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

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">Z</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">Z</font> &amp;  ex <font color="Olive">X</font>, <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">SFX</font> &amp; <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">SFY</font> &amp; <font color="Olive">Z</font> <a href="hidden.html#R1">=</a> <font color="Olive">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Olive">Y</font> ) );<br/>
<b>consider </b><font color="Maroon">XX</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:21_1_1"/><i><font color="Green">E17</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">XX</font> iff ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">SFX</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">SFY</font></span>)</span></span>)</span> &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>
<b>take </b>
<font color="Maroon">XX</font>
;<br/>

<a NAME="E3:21_1_1"/>
for <font color="Olive">Z</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">XX</font> iff  ex <font color="Olive">X</font>, <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">SFX</font> &amp; <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">SFY</font> &amp; <font color="Olive">Z</font> <a href="hidden.html#R1">=</a> <font color="Olive">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Olive">Y</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_1"><b>proof </b></a><div class="add">

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

<div><i><font color="Green">E18</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:21_1_1_1_1"/>
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">XX</font>
 ;<br/><a NAME="E2:21_1_1_1_1"/><b>then </b>
 ex <font color="Olive">Z1</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Maroon">Z</font> <a href="hidden.html#R1">=</a> <font color="Olive">Z1</font> &amp;  ex <font color="Olive">X</font>, <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">SFX</font> &amp; <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">SFY</font> &amp; <font color="Olive">Z1</font> <a href="hidden.html#R1">=</a> <font color="Olive">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Olive">Y</font> ) )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:21_1_1_1_1"/>
 ex <font color="Olive">X</font>, <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">SFX</font> &amp; <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">SFY</font> &amp; <font color="Maroon">Z</font> <a href="hidden.html#R1">=</a> <font color="Olive">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Olive">Y</font> )
 ;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_1_2"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">X</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">Y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E1:21_1_1_1_2"/><i><font color="Green">E19</font></i>: 
( <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">SFX</font> &amp; <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">SFY</font> &amp; <font color="Maroon">Z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Y</font> )
 ;<br/><a NAME="E2:21_1_1_1_2"/>
( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">SFX</font> &amp; <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">SFY</font> )
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/><a NAME="E3:21_1_1_1_2"/><b>then </b>
<font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">SFX</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">SFY</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T13">XBOOLE_1:13</a></i>;<br/><b>hence </b><a NAME="E4:21_1_1_1_2"/>
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">XX</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:21_1_1_1"/>
( <font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">XX</font> iff  ex <font color="Olive">X</font>, <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">SFX</font> &amp; <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">SFY</font> &amp; <font color="Maroon">Z</font> <a href="hidden.html#R1">=</a> <font color="Olive">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Olive">Y</font> ) )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:21_1_1"/>
for <font color="Olive">Z</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">XX</font> iff  ex <font color="Olive">X</font>, <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">SFX</font> &amp; <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">SFY</font> &amp; <font color="Olive">Z</font> <a href="hidden.html#R1">=</a> <font color="Olive">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Olive">Y</font> ) )
 ;<br/>


</div>
