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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">a<sub>1</sub></font> <a href="series_1.html#K3">to_power</a> 6;<br/>
<a NAME="E1:70"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[30]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1"><b>proof </b></a><div class="add">

2 <a href="series_1.html#K3">to_power</a> 30 = 
2 <a href="series_1.html#K3">to_power</a> <span class="p1">(<span class="default">5 <a href="nat_1.html#K2">*</a> 6</span>)</span>

<br/>.= 
32 <a href="series_1.html#K3">to_power</a> 6
<b>by </b><i>, <a class="ref" href="power.html#T38">POWER:38</a></i>
;<br/>
<b>hence </b><a NAME="E2:70_1"/>
<font color="Maroon">S<sub>1</sub></font>[30]
 <b>by </b><i><a class="ref" href="power.html#T42">POWER:42</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:70"/><i><font color="Green">E50</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> 30 &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_2"><b>proof </b></a><div class="add">

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

<b>assume </b><b>that </b><br/><a NAME="E1:70_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 30
 <b>and </b><br/><a NAME="E2:70_2"/><i><font color="Green">E52</font></i>: 
2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> 6
 ;<br/>

<i><font color="Green">E59</font></i>: 2 <a href="series_1.html#K3">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="power.html#T32">POWER:32</a></i>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K2">*</a> 2
<b>by </b><i><a class="ref" href="power.html#T30">POWER:30</a></i>
;<br/>
<a NAME="E4:70_2"/><i><font color="Green">E60</font></i>: 
<span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K2">*</a> 2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> 6</span>)</span> <a href="nat_1.html#K2">*</a> 2
 
<b>by </b><i><a class="txt" href="asympt_1.html#E1"><i><font color="Green">Lemma47</font></i></a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<a NAME="E5:70_2"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 9
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E6:70_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="series_1.html#K3">to_power</a> 6 <a href="xxreal_0.html#R1">&lt;</a> 2 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> 6</span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:70_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i>, <a class="txt" href="asympt_1.html#E1:70"><i><font color="Green">E49</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:70"/>
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> 30 holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S9">NAT_1:sch 9</a>(, );<br/></i>
<b>hence </b><a NAME="E4:70"/>
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> 30 holds <br/>2 <a href="series_1.html#K3">to_power</a> <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Olive">n</font> <a href="series_1.html#K3">to_power</a> 6
 ;<br/>


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