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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> $1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> $1 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> 6;<br/>
<a NAME="E1:70"/><i><font color="Green" title="E44">A1</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" title="SERIES_1:func.3">to_power</a> 30 = 
2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p1">(<span class="default">5 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 6</span>)</span>

<br/>.= 
32 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> 6
<b>by </b><i><a class="txt" href="asympt_1.html#E17"><i><font color="Green" title="E14">Lm10</font></i></a>, <a class="ref" href="power.html#T38" title="POWER:th.38">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" title="POWER:th.42">POWER:42</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:70"/><i><font color="Green" title="E45">A2</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 30 &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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" title="c1">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:70_2"/><i><font color="Green" title="E46">A3</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 30
 <b>and </b><br/><a NAME="E2:70_2"/><i><font color="Green" title="E47">A4</font></i>: 
2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">n</font> <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> 6
 ;<br/>

<i><font color="Green" title="E48">A5</font></i>: 2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="power.html#T32" title="POWER:th.32">POWER:32</a></i>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 2
<b>by </b><i><a class="ref" href="power.html#T30" title="POWER:th.30">POWER:30</a></i>
;<br/>
<a NAME="E4:70_2"/><i><font color="Green" title="E49">A6</font></i>: 
<span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> 6</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 2
 
<b>by </b><i><a class="txt" href="asympt_1.html#E2:70_2"><i><font color="Green" title="E47">A4</font></i></a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></i>;<br/>
<a NAME="E5:70_2"/>
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 9
 
<b>by </b><i><a class="txt" href="asympt_1.html#E1:70_2"><i><font color="Green" title="E46">A3</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E6:70_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> 6 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> 6</span>)</span>
 
<b>by </b><i><a class="txt" href="asympt_1.html#E65"><i><font color="Green" title="E42">Lm35</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:70_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <b>by </b><i><a class="txt" href="asympt_1.html#E3:70_2"><i><font color="Green" title="E48">A5</font></i></a>, <a class="txt" href="asympt_1.html#E4:70_2"><i><font color="Green" title="E49">A6</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:70"/>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 30 holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S8" title="NAT_1:sch.8">NAT_1:sch 8</a>(<a class="txt" href="asympt_1.html#E1:70"><i><font color="Green" title="E44">A1</font></i></a>, <a class="txt" href="asympt_1.html#E2:70"><i><font color="Green" title="E45">A2</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:70"/>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 30 holds <br/>2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b1">n</font> <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> 6
 ;<br/>


</div>
