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

<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">A</font> be   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font>;<br/>



<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</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="E1:33_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="cantor_1.html#K1">UniCl</a> <span class="p1">(<span class="default"><a href="cantor_1.html#K1">UniCl</a> <font color="Maroon">A</font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">Y</font> being   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:33_1"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a>  <a href="cantor_1.html#K1">UniCl</a> <font color="Maroon">A</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">Y</font> )
 <b>by </b><i><a class="ref" href="cantor_1.html#D1">CANTOR_1:def 1</a></i>;<br/><b>set </b><font color="Maroon">Z</font> = <span class="p1">{<span class="default"> <font color="Olive">y</font> where <font color="Olive">y</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> : ( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">x</font> ) </span>}</span> ;<br/><a NAME="E4:33_1"/>
<span class="p1">{<span class="default"> <font color="Olive">y</font> where <font color="Olive">y</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> : ( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">x</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:33_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">y</font> where <font color="Olive">y</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> : ( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">x</font> ) </span>}</span> 
 ;<br/>

<a NAME="E2:33_1_1"/><b>then </b>
 ex <font color="Olive">y</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">x</font> )
 
;<br/>
<b>hence </b><a NAME="E3:33_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font>
 ;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">Z</font> = <span class="p1">{<span class="default"> <font color="Olive">y</font> where <font color="Olive">y</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> : ( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">x</font> ) </span>}</span>  as   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font> ;<br/><a NAME="E6:33_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">Z</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_2"><b>proof </b></a><div class="add">

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a>
<div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:33_1_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font>
 ;<br/><b>then consider </b><font color="Maroon">s</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:33_1_2_1"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">s</font> &amp; <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon">t</font> being   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E5:33_1_2_1"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">t</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> &amp; <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">t</font> )
 <b>by </b><i>, , <a class="ref" href="cantor_1.html#D1">CANTOR_1:def 1</a></i>;<br/><b>consider </b><font color="Maroon">u</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:33_1_2_1"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">u</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">t</font> )
 <b>by </b><i>, , <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>reconsider </b><font color="Maroon">u</font> = <font color="Maroon">u</font> as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> <b>by </b><i/>;<br/><a NAME="E9:33_1_2_1"/>
( <font color="Maroon">u</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">s</font> &amp; <font color="Maroon">s</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">x</font> )
 <b>by </b><i>, , , , <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/><a NAME="E10:33_1_2_1"/><b>then </b>
<font color="Maroon">u</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E11:33_1_2_1"/><b>then </b>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E12:33_1_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">Z</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:33_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">Z</font>
 ;<br/>

<b>then consider </b><font color="Maroon">u</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:33_1_2"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">u</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E5:33_1_2"/>
 ex <font color="Olive">y</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">x</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:33_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><a NAME="E7:33_1"/>
<font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:33_1_3"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 ;<br/>

<a NAME="E2:33_1_3"/><b>then </b>
 ex <font color="Olive">y</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">x</font> )
 
;<br/>
<b>hence </b><a NAME="E3:33_1_3"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E8:33_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="cantor_1.html#K1">UniCl</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="cantor_1.html#D1">CANTOR_1:def 1</a></i>;<br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E2:33"/>
 <a href="cantor_1.html#K1">UniCl</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="cantor_1.html#K1">UniCl</a> <span class="p1">(<span class="default"><a href="cantor_1.html#K1">UniCl</a> <font color="Maroon">A</font></span>)</span>
 <b>by </b><i><a class="ref" href="cantor_1.html#T1">CANTOR_1:1</a></i>;<br/>


</div>
