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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>

<b>assume </b><a NAME="E1:64"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></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"><font color="Olive">b<sub>1</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="power.html#K4">to_power</a> 4
 ;<br/>

<b>set </b><font color="Maroon">c<sub>2</sub></font> =  <a href="asympt_1.html#K3" target="_self">seq_n^</a> 8;<br/>
<a NAME="E2:64"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="asympt_0.html#V4">eventually-positive</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1"><b>proof </b></a><div class="add">

<b>take </b>
0
; <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">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:64_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/>

<a NAME="E2:64_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></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"><font color="Maroon">c<sub>3</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="power.html#K4">to_power</a> 4
 
<b>by </b><i><a class="txt" href="asympt_1.html#E1:64"><i><font color="Green">E39</font></i></a></i>;<br/>
<a NAME="E3:64_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K3">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="asympt_1.html#E52"><i><font color="Green">Lemma34</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:64_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="asympt_1.html#E2:64_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="power.html#T39">POWER:39</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>1</sub></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">c<sub>3</sub></font>
;<br/>

<a NAME="E4:64"/><i><font color="Green">E40</font></i>: 
 <a href="asympt_1.html#K3" target="_self">seq_n^</a> 8 is    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <a href="numbers.html#K5">NAT</a> ,<a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/>
<a NAME="E5:64"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>3</sub></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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:64_2"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 ;<br/><b>then </b><i><font color="Green">E43</font></i>: <span class="p1">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> 8</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> = 
<font color="Maroon">c<sub>4</sub></font> <a href="series_1.html#K3">to_power</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 4</span>)</span>
<b>by </b><i><a class="ref" href="asympt_1.html#D3" target="_self">Def3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="series_1.html#K3">to_power</a> 2</span>)</span> <a href="series_1.html#K3">to_power</a> 4
<b>by </b><i><a class="txt" href="asympt_1.html#E1:64_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="power.html#T38">POWER:38</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="power.html#K4">to_power</a> 4
<b>by </b><i><a class="ref" href="power.html#T53">POWER:53</a></i>
;<br/><a NAME="E3:64_2"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></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"><font color="Maroon">c<sub>4</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="power.html#K4">to_power</a> 4
 <b>by </b><i><a class="txt" href="asympt_1.html#E1:64"><i><font color="Green">E39</font></i></a></i>;<br/><a NAME="E4:64_2"/><i><font color="Green">E45</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K3">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="square_1.html#K5">^2</a> 
 <b>by </b><i><a class="txt" href="asympt_1.html#E1:64_2"><i><font color="Green">E42</font></i></a>, <a class="txt" href="asympt_1.html#E54"><i><font color="Green">Lemma36</font></i></a></i>;<br/><a NAME="E5:64_2"/><i><font color="Green">E46</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K3">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="asympt_1.html#E52"><i><font color="Green">Lemma34</font></i></a></i>;<br/><b>hence </b><a NAME="E6:64_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> 8</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E2:64_2"><i><font color="Green">E43</font></i></a>, <a class="txt" href="asympt_1.html#E3:64_2"><i><font color="Green">E44</font></i></a>, <a class="txt" href="asympt_1.html#E4:64_2"><i><font color="Green">E45</font></i></a>, <a class="txt" href="asympt_1.html#E13"><i><font color="Green">Lemma10</font></i></a></i>;<br/><b>thus </b><a NAME="E7:64_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="asympt_1.html#E3:64_2"><i><font color="Green">E44</font></i></a>, <a class="txt" href="asympt_1.html#E5:64_2"><i><font color="Green">E46</font></i></a>, <a class="ref" href="power.html#T39">POWER:39</a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:64"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="asympt_0.html#K9">Big_Oh</a> <span class="p1">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> 8</span>)</span>
 
<b>by </b><i><a class="txt" href="asympt_1.html#E5:64"><i><font color="Green">E41</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:64_3"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 ;<br/><b>then </b><i><font color="Green">E44</font></i>: <span class="p1">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> 8</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> = 
<font color="Maroon">c<sub>4</sub></font> <a href="series_1.html#K3">to_power</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 4</span>)</span>
<b>by </b><i><a class="ref" href="asympt_1.html#D3" target="_self">Def3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="series_1.html#K3">to_power</a> 2</span>)</span> <a href="series_1.html#K3">to_power</a> 4
<b>by </b><i><a class="txt" href="asympt_1.html#E1:64_3"><i><font color="Green">E43</font></i></a>, <a class="ref" href="power.html#T38">POWER:38</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="power.html#K4">to_power</a> 4
<b>by </b><i><a class="ref" href="power.html#T53">POWER:53</a></i>
;<br/><a NAME="E3:64_3"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></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"><font color="Maroon">c<sub>4</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="power.html#K4">to_power</a> 4
 <b>by </b><i><a class="txt" href="asympt_1.html#E1:64"><i><font color="Green">E39</font></i></a></i>;<br/><a NAME="E4:64_3"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 2 <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"><font color="Maroon">c<sub>4</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E1:64_3"><i><font color="Green">E43</font></i></a>, <a class="txt" href="asympt_1.html#E55"><i><font color="Green">Lemma37</font></i></a></i>;<br/><a NAME="E5:64_3"/><i><font color="Green">E47</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K3">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="asympt_1.html#E52"><i><font color="Green">Lemma34</font></i></a></i>;<br/><a NAME="E6:64_3"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K2">*</a> 0
 <b>by </b><i><a class="txt" href="asympt_1.html#E1:64_3"><i><font color="Green">E43</font></i></a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E7:64_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> 8</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <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"><font color="Maroon">c<sub>4</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span></span>)</span> <a href="power.html#K4">to_power</a> 4
 <b>by </b><i><a class="txt" href="asympt_1.html#E2:64_3"><i><font color="Green">E44</font></i></a>, <a class="txt" href="asympt_1.html#E4:64_3"><i><font color="Green">E46</font></i></a>, <a class="txt" href="asympt_1.html#E13"><i><font color="Green">Lemma10</font></i></a></i>;<br/><b>hence </b><a NAME="E8:64_3"/>
<span class="p1">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> 8</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 16 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E3:64_3"><i><font color="Green">E45</font></i></a>, <a class="txt" href="asympt_1.html#E5:64_3"><i><font color="Green">E47</font></i></a>, <a class="txt" href="asympt_1.html#E16"><i><font color="Green">Lemma13</font></i></a>, <a class="ref" href="power.html#T35">POWER:35</a></i>;<br/><b>thus </b><a NAME="E9:64_3"/>
<span class="p1">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> 8</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="asympt_1.html#E2:64_3"><i><font color="Green">E44</font></i></a>, <a class="txt" href="asympt_1.html#E6:64_3"><i><font color="Green">E48</font></i></a>, <a class="ref" href="power.html#T39">POWER:39</a></i>;<br/></div><b>end;</b></div>
<a NAME="E9:64"/><b>then </b>
 <a href="asympt_1.html#K3" target="_self">seq_n^</a> 8 <a href="hidden.html#R2">in</a>  <a href="asympt_0.html#K9">Big_Oh</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="asympt_1.html#E4:64"><i><font color="Green">E40</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:64"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="asympt_0.html#K9">Big_Oh</a> <span class="p1">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> 8</span>)</span> <a href="hidden.html#R1">=</a>  <a href="asympt_0.html#K9">Big_Oh</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="asympt_1.html#E7:64"><i><font color="Green">E42</font></i></a>, <a class="txt" href="asympt_1.html#E11"><i><font color="Green">Lemma8</font></i></a></i>;<br/>


</div>
