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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">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#NR2" title="XXREAL_0:NR.2">&gt;=</a> $1;<br/>
<a NAME="E1:2"/><i><font color="Green" title="E3">A1</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<a NAME="E2:2"/><i><font color="Green" title="E4">A2</font></i>: 
for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">m</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c1">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:2_1"/><i><font color="Green" title="E5">A3</font></i>: 
2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c1">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c1">m</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_1_1_1"/>
( <font color="Maroon" title="c1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Maroon" title="c1">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_1_1"><b>suppose </b></a><a NAME="E1:2_1_1_1_1"/><i><font color="Green" title="E6">A4</font></i>: 
<font color="Maroon" title="c1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:2_1_1_1_1"/><b>then </b>
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">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 <b>by </b><i><a class="ref" href="power.html#T30" title="POWER:th.30">POWER:30</a></i>;<br/><b>hence </b><a NAME="E3:2_1_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <b>by </b><i><a class="txt" href="binari_4.html#E1:2_1_1_1_1"><i><font color="Green" title="E6">A4</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_1_2"><b>suppose </b></a><a NAME="E1:2_1_1_1_2"/><i><font color="Green" title="E6">A5</font></i>: 
<font color="Maroon" title="c1">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:2_1_1_1_2"/><i><font color="Green" title="E7">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">m</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 2 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c1">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 2
 <b>by </b><i><a class="txt" href="binari_4.html#E1:2_1"><i><font color="Green" title="E5">A3</font></i></a>, <a class="ref" href="nat_1.html#T4" title="NAT_1:th.4">NAT_1:4</a></i>;<br/><a NAME="E3:2_1_1_1_2"/>
2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 <b>by </b><i><a class="ref" href="power.html#T30" title="POWER:th.30">POWER:30</a></i>;<br/><a NAME="E4:2_1_1_1_2"/><b>then </b><i><font color="Green" title="E8">A7</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">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c1">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 2
 <b>by </b><i><a class="txt" href="binari_4.html#E2:2_1_1_1_2"><i><font color="Green" title="E7">A6</font></i></a>, <a class="ref" href="power.html#T32" title="POWER:th.32">POWER:32</a></i>;<br/><a NAME="E5:2_1_1_1_2"/>
<font color="Maroon" title="c1">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 2 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c1">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="binari_4.html#E1:2_1_1_1_2"><i><font color="Green" title="E6">A5</font></i></a>, <a class="ref" href="binari_4.html#T1" target="_self" title="BINARI_4:th.1">Th1</a></i>;<br/><b>hence </b><a NAME="E6:2_1_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <b>by </b><i><a class="txt" href="binari_4.html#E4:2_1_1_1_2"><i><font color="Green" title="E8">A7</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></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:2_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 ;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E3:2"/>
for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">m</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="binari_4.html#E1:2"><i><font color="Green" title="E3">A1</font></i></a>, <a class="txt" href="binari_4.html#E2:2"><i><font color="Green" title="E4">A2</font></i></a>);</i><br/>


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