">max</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span>
<b>by </b><i><a class="ref" href="asympt_0.html#T9">ASYMPT_0:9</a></i>
;<br/>
4 = 
2 <a href="square_1.html#K5">^2</a> 

<br/>.= 
2 <a href="series_1.html#K3">to_power</a> 2
<b>by </b><i><a class="ref" href="power.html#T53">POWER:53</a></i>
;<br/>
<b>then </b><i><font color="Green">E81</font></i>:  <a href="power.html#K6">log</a> 2,4 = 
2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,2</span>)</span>
<b>by </b><i><a class="ref" href="power.html#T63">POWER:63</a></i>
<br/>.= 
2 <a href="nat_1.html#K2">*</a> 1
<b>by </b><i><a class="ref" href="power.html#T60">POWER:60</a></i>
<br/>.= 
2

;<br/>
<a NAME="E21:2"/><i><font color="Green">E82</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>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 4 holds <br/>7 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">q</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_11"><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> ] <b>means </b>7 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">q</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E1:2_11"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[4]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_11_1"><b>proof </b></a><div class="add">

<font color="Maroon">q</font> <a href="seq_1.html#K2">.</a> 4 = 
<span class="p1">(<span class="default">2 <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> 36
<b>by </b><i>, </i>
<br/>.= 
40

;<br/>
<b>hence </b><a NAME="E2:2_11_1"/>
<font color="Maroon">S<sub>1</sub></font>[4]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:2_11"/><i><font color="Green">E84</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="Olive">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 4 &amp; <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="2_11_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><b>that </b><br/><a NAME="E1:2_11_2"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 4
 <b>and </b><br/><a NAME="E2:2_11_2"/><i><font color="Green">E86</font></i>: 
7 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">q</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font>
 ;<br/>

<a NAME="E3:2_11_2"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">q</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">k</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> 36
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:2_11_2"/>
7 <a href="real_1.html#K4">*</a> <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="square_1.html#K5">^2</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">7 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">7 <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default">7 <a href="nat_1.html#K2">*</a> 1</span>)</span></span>)</span>
 
;<br/>
<a NAME="E5:2_11_2"/><b>then </b><i><font color="Green">E88</font></i>: 
7 <a href="real_1.html#K4">*</a> <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="square_1.html#K5">^2</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">7 <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default">7 <a href="nat_1.html#K2">*</a> 1</span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E6:2_11_2"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">q</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="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="power.html#K6">log</a> 2,<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="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> 36
 
<b>by </b><i/>;<br/>
<a NAME="E7:2_11_2"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E8:2_11_2"/><b>then </b>
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E9:2_11_2"/><b>then </b><i><font color="Green">E90</font></i>: 
 <a href="power.html#K6">log</a> 2,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="power.html#K6">log</a> 2,<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="pre_ff.html#T12">PRE_FF:12</a></i>;<br/>
<a NAME="E10:2_11_2"/>
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 4 <a href="nat_1.html#K1">+</a> 0
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E11:2_11_2"/><b>then </b>
 <a href="power.html#K6">log</a> 2,<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#NR2" target="_self">&gt;=</a> 2
 
<b>by </b><i>, <a class="ref" href="pre_ff.html#T12">PRE_FF:12</a></i>;<br/>
<a NAME="E12:2_11_2"/><b>then </b>
<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> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</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="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> 
 
<b>by </b><i>, <a class="ref" href="square_1.html#T77">SQUARE_1:77</a></i>;<br/>
<a NAME="E13:2_11_2"/><b>then </b><i><font color="Green">E91</font></i>: 
<font color="Maroon">q</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="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="power.html#K6">log</a> 2,<span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> 36
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E14:2_11_2"/>
 <a href="power.html#K6">log</a> 2,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="power.html#K6">log</a> 2,<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span>
 
;<br/>
<a NAME="E15:2_11_2"/><b>then </b>
 <a href="power.html#K6">log</a> 2,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">k</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,2</span>)</span>
 
<b>by </b><i>, <a class="ref" href="power.html#T61">POWER:61</a></i>;<br/>
<b>then </b><i><font color="Green">E92</font></i>: <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> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">k</font></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="square_1.html#K5">^2</a> 
<b>by </b><i><a class="ref" href="power.html#T60">POWER:60</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">k</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1

;<br/>
<a NAME="E17:2_11_2"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E18:2_11_2"/><b>then </b><i><font color="Green">E93</font></i>: 
2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i/>;<br/>
<a NAME="E19:2_11_2"/><i><font color="Green">E94</font></i>: 
 <a href="power.html#K6">log</a> 2,<font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2
 
<b>by </b><i>, , <a class="ref" href="pre_ff.html#T12">PRE_FF:12</a></i>;<br/>
<a NAME="E20:2_11_2"/>
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E21:2_11_2"/><b>then </b>
2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">k</font>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E22:2_11_2"/><b>then </b>
 <a href="power.html#K6">log</a> 2,<span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">k</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="power.html#K6">log</a> 2,<font color="Maroon">k</font>
 
<b>by </b><i>, <a class="ref" href="power.html#T65">POWER:65</a></i>;<br/>
<a NAME="E23:2_11_2"/><b>then </b>
<font color="Maroon">k</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,2</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="power.html#K6">log</a> 2,<font color="Maroon">k</font>
 
<b>by </b><i><a class="ref" href="power.html#T63">POWER:63</a></i>;<br/>
<a NAME="E24:2_11_2"/><b>then </b>
<font color="Maroon">k</font> <a href="nat_1.html#K2">*</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="power.html#K6">log</a> 2,<font color="Maroon">k</font>
 
<b>by </b><i><a class="ref" href="power.html#T60">POWER:60</a></i>;<br/>
<a NAME="E25:2_11_2"/><b>then </b>
14 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">k</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="real_2.html#T199">REAL_2:199</a></i>;<br/>
<a NAME="E26:2_11_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">7 <a href="nat_1.html#K2">*</a> 2</span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 7 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E27:2_11_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">k</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">k</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">7 <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 7</span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E28:2_11_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="power.html#K6">log</a> 2,<span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> 36 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">k</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">7 <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 7</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 36
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E29:2_11_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">7 <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default">7 <a href="nat_1.html#K2">*</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">q</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>
 
<b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E30:2_11_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:2_11"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 4 holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S9">NAT_1:sch 9</a>(, );<br/></i>
<b>hence </b><a NAME="E4:2_11"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 4 holds <br/>7 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">q</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E22:2"/><i><font color="Green">E95</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>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 4 holds <br/><font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 7 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="square_1.html#K5">^2</a> </span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_12"><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/>

<b>assume </b><a NAME="E1:2_12"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 4
 ;<br/>

<a NAME="E2:2_12"/><b>then </b><i><font color="Green">E97</font></i>: 
<font color="Maroon">p</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"><span class="p2">(<span class="default">12 <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> 3</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">5 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E3:2_12"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E4:2_12"/><b>then </b><i><font color="Green">E98</font></i>: 
<font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> 3 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> 2
 
<b>by </b><i><a class="ref" href="power.html#T44">POWER:44</a></i>;<br/>
<a NAME="E5:2_12"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> 2 <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="E6:2_12"/>
 <a href="power.html#K6">log</a> 2,<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="power.html#K6">log</a> 2,4
 
<b>by </b><i>, <a class="ref" href="pre_ff.html#T12">PRE_FF:12</a></i>;<br/>
<a NAME="E7:2_12"/><b>then </b>
 <a href="power.html#K6">log</a> 2,<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E8:2_12"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> 3</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> 2</span>)</span> <a href="nat_1.html#K2">*</a> 1
 
<b>by </b><i>, , <a class="ref" href="real_2.html#T199">REAL_2:199</a></i>;<br/>
<a NAME="E9:2_12"/><b>then </b>
12 <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="series_1.html#K3">to_power</a> 3</span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 12 <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> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<a NAME="E10:2_12"/><b>then </b>
<span class="p1">(<span class="default">12 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> 3</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 12 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="square_1.html#K5">^2</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="power.html#T53">POWER:53</a></i>;<br/>
<a NAME="E11:2_12"/><b>then </b>
<font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default">12 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">5 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<b>hence </b><a NAME="E12:2_12"/>
<font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 7 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="square_1.html#K5">^2</a> </span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E23:2"/><i><font color="Green">E100</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>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 4 holds <br/><font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">q</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_13"><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/>

<b>assume </b><a NAME="E1:2_13"/><i><font color="Green">E101</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 4
 ;<br/>

<a NAME="E2:2_13"/><b>then </b><i><font color="Green">E102</font></i>: 
<font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 7 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="square_1.html#K5">^2</a> </span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E3:2_13"/>
7 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">q</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E4:2_13"/>
<font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">q</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/>


</div><b>end;</b></div>
<a NAME="E24:2"/><i><font color="Green">E103</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>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 4 holds <br/><span class="p1">(<span class="default"><a href="asympt_0.html#K8">max</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_14"><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/>

<b>assume </b><a NAME="E1:2_14"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 4
 ;<br/>

<a NAME="E2:2_14"/><b>then </b><i><font color="Green">E104</font></i>: 
<font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">q</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 
<b>by </b><i/>;<br/>
<b>thus </b><span class="p1">(<span class="default"><a href="asympt_0.html#K8">max</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> = 
 <a href="asympt_0.html#K2">max</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="asympt_0.html#D10">ASYMPT_0:def 10</a></i>
<br/>.= 
<font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
<b>by </b><i>, <a class="ref" href="xxreal_0.html#D9">XXREAL_0:def 9</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E25:2"/><i><font color="Green">E105</font></i>: 
 <a href="asympt_0.html#K8">max</a> <font color="Maroon">p</font>,<font color="Maroon">q</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/>
<b>consider </b><font color="Maroon">N</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E27:2"/><i><font color="Green">E106</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>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">N</font> holds <br/><span class="p1">(<span class="default"><a href="asympt_0.html#K8">max</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="asympt_0.html#D6">ASYMPT_0:def 6</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_15"><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="E1:2_15"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="asympt_0.html#K4">max</a> 4,<font color="Maroon">N</font>
 ;<br/><a NAME="E2:2_15"/><i><font color="Green">E108</font></i>: 
(  <a href="asympt_0.html#K4">max</a> 4,<font color="Maroon">N</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 4 &amp;  <a href="asympt_0.html#K4">max</a> 4,<font color="Maroon">N</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">N</font> )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/><a NAME="E3:2_15"/><b>then </b><i><font color="Green">E109</font></i>: 
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 4 &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</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>then </b><i><font color="Green">E110</font></i>: <span class="p1">(<span class="default"><a href="asympt_0.html#K8">max</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> = 
<font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">12 <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> 3</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">5 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
<b>by </b><i>, , </i>
;<br/><a NAME="E5:2_15"/>
<font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> 2 <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="E6:2_15"/><b>then </b>
<font color="Maroon">n</font> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="power.html#T53">POWER:53</a></i>;<br/><a NAME="E7:2_15"/><b>then </b>
5 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 5 <a href="nat_1.html#K2">*</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E8:2_15"/><b>then </b>
<span class="p1">(<span class="default"><a href="asympt_0.html#K8">max</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">12 <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> 3</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 0
 <b>by </b><i>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E9:2_15"/><b>then </b>
<span class="p1">(<span class="default"><a href="asympt_0.html#K8">max</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 12 <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="series_1.html#K3">to_power</a> 3</span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">n</font></span>)</span></span>)</span>
 ;<br/><b>hence </b><a NAME="E10:2_15"/>
<span class="p1">(<span class="default"><a href="asympt_0.html#K8">max</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 12 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, , </i>;<br/><b>thus </b><a NAME="E11:2_15"/>
<span class="p1">(<span class="default"><a href="asympt_0.html#K8">max</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <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="E29:2"/><b>then </b><i><font color="Green">E111</font></i>: 
 <a href="asympt_0.html#K8">max</a> <font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="asympt_0.html#K9">Big_Oh</a> <font color="Maroon">g</font>
 
<b>by </b><i/>;<br/>
<a NAME="E30:2"/>
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="asympt_0.html#K9">Big_Oh</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="asympt_0.html#T10">ASYMPT_0:10</a></i>;<br/>
<b>hence </b><a NAME="E31:2"/>
( <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> &amp; <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> &amp; <font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="asympt_0.html#K9">Big_Oh</a> <font color="Maroon">g</font> )
 <b>by </b><i>, , <a class="ref" href="asympt_0.html#T12">ASYMPT_0:12</a></i>;<br/>


</div>
