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<div class="add">

<b>let </b><font color="Maroon">X1</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">Y1</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">X2</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">Y2</font> be    <a href="hidden.html#M1">set</a> ;<br/>







<b>assume </b><a NAME="E1:120"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">X1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:120"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<b>assume </b><a NAME="E4:120"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">Y1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:120"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y1</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<b>assume </b><a NAME="E7:120"/><i><font color="Green">E35</font></i>: 
<span class="p1"><a href="zfmisc_1.html#K2" target="_self">[:</a><span class="default"><font color="Maroon">X1</font>,<font color="Maroon">Y1</font></span><a href="zfmisc_1.html#K2" target="_self">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" target="_self">[:</a><span class="default"><font color="Maroon">X2</font>,<font color="Maroon">Y2</font></span><a href="zfmisc_1.html#K2" target="_self">:]</a></span>
 ;<br/>

<a NAME="E8:120"/><b>then </b>
<span class="p1"><a href="zfmisc_1.html#K2" target="_self">[:</a><span class="default"><font color="Maroon">X2</font>,<font color="Maroon">Y2</font></span><a href="zfmisc_1.html#K2" target="_self">:]</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, , <a class="txt" href="zfmisc_1.html#E28"><i><font color="Green">Lemma45</font></i></a></i>;<br/>
<a NAME="E9:120"/><b>then </b><i><font color="Green">E36</font></i>: 
( <font color="Maroon">X2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">Y2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="txt" href="zfmisc_1.html#E28"><i><font color="Green">Lemma45</font></i></a></i>;<br/>
<a NAME="E10:120"/>
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">X1</font> iff <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X2</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="120_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/>

<b>thus </b><a NAME="E1:120_1"/>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font> implies <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="120_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:120_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font>
 ;<br/>

<a NAME="E2:120_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" target="_self">[:</a><span class="default"><font color="Maroon">X2</font>,<font color="Maroon">Y2</font></span><a href="zfmisc_1.html#K2" target="_self">:]</a></span>
 
<b>by </b><i>, , <a class="txt" href="zfmisc_1.html#E4"><i><font color="Green">Lemma28</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:120_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X2</font>
 <b>by </b><i><a class="txt" href="zfmisc_1.html#E4"><i><font color="Green">Lemma28</font></i></a></i>;<br/>


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

<b>consider </b><font color="Maroon">y2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:120_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">y2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y2</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<b>assume </b><a NAME="E4:120_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X2</font>
 ;<br/>

<a NAME="E5:120_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">y2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" target="_self">[:</a><span class="default"><font color="Maroon">X2</font>,<font color="Maroon">Y2</font></span><a href="zfmisc_1.html#K2" target="_self">:]</a></span>
 
<b>by </b><i><a class="txt" href="zfmisc_1.html#E50"><i><font color="Green">Lemma61</font></i></a>, <a class="txt" href="zfmisc_1.html#E4"><i><font color="Green">Lemma28</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:120_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font>
 <b>by </b><i>, <a class="txt" href="zfmisc_1.html#E4"><i><font color="Green">Lemma28</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E11:120"/>
<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/>

<a NAME="E12:120"/>
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">Y1</font> iff <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y2</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="120_2"><b>proof </b></a><div class="add">

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

<b>thus </b><a NAME="E1:120_2"/>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y1</font> implies <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="120_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:120_2_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y1</font>
 ;<br/>

<a NAME="E2:120_2_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" target="_self">[:</a><span class="default"><font color="Maroon">X2</font>,<font color="Maroon">Y2</font></span><a href="zfmisc_1.html#K2" target="_self">:]</a></span>
 
<b>by </b><i><a class="txt" href="zfmisc_1.html#E3:120_2"><i><font color="Green">E56</font></i></a>, , <a class="txt" href="zfmisc_1.html#E4"><i><font color="Green">Lemma28</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:120_2_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y2</font>
 <b>by </b><i><a class="txt" href="zfmisc_1.html#E4"><i><font color="Green">Lemma28</font></i></a></i>;<br/>


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

<b>consider </b><font color="Maroon">x2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:120_2"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X2</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<b>assume </b><a NAME="E4:120_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y2</font>
 ;<br/>

<a NAME="E5:120_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x2</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" target="_self">[:</a><span class="default"><font color="Maroon">X2</font>,<font color="Maroon">Y2</font></span><a href="zfmisc_1.html#K2" target="_self">:]</a></span>
 
<b>by </b><i><a class="txt" href="zfmisc_1.html#E50"><i><font color="Green">Lemma61</font></i></a>, <a class="txt" href="zfmisc_1.html#E4"><i><font color="Green">Lemma28</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:120_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y1</font>
 <b>by </b><i>, <a class="txt" href="zfmisc_1.html#E4"><i><font color="Green">Lemma28</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E13:120"/>
<font color="Maroon">Y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y2</font>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


</div>
