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

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>0 <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font> <a href="radix_1.html#K2" target="_self">-SD</a> ;<br/>
<a NAME="E1:16"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i/>;<br/>
<a NAME="E2:16"/><i><font color="Green">E30</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">kk</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:16_1"/><i><font color="Green">E37</font></i>: 
0 <a href="hidden.html#R2">in</a> <font color="Maroon">kk</font> <a href="radix_1.html#K2" target="_self">-SD</a> 
 ;<br/>

<a NAME="E2:16_1"/>
<font color="Maroon">kk</font> <a href="radix_1.html#K2" target="_self">-SD</a>  <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="radix_1.html#K2" target="_self">-SD</a> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:16_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">kk</font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E3:16"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<b>hence </b><a NAME="E4:16"/>
0 <a href="hidden.html#R2">in</a> <font color="Maroon">k</font> <a href="radix_1.html#K2" target="_self">-SD</a> 
 ;<br/>


</div>
