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

<b>let </b><font color="Maroon">A</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">B</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">X</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">A</font>;<br/><b>let </b><font color="Maroon">R1</font> be   <a href="subset_1.html#NM2">Subset</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/><b>let </b><font color="Maroon">R2</font> be   <a href="subset_1.html#NM2">Subset</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/><b>let </b><font color="Maroon">y</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="E1:49"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">R1</font> <a href="relset_2.html#K5" target="_self">.:^</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">R2</font> <a href="relset_2.html#K5" target="_self">.:^</a> <font color="Maroon">X</font></span>)</span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:49_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R1</font> <a href="relset_2.html#K5" target="_self">.:^</a> <font color="Maroon">X</font> or <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R2</font> <a href="relset_2.html#K5" target="_self">.:^</a> <font color="Maroon">X</font> )
 </span><b>by </b><i><a class="txt" href="relset_2.html#E1:49"><i><font color="Green">E21</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1"><b>suppose </b></a><a NAME="E1:49_1_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R1</font> <a href="relset_2.html#K5" target="_self">.:^</a> <font color="Maroon">X</font>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">B</font> = <font color="Maroon">B</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/><b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font> <b>by </b><i><a class="txt" href="relset_2.html#E1:49_1_1"><i><font color="Green">E22</font></i></a></i>;<br/><a NAME="E4:49_1_1"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">R1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">R2</font></span>)</span> <a href="relat_1.html#K9">.:</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_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="E1:49_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E2:49_1_1_1"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R1</font> <a href="relat_1.html#K9">.:</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="relset_2.html#E1:49_1_1"><i><font color="Green">E22</font></i></a>, </i>;<br/>
<a NAME="E3:49_1_1_1"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">R1</font> <a href="relat_1.html#K9">.:</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">R2</font> <a href="relat_1.html#K9">.:</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E4:49_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">R1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">R2</font></span>)</span> <a href="relat_1.html#K9">.:</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E5:49_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">R1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">R2</font></span>)</span> <a href="relset_2.html#K5" target="_self">.:^</a> <font color="Maroon">X</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_2"><b>suppose </b></a><a NAME="E1:49_1_2"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R2</font> <a href="relset_2.html#K5" target="_self">.:^</a> <font color="Maroon">X</font>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">B</font> = <font color="Maroon">B</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/><b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font> <b>by </b><i/>;<br/><a NAME="E4:49_1_2"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">R1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">R2</font></span>)</span> <a href="relat_1.html#K9">.:</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_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/>

<b>assume </b><a NAME="E1:49_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E2:49_1_2_1"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R2</font> <a href="relat_1.html#K9">.:</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:49_1_2_1"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">R1</font> <a href="relat_1.html#K9">.:</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">R2</font> <a href="relat_1.html#K9">.:</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E4:49_1_2_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">R1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">R2</font></span>)</span> <a href="relat_1.html#K9">.:</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E5:49_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">R1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">R2</font></span>)</span> <a href="relset_2.html#K5" target="_self">.:^</a> <font color="Maroon">X</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
