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

<b>let </b><font color="Maroon">X1</font> be   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span> <a href="afinsq_1.html#K10">^omega</a> </span>)</span>, <font color="Maroon">X2</font> be   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span> <a href="afinsq_1.html#K10">^omega</a> </span>)</span>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:30_1_2"/><i><font color="Green">E58</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font> iff  ex <font color="Olive">p</font> being  <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> &amp; <font color="Olive">p</font> is <a href="catalan2.html#V1" target="_self">dominated_by_0</a> &amp;  <a href="afinsq_1.html#K2">dom</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="stirl2_1.html#K7">Sum</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> ) )
 <b>and </b><br/><a NAME="E2:30_1_2"/><i><font color="Green">E59</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X2</font> iff  ex <font color="Olive">p</font> being  <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> &amp; <font color="Olive">p</font> is <a href="catalan2.html#V1" target="_self">dominated_by_0</a> &amp;  <a href="afinsq_1.html#K2">dom</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="stirl2_1.html#K7">Sum</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> ) )
 ;<br/>

<a NAME="E3:30_1_2"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font> iff <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X2</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<a NAME="E1:30_1_2_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font> iff  ex <font color="Olive">p</font> being  <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Olive">p</font> is <a href="catalan2.html#V1" target="_self">dominated_by_0</a> &amp;  <a href="afinsq_1.html#K2">dom</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="stirl2_1.html#K7">Sum</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> ) )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:30_1_2_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font> iff <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X2</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:30_1_2"/>
<font color="Maroon">X1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X2</font>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


</div>
