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

<b>thus </b><a NAME="E1:1_1_1"/>
( <font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  implies  ex <font color="Olive">Z1</font> being   <a href="hidden.html#M1">set</a>  st <br/>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="Olive">Z1</font> iff for <font color="Olive">Z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">Z</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:1_1_1_1"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b>for <font color="Olive">Z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">Z</font>;<br/>
<b>consider </b><font color="Maroon">Y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:1_1_1_1"/><i><font color="Green">E18</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">Y</font> iff ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">X</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<a NAME="E4:1_1_1_1"/><i><font color="Green">E19</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">Y</font> iff for <font color="Olive">Z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">Z</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:1_1_1_1_1"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st ( for <font color="Olive">Z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">Z</font> ) holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_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/>

<b>assume </b><a NAME="E1:1_1_1_1_1_1"/><i><font color="Green">E20</font></i>: 
for <font color="Olive">Z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">Z</font>
 ;<br/>

<b>consider </b><font color="Maroon">y</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>;<br/>
<a NAME="E3:1_1_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">X</font>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>hence </b><a NAME="E5:1_1_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:1_1_1_1_1"/>
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">Y</font> iff for <font color="Olive">Z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">Z</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>take </b>
<font color="Maroon">Y</font>
;<br/>

<b>thus </b><a NAME="E5:1_1_1_1"/>
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">Y</font> iff for <font color="Olive">Z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">Z</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:1_1_1"/>
( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies  ex <font color="Olive">Z</font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">Z</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/>


</div>
