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

<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="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> $1;<br/>
<a NAME="E1:4"/>
 <a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</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="E2:4"/><b>then </b><i><font color="Green" title="E5">A1</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[1]
 
;<br/>
<a NAME="E3:4"/><i><font color="Green" title="E6">A2</font></i>: 
for <font color="Olive" title="b1">kk</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="Olive" title="b1">kk</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">kk</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c1">kk</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="E1:4_1"/><i><font color="Green" title="E7">A3</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">kk</font> &amp; 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c1">kk</font> )
 ;<br/>

<i><font color="Green" title="E8">A4</font></i>:  <a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">kk</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="power.html#K4" title="POWER:func.4">to_power</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default">2 <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c1">kk</font></span>)</span>
<b>by </b><i><a class="ref" href="power.html#T32" title="POWER:th.32">POWER:32</a></i>
<br/>.= 
2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c1">kk</font></span>)</span>
<b>by </b><i><a class="ref" href="power.html#T30" title="POWER:th.30">POWER:30</a></i>
;<br/>
<a NAME="E3:4_1"/>
 <a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c1">kk</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 
<b>by </b><i><a class="txt" href="radix_3.html#E1:4_1"><i><font color="Green" title="E7">A3</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E4:4_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <b>by </b><i><a class="txt" href="radix_3.html#E2:4_1"><i><font color="Green" title="E8">A4</font></i></a>, <a class="ref" href="real_2.html#T144" title="REAL_2:th.144">REAL_2:144</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:4"/>
for <font color="Olive" title="b1">k</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 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</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="radix_3.html#E2:4"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="radix_3.html#E3:4"><i><font color="Green" title="E6">A2</font></i></a>);<br/></i>
<b>hence </b><a NAME="E5:4"/>
for <font color="Olive" title="b1">k</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 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> holds <br/>2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Olive" title="b1">k</font>
 ;<br/>


</div>
