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

<b>reconsider </b><font color="Maroon">N</font> = <font color="Maroon">F<sub>1</sub></font>() as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b> ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">a<sub>1</sub></font>,<font color="Olive">n</font>];<br/>
<a NAME="E2:16_1"/>
<font color="Maroon">P<sub>1</sub></font>[0,<font color="Maroon">N</font>]
 
<b>by </b><i/>;<br/>
<a NAME="E3:16_1"/><b>then </b><i><font color="Green">E32</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<a NAME="E4:16_1"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">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" target="_self">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>given </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><a NAME="E1:16_1_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">k</font>,<font color="Maroon">n</font>]
 ;<br/>

<b>reconsider </b><font color="Maroon">F</font> = <font color="Maroon">F<sub>2</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1" target="_self">+</a> 1</span>)</span>,<font color="Maroon">n</font>) as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<b>take </b>
<font color="Maroon">F</font>
;<br/>

<b>thus </b><a NAME="E3:16_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">k</font> <a href="nat_1.html#K1" target="_self">+</a> 1,<font color="Maroon">F</font>]
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E5:16_1"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">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" target="_self">NAT_1:sch 1</a>(, );</i><br/>

<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b>for <font color="Olive">n</font>, <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">a<sub>1</sub></font>,<font color="Olive">n</font>] &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">a<sub>1</sub></font>,<font color="Olive">m</font>] holds <br/><font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Olive">m</font>;<br/>
<a NAME="E6:16_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:16_1_2"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">P<sub>1</sub></font>[0,<font color="Maroon">n</font>] &amp; <font color="Maroon">P<sub>1</sub></font>[0,<font color="Maroon">m</font>] )
 ;<br/>

<a NAME="E2:16_1_2"/>
( ( for <font color="Olive">m</font>, <font color="Olive">l</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   holds <br/>(  not 0 <a href="hidden.html#R1">=</a> <font color="Olive">m</font> <a href="nat_1.html#K1" target="_self">+</a> 1 or  not <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">m</font>,<font color="Olive">l</font>] or  not <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(0,<font color="Olive">l</font>) ) ) &amp; ( for <font color="Olive">n</font>, <font color="Olive">l</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   holds <br/>(  not 0 <a href="hidden.html#R1">=</a> <font color="Olive">n</font> <a href="nat_1.html#K1" target="_self">+</a> 1 or  not <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Olive">l</font>] or  not <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(0,<font color="Olive">l</font>) ) ) )
 
;<br/>
<a NAME="E3:16_1_2"/><b>then </b>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>() )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E4:16_1_2"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E7:16_1"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1" target="_self">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:16_1_3"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">n</font>, <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">k</font>,<font color="Olive">n</font>] &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">k</font>,<font color="Olive">m</font>] holds <br/><font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Olive">m</font>
 ;<br/>

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E2:16_1_3"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">k</font> <a href="nat_1.html#K1" target="_self">+</a> 1,<font color="Maroon">n</font>] &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">k</font> <a href="nat_1.html#K1" target="_self">+</a> 1,<font color="Maroon">m</font>] )
 ;<br/>

<b>consider </b><font color="Maroon">l</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">u</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:16_1_3"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">k</font> <a href="nat_1.html#K1" target="_self">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="nat_1.html#K1" target="_self">+</a> 1 &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">l</font>,<font color="Maroon">u</font>] &amp; <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1" target="_self">+</a> 1</span>)</span>,<font color="Maroon">u</font>) )
 <b>by </b><i>, </i>;<br/>
<a NAME="E5:16_1_3"/>
 ex <font color="Olive">v</font>, <font color="Olive">w</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">k</font> <a href="nat_1.html#K1" target="_self">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Olive">v</font> <a href="nat_1.html#K1" target="_self">+</a> 1 &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">v</font>,<font color="Olive">w</font>] &amp; <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1" target="_self">+</a> 1</span>)</span>,<font color="Olive">w</font>) )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E6:16_1_3"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E8:16_1"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1" target="_self">NAT_1:sch 1</a>(, );</i><br/>


</div>
