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

<b>let </b><font color="Maroon">F</font> be  <a href="asympt_0.html#V2">eventually-nonnegative</a> <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>

<b>assume </b><a NAME="E1:117"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> 1</span>)</span> <a href="seq_1.html#K10">-</a> <span class="p1">(<span class="default"><a href="asympt_1.html#K4" target="_self">seq_const</a> 1</span>)</span>
 ;<br/>

<b>set </b><font color="Maroon">p</font> =  <a href="asympt_1.html#K3" target="_self">seq_n^</a> 1;<br/>
<b>set </b><font color="Maroon">q</font> =  <a href="asympt_1.html#K4" target="_self">seq_const</a> 1;<br/>
<b>set </b><font color="Maroon">f</font> = <span class="p1">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> 1</span>)</span> <a href="seq_1.html#K10">-</a> <span class="p1">(<span class="default"><a href="asympt_1.html#K4" target="_self">seq_const</a> 1</span>)</span>;<br/>
<b>set </b><font color="Maroon">g</font> =  <a href="asympt_1.html#K3" target="_self">seq_n^</a> 1;<br/>
<a NAME="E2:117"/><i><font color="Green">E50</font></i>: 
 <a href="asympt_0.html#K11">Big_Theta</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">t</font> where <font color="Olive">t</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>  :  ex <font color="Olive">c</font>, <font color="Olive">d</font> being  <a href="real_1.html#NM1">Real</a> ex <font color="Olive">N</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">c</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Olive">d</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</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#NR2" target="_self">&gt;=</a> <font color="Olive">N</font> holds <br/>( <font color="Olive">d</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> 1</span>)</span> <a href="seq_1.html#K10">-</a> <span class="p3">(<span class="default"><a href="asympt_1.html#K4" target="_self">seq_const</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">t</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> &amp; <font color="Olive">t</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">c</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> 1</span>)</span> <a href="seq_1.html#K10">-</a> <span class="p3">(<span class="default"><a href="asympt_1.html#K4" target="_self">seq_const</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> ) ) ) </span>}</span> 
 
<b>by </b><i>, <a class="ref" href="asympt_0.html#T27">ASYMPT_0:27</a></i>;<br/>
<a NAME="E3:117"/><i><font color="Green">E51</font></i>: 
 <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> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">t</font> where <font color="Olive">t</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>  :  ex <font color="Olive">c</font>, <font color="Olive">d</font> being  <a href="real_1.html#NM1">Real</a> ex <font color="Olive">N</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">c</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Olive">d</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</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#NR2" target="_self">&gt;=</a> <font color="Olive">N</font> holds <br/>( <font color="Olive">d</font> <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> 1</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">t</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> &amp; <font color="Olive">t</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">c</font> <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> 1</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> ) ) ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="asympt_0.html#T27">ASYMPT_0:27</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="117_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:117_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K11">Big_Theta</a> <font color="Maroon">F</font></span>)</span> <a href="asympt_0.html#K16">+</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K11">Big_Theta</a> <span class="p2">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> 1</span>)</span></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">t</font> being    <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> such that </b><br/><a NAME="E3:117_1_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E4:117_1_1"/><i><font color="Green">E59</font></i>: 
 ex <font color="Olive">f'</font>, <font color="Olive">g'</font> being   <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>  st <br/>( <font color="Olive">f'</font> <a href="hidden.html#R2">in</a>  <a href="asympt_0.html#K11">Big_Theta</a> <font color="Maroon">F</font> &amp; <font color="Olive">g'</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> 1</span>)</span> &amp; ( 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">t</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">f'</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="Olive">g'</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> ) )
 ;<br/><b>consider </b><font color="Maroon">f'</font> being    <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> , <font color="Maroon">g'</font> being    <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> such that </b><br/><a NAME="E6:117_1_1"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">f'</font> <a href="hidden.html#R2">in</a>  <a href="asympt_0.html#K11">Big_Theta</a> <font color="Maroon">F</font> &amp; <font color="Maroon">g'</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> 1</span>)</span> )
 <b>and </b><br/><a NAME="E7:117_1_1"/><i><font color="Green">E61</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">t</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">f'</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">g'</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span>
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">r</font> being    <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> such that </b><br/><a NAME="E9:117_1_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f'</font>
 <b>and </b><br/><a NAME="E10:117_1_1"/><i><font color="Green">E63</font></i>: 
 ex <font color="Olive">c</font>, <font color="Olive">d</font> being  <a href="real_1.html#NM1">Real</a> ex <font color="Olive">N</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">c</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Olive">d</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</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#NR2" target="_self">&gt;=</a> <font color="Olive">N</font> holds <br/>( <font color="Olive">d</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> 1</span>)</span> <a href="seq_1.html#K10">-</a> <span class="p3">(<span class="default"><a href="asympt_1.html#K4" target="_self">seq_const</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> &amp; <font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">c</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> 1</span>)</span> <a href="seq_1.html#K10">-</a> <span class="p3">(<span class="default"><a href="asympt_1.html#K4" target="_self">seq_const</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> ) ) )
 <b>by </b><i><a class="txt" href="asympt_1.html#E21:117_1_1"><i><font color="Green">E74</font></i></a>, </i>;<br/><b>consider </b><font color="Maroon">s</font> being    <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> such that </b><br/><a NAME="E12:117_1_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g'</font>
 <b>and </b><br/><a NAME="E13:117_1_1"/><i><font color="Green">E67</font></i>: 
 ex <font color="Olive">c</font>, <font color="Olive">d</font> being  <a href="real_1.html#NM1">Real</a> ex <font color="Olive">N</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">c</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Olive">d</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</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#NR2" target="_self">&gt;=</a> <font color="Olive">N</font> holds <br/>( <font color="Olive">d</font> <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> 1</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> &amp; <font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">c</font> <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> 1</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> ) ) )
 <b>by </b><i><a class="txt" href="asympt_1.html#E22:117_1_1"><i><font color="Green">E75</font></i></a>, </i>;<br/><b>consider </b><font color="Maroon">c1</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">d1</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">N1</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="E15:117_1_1"/><i><font color="Green">E69</font></i>: 
( <font color="Maroon">c1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">d1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 <b>and </b><br/><a NAME="E16:117_1_1"/><i><font color="Green">E70</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">N1</font> holds <br/>( <font color="Maroon">d1</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> 1</span>)</span> <a href="seq_1.html#K10">-</a> <span class="p3">(<span class="default"><a href="asympt_1.html#K4" target="_self">seq_const</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">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="Olive">n</font> &amp; <font color="Maroon">f'</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c1</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> 1</span>)</span> <a href="seq_1.html#K10">-</a> <span class="p3">(<span class="default"><a href="asympt_1.html#K4" target="_self">seq_const</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> )
 <b>by </b><i>, </i>;<br/><b>consider </b><font color="Maroon">c2</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">d2</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">N2</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="E18:117_1_1"/><i><font color="Green">E71</font></i>: 
( <font color="Maroon">c2</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">d2</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 <b>and </b><br/><a NAME="E19:117_1_1"/><i><font color="Green">E72</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">N2</font> holds <br/>( <font color="Maroon">d2</font> <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> 1</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">g'</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> &amp; <font color="Maroon">g'</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c2</font> <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> 1</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> )
 <b>by </b><i>, </i>;<br/><b>set </b><font color="Maroon">N</font> =  <a href="asympt_0.html#K4">max</a> 1,<span class="p1">(<span class="default"><a href="asympt_0.html#K4">max</a> <font color="Maroon">N1</font>,<font color="Maroon">N2</font></span>)</span>;<br/><a NAME="E20:117_1_1"/><i><font color="Green">E73</font></i>: 
(  <a href="asympt_0.html#K4">max</a> 1,<span class="p1">(<span class="default"><a href="asympt_0.html#K4">max</a> <font color="Maroon">N1</font>,<font color="Maroon">N2</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp;  <a href="asympt_0.html#K4">max</a> 1,<span class="p1">(<span class="default"><a href="asympt_0.html#K4">max</a> <font color="Maroon">N1</font>,<font color="Maroon">N2</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="asympt_0.html#K4">max</a> <font color="Maroon">N1</font>,<font color="Maroon">N2</font> &amp;  <a href="asympt_0.html#K4">max</a> <font color="Maroon">N1</font>,<font color="Maroon">N2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">N1</font> &amp;  <a href="asympt_0.html#K4">max</a> <font color="Maroon">N1</font>,<font color="Maroon">N2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">N2</font> )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/><a NAME="E21:117_1_1"/><b>then </b><i><font color="Green">E74</font></i>: 
(  <a href="asympt_0.html#K4">max</a> 1,<span class="p1">(<span class="default"><a href="asympt_0.html#K4">max</a> <font color="Maroon">N1</font>,<font color="Maroon">N2</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp;  <a href="asympt_0.html#K4">max</a> 1,<span class="p1">(<span class="default"><a href="asympt_0.html#K4">max</a> <font color="Maroon">N1</font>,<font color="Maroon">N2</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">N1</font> &amp;  <a href="asympt_0.html#K4">max</a> 1,<span class="p1">(<span class="default"><a href="asympt_0.html#K4">max</a> <font color="Maroon">N1</font>,<font color="Maroon">N2</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">N2</font> )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>set </b><font color="Maroon">d</font> = <font color="Maroon">d2</font>;<br/><b>set </b><font color="Maroon">c</font> = <font color="Maroon">c1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">c2</font>;<br/><a NAME="E22:117_1_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">c1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">c2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="real_1.html#K3">+</a> <font color="Maroon">c2</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="117_1_1_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="E1:117_1_1_1"/><i><font color="Green">E77</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> 1,<span class="p1">(<span class="default"><a href="asympt_0.html#K4">max</a> <font color="Maroon">N1</font>,<font color="Maroon">N2</font></span>)</span>
 ;<br/><a NAME="E2:117_1_1_1"/><b>then </b><i><font color="Green">E78</font></i>: 
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">N1</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">N2</font> )
 <b>by </b><i><a class="txt" href="asympt_1.html#E9:117_1"><i><font color="Green">E89</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E3:117_1_1_1"/><b>then </b>
<font color="Maroon">d1</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> 1</span>)</span> <a href="seq_1.html#K10">-</a> <span class="p3">(<span class="default"><a href="asympt_1.html#K4" target="_self">seq_const</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="txt" href="asympt_1.html#E4:117_1"><i><font color="Green">E85</font></i></a></i>;<br/><a NAME="E4:117_1_1_1"/><b>then </b><i><font color="Green">E79</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">d1</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> 1</span>)</span> <a href="seq_1.html#K10">-</a> <span class="p4">(<span class="default"><a href="asympt_1.html#K4" target="_self">seq_const</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</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> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">f'</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">g'</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E5:117_1_1_1"/>
<font color="Maroon">d2</font> <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> 1</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">g'</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="asympt_1.html#E7:117_1"><i><font color="Green">E87</font></i></a>, </i>;<br/><a NAME="E6:117_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">d1</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> 1</span>)</span> <a href="seq_1.html#K10">-</a> <span class="p4">(<span class="default"><a href="asympt_1.html#K4" target="_self">seq_const</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">d2</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> 1</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">d1</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> 1</span>)</span> <a href="seq_1.html#K10">-</a> <span class="p4">(<span class="default"><a href="asympt_1.html#K4" target="_self">seq_const</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</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><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E7:117_1_1_1"/><b>then </b><i><font color="Green">E80</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">d1</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> 1</span>)</span> <a href="seq_1.html#K10">-</a> <span class="p4">(<span class="default"><a href="asympt_1.html#K4" target="_self">seq_const</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">d2</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> 1</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">f'</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">g'</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><i><font color="Green">E81</font></i>: <span class="p1">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> 1</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> = 
<font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> 1
<b>by </b><i><a class="txt" href="asympt_1.html#E8:117_1"><i><font color="Green">E88</font></i></a>, <a class="txt" href="asympt_1.html#E11:117_1"><i><font color="Green">E91</font></i></a>, </i>
<br/>.= 
<font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="power.html#T30">POWER:30</a></i>
;<br/><a NAME="E9:117_1_1_1"/><i><font color="Green">E82</font></i>: 
<span class="p1">(<span class="default"><a href="asympt_1.html#K4" target="_self">seq_const</a> 1</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/><i><font color="Green">E83</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> 1</span>)</span> <a href="seq_1.html#K10">-</a> <span class="p2">(<span class="default"><a href="asympt_1.html#K4" target="_self">seq_const</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> 1</span>)</span> <a href="seq_1.html#K9">+</a> <span class="p2">(<span class="default"><a href="seq_1.html#K17">-</a> <span class="p3">(<span class="default"><a href="asympt_1.html#K4" target="_self">seq_const</a> 1</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="seq_1.html#T15">SEQ_1:15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> 1</span>)</span> <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"><span class="p2">(<span class="default"><a href="seq_1.html#K17">-</a> <span class="p3">(<span class="default"><a href="asympt_1.html#K4" target="_self">seq_const</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T11">SEQ_1:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> 1</span>)</span> <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"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="asympt_1.html#K4" target="_self">seq_const</a> 1</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T14">SEQ_1:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> 1</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="asympt_1.html#K4" target="_self">seq_const</a> 1</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="asympt_1.html#E8:117_1"><i><font color="Green">E88</font></i></a>, <a class="txt" href="asympt_1.html#E11:117_1"><i><font color="Green">E91</font></i></a>, </i>
<br/>.= 
<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>
<b>by </b><i>, <a class="ref" href="power.html#T30">POWER:30</a></i>
;<br/><a NAME="E11:117_1_1_1"/>
 <a href="real_1.html#K1">-</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> 1
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><a NAME="E12:117_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">d1</font> <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#K4">*</a> <font color="Maroon">d1</font>
 <b>by </b><i><a class="txt" href="asympt_1.html#E18:117_1_1_1"><i><font color="Green">E84</font></i></a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E13:117_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">d1</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">d2</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">d1</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">d2</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E14:117_1_1_1"/><b>then </b>
<font color="Maroon">d2</font> <a href="real_1.html#K4">*</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">f'</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">g'</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></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="E15:117_1_1_1"/>
<font color="Maroon">d2</font> <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> 1</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">t</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="asympt_1.html#E4:117_1_1_1"><i><font color="Green">E79</font></i></a>, </i>;<br/><a NAME="E16:117_1_1_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> <font color="Maroon">c1</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> 1</span>)</span> <a href="seq_1.html#K10">-</a> <span class="p3">(<span class="default"><a href="asympt_1.html#K4" target="_self">seq_const</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">g'</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c2</font> <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> 1</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> )
 <b>by </b><i><a class="txt" href="asympt_1.html#E4:117_1"><i><font color="Green">E85</font></i></a>, <a class="txt" href="asympt_1.html#E7:117_1"><i><font color="Green">E87</font></i></a>, </i>;<br/><a NAME="E17:117_1_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">f'</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">g'</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c1</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> 1</span>)</span> <a href="seq_1.html#K10">-</a> <span class="p4">(<span class="default"><a href="asympt_1.html#K4" target="_self">seq_const</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</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> &amp; <span class="p1">(<span class="default"><font color="Maroon">c1</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> 1</span>)</span> <a href="seq_1.html#K10">-</a> <span class="p4">(<span class="default"><a href="asympt_1.html#K4" target="_self">seq_const</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</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> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c1</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> 1</span>)</span> <a href="seq_1.html#K10">-</a> <span class="p4">(<span class="default"><a href="asympt_1.html#K4" target="_self">seq_const</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c2</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> 1</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E18:117_1_1_1"/><b>then </b><i><font color="Green">E84</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f'</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">g'</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c1</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> 1</span>)</span> <a href="seq_1.html#K10">-</a> <span class="p4">(<span class="default"><a href="asympt_1.html#K4" target="_self">seq_const</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c2</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> 1</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E19:117_1_1_1"/>
 <a href="real_1.html#K1">-</a> <font color="Maroon">c1</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="real_1.html#K1">-</a> 0
 <b>by </b><i><a class="txt" href="asympt_1.html#E18:117_1_1_1"><i><font color="Green">E84</font></i></a>, <a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><a NAME="E20:117_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">c1</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">c2</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">c2</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E21:117_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f'</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">g'</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">c2</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">n</font>
 <b>by </b><i>, , <a class="txt" href="asympt_1.html#E14:117_1"><i><font color="Green">E98</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E22:117_1_1_1"/>
<font color="Maroon">t</font> <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"><font color="Maroon">c1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">c2</font></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> 1</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E4:117_1_1_1"><i><font color="Green">E79</font></i></a>, </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E24:117_1_1"/>
<font color="Maroon">x</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> 1</span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E22:117_1_1"><i><font color="Green">E75</font></i></a>, , <a class="txt" href="asympt_1.html#E10:117_1"><i><font color="Green">E90</font></i></a>, <a class="txt" href="asympt_1.html#E5:117_1"><i><font color="Green">E86</font></i></a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:117_1"/>
<font color="Maroon">x</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> 1</span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">t</font> being    <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> such that </b><br/><a NAME="E4:117_1"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E5:117_1"/><i><font color="Green">E86</font></i>: 
 ex <font color="Olive">c</font>, <font color="Olive">d</font> being  <a href="real_1.html#NM1">Real</a> ex <font color="Olive">N</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">c</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Olive">d</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</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#NR2" target="_self">&gt;=</a> <font color="Olive">N</font> holds <br/>( <font color="Olive">d</font> <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> 1</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> &amp; <font color="Maroon">t</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">c</font> <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> 1</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> ) ) )
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">c</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">d</font> being   <a href="real_1.html#NM1">Real</a>, <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="E7:117_1"/><i><font color="Green">E87</font></i>: 
( <font color="Maroon">c</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">d</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 <b>and </b><br/><a NAME="E8:117_1"/><i><font color="Green">E88</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/>( <font color="Maroon">d</font> <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> 1</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> &amp; <font color="Maroon">t</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c</font> <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> 1</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> )
 <b>by </b><i/>;<br/><a NAME="E9:117_1"/><i><font color="Green">E89</font></i>: 
( <span class="p1">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> 0 &amp; <span class="p1">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">d</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> 0 )
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><b>set </b><font color="Maroon">f'</font> = <span class="p1">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">t</font>;<br/><b>set </b><font color="Maroon">g'</font> = <span class="p1">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">t</font>;<br/><a NAME="E10:117_1"/><i><font color="Green">E90</font></i>: 
( <span class="p1">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">t</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>  &amp; <span class="p1">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">t</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>set </b><font color="Maroon">N0</font> =  <a href="asympt_0.html#K4">max</a> <font color="Maroon">N</font>,2;<br/><a NAME="E11:117_1"/><i><font color="Green">E91</font></i>: 
(  <a href="asympt_0.html#K4">max</a> <font color="Maroon">N</font>,2 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">N</font> &amp;  <a href="asympt_0.html#K4">max</a> <font color="Maroon">N</font>,2 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/><b>reconsider </b><font color="Maroon">N0</font> =  <a href="asympt_0.html#K4">max</a> <font color="Maroon">N</font>,2 as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="117_1_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="E1:117_1_2"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">N0</font>
 ;<br/><a NAME="E2:117_1_2"/><b>then </b><i><font color="Green">E93</font></i>: 
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">N</font> &amp; <font color="Maroon">n</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="E3:117_1_2"/><b>then </b>
( <font color="Maroon">d</font> <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> 1</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">t</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> &amp; <font color="Maroon">t</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c</font> <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> 1</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> )
 <b>by </b><i/>;<br/><a NAME="E4:117_1_2"/><b>then </b><i><font color="Green">E94</font></i>: 
( <span class="p1">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> 1</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> &amp; <span class="p1">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> 1</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E5:117_1_2"/><i><font color="Green">E95</font></i>: 
<span class="p1">(<span class="default"><a href="asympt_1.html#K4" target="_self">seq_const</a> 1</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/><i><font color="Green">E96</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> 1</span>)</span> <a href="seq_1.html#K10">-</a> <span class="p2">(<span class="default"><a href="asympt_1.html#K4" target="_self">seq_const</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> 1</span>)</span> <a href="seq_1.html#K9">+</a> <span class="p2">(<span class="default"><a href="seq_1.html#K17">-</a> <span class="p3">(<span class="default"><a href="asympt_1.html#K4" target="_self">seq_const</a> 1</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="seq_1.html#T15">SEQ_1:15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> 1</span>)</span> <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"><span class="p2">(<span class="default"><a href="seq_1.html#K17">-</a> <span class="p3">(<span class="default"><a href="asympt_1.html#K4" target="_self">seq_const</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T11">SEQ_1:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> 1</span>)</span> <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"><a href="real_1.html#K1">-</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="asympt_1.html#E5:117_1"><i><font color="Green">E86</font></i></a>, <a class="ref" href="seq_1.html#T14">SEQ_1:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> 1</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> 1

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1
<b>by </b><i>, , </i>
<br/>.= 
<font color="Maroon">n</font> <a href="real_1.html#K5">-</a> 1
<b>by </b><i><a class="ref" href="power.html#T30">POWER:30</a></i>
;<br/><i><font color="Green">E97</font></i>: <span class="p1">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> 1</span>)</span> <a href="seq_1.html#K2">.</a> <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> 1</span>)</span> <a href="real_1.html#K5">-</a> 0
<b>by </b><i>, , </i>
<br/>.= 
<font color="Maroon">n</font> <a href="real_1.html#K5">-</a> 0
<b>by </b><i><a class="ref" href="power.html#T30">POWER:30</a></i>
;<br/><a NAME="E8:117_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> 1</span>)</span> <a href="seq_1.html#K10">-</a> <span class="p2">(<span class="default"><a href="asympt_1.html#K4" target="_self">seq_const</a> 1</span>)</span></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"><a href="asympt_1.html#K3" target="_self">seq_n^</a> 1</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="asympt_1.html#E7:117_1"><i><font color="Green">E87</font></i></a>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E9:117_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">d</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> 1</span>)</span> <a href="seq_1.html#K10">-</a> <span class="p3">(<span class="default"><a href="asympt_1.html#K4" target="_self">seq_const</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">d</font></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> 1</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E10:117_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">d</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> 1</span>)</span> <a href="seq_1.html#K10">-</a> <span class="p3">(<span class="default"><a href="asympt_1.html#K4" target="_self">seq_const</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> <span class="p1">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E4:117_1"><i><font color="Green">E85</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E11:117_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">d</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> 1</span>)</span> <a href="seq_1.html#K10">-</a> <span class="p3">(<span class="default"><a href="asympt_1.html#K4" target="_self">seq_const</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">t</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="seq_1.html#T13">SEQ_1:13</a></i>;<br/><a NAME="E12:117_1_2"/>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E13:117_1_2"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E14:117_1_2"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E15:117_1_2"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</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">2 <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K5">-</a> 1</span>)</span>
 ;<br/><a NAME="E16:117_1_2"/><b>then </b>
<font color="Maroon">c</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K5">-</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E17:117_1_2"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <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">c</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> 1</span>)</span> <a href="seq_1.html#K10">-</a> <span class="p3">(<span class="default"><a href="asympt_1.html#K4" target="_self">seq_const</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E4:117_1"><i><font color="Green">E85</font></i></a>, <a class="txt" href="asympt_1.html#E7:117_1"><i><font color="Green">E87</font></i></a>, <a class="txt" href="asympt_1.html#E8:117_1"><i><font color="Green">E88</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E18:117_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">t</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> 1</span>)</span> <a href="seq_1.html#K10">-</a> <span class="p3">(<span class="default"><a href="asympt_1.html#K4" target="_self">seq_const</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="seq_1.html#T13">SEQ_1:13</a></i>;<br/></div><b>end;</b></div><a NAME="E14:117_1"/><b>then </b><i><font color="Green">E98</font></i>: 
<span class="p1">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="asympt_0.html#K11">Big_Theta</a> <font color="Maroon">F</font>
 <b>by </b><i>, , , </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="117_1_3"><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="E1:117_1_3"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">N0</font>
 ;<br/><a NAME="E2:117_1_3"/><b>then </b>
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">N</font> &amp; <font color="Maroon">n</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="E3:117_1_3"/><b>then </b>
( <font color="Maroon">d</font> <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> 1</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">t</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> &amp; <font color="Maroon">t</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c</font> <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> 1</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> )
 <b>by </b><i/>;<br/><a NAME="E4:117_1_3"/><b>then </b>
( <span class="p1">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> 1</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> &amp; <span class="p1">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> 1</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><b>hence </b><a NAME="E5:117_1_3"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">d</font></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> 1</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">t</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">t</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">2 <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c</font></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> 1</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> )
 <b>by </b><i><a class="ref" href="seq_1.html#T13">SEQ_1:13</a></i>;<br/></div><b>end;</b></div><a NAME="E16:117_1"/><b>then </b><i><font color="Green">E99</font></i>: 
<span class="p1">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">t</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> 1</span>)</span>
 <b>by </b><i>, , </i>;<br/><a NAME="E17:117_1"/>
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">t</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">2 <a href="real_1.html#K2">"</a> </span>)</span> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">t</font></span>)</span> <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"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">t</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="117_1_4"><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="E1:117_1_4"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">t</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="seq_1.html#T13">SEQ_1:13</a></i>;<br/>
<a NAME="E2:117_1_4"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">t</font></span>)</span> <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"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">t</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E3:117_1_4"/>
<font color="Maroon">t</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"><span class="p3">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">t</font></span>)</span> <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"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">t</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E18:117_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K11">Big_Theta</a> <font color="Maroon">F</font></span>)</span> <a href="asympt_0.html#K16">+</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K11">Big_Theta</a> <span class="p2">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, , , </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:117"/>
<span class="p1">(<span class="default"><a href="asympt_0.html#K11">Big_Theta</a> <font color="Maroon">F</font></span>)</span> <a href="asympt_0.html#K16">+</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K11">Big_Theta</a> <span class="p2">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</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> 1</span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


</div>
