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

<b>let </b><font color="Maroon" title="c2">n</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>let </b><font color="Maroon" title="c3">A</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</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><font color="Maroon" title="c3">A</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> $1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> $1;<br/>
<a NAME="E1:21"/><i><font color="Green" title="E17">A1</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><b>proof </b></a><div class="add">

<b>thus </b><font color="Maroon" title="c3">A</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> 0</span>)</span> = 
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c3">A</font>
<b>by </b><i><a class="ref" href="ordinal2.html#T48" title="ORDINAL2:th.48">ORDINAL2:48</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> 0
<b>by </b><i><a class="ref" href="ordinal2.html#T44" title="ORDINAL2:th.44">ORDINAL2:44</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E2:21"/><i><font color="Green" title="E18">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="21_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">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>assume </b><a NAME="E1:21_2"/><i><font color="Green" title="E19">A3</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">k</font>]
 ;<br/>

<a NAME="E2:21_2"/><i><font color="Green" title="E20">A4</font></i>: 
<font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">k</font>
 
<b>by </b><i><a class="ref" href="card_1.html#T52" title="CARD_1:th.52">CARD_1:52</a></i>;<br/>
<b>hence </b><font color="Maroon" title="c3">A</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> = 
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c4">k</font></span>)</span>
<b>by </b><i><a class="txt" href="card_4.html#E1:21_2"><i><font color="Green" title="E19">A3</font></i></a>, <a class="ref" href="ordinal2.html#T45" title="ORDINAL2:th.45">ORDINAL2:45</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> 1
<b>by </b><i><a class="ref" href="ordinal2.html#T48" title="ORDINAL2:th.48">ORDINAL2:48</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="ordinal3.html#T33" title="ORDINAL3:th.33">ORDINAL3:33</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="card_4.html#E2:21_2"><i><font color="Green" title="E20">A4</font></i></a>, <a class="ref" href="ordinal2.html#T48" title="ORDINAL2:th.48">ORDINAL2:48</a></i>
;<br/><br/>


</div><b>end;</b></div>
<a NAME="E3:21"/>
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="card_4.html#E1:21"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="card_4.html#E2:21"><i><font color="Green" title="E18">A2</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:21"/>
<font color="Maroon" title="c3">A</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">n</font>
 ;<br/>

<b>hence </b><a NAME="E5:21"/>
<font color="Maroon" title="c3">A</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</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> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="ref" href="card_1.html#T52" title="CARD_1:th.52">CARD_1:52</a></i>;<br/>


</div>
