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

<b>let </b><font color="Maroon" title="c1">k</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>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>0 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> $1 <a href="radix_3.html#K1" title="RADIX_3:func.1">-SD_Sub_S</a> ;<br/>
<a NAME="E1:12"/><i><font color="Green" title="E11">A1</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i><a class="ref" href="radix_3.html#T6" target="_self" title="RADIX_3:th.6">Th6</a></i>;<br/>
<a NAME="E2:12"/><i><font color="Green" title="E12">A2</font></i>: 
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 <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">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="E1:12_1"/><i><font color="Green" title="E13">A3</font></i>: 
0 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">kk</font> <a href="radix_3.html#K1" title="RADIX_3:func.1">-SD_Sub_S</a> 
 ;<br/>

<a NAME="E2:12_1"/>
<font color="Maroon" title="c2">kk</font> <a href="radix_3.html#K1" title="RADIX_3:func.1">-SD_Sub_S</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="radix_3.html#K1" title="RADIX_3:func.1">-SD_Sub_S</a> 
 
<b>by </b><i><a class="ref" href="radix_3.html#T4" target="_self" title="RADIX_3:th.4">Th4</a></i>;<br/>
<b>hence </b><a NAME="E3:12_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">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#E1:12_1"><i><font color="Green" title="E13">A3</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:12"/>
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>  holds  <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#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="radix_3.html#E1:12"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="radix_3.html#E2:12"><i><font color="Green" title="E12">A2</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:12"/>
0 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">k</font> <a href="radix_3.html#K1" title="RADIX_3:func.1">-SD_Sub_S</a> 
 ;<br/>


</div>
