<?xml version="1.0"?>
<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">X2</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">Y2</font> be    <a href="hidden.html#M1">set</a> ;<br/>







<a NAME="E1:113"/>
( <span class="p1"><a href="zfmisc_1.html#K2" target="_self">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="xboole_0.html#K4">\</a> <font color="Maroon">Y1</font></span>)</span>,<font color="Maroon">X2</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">X1</font>,<font color="Maroon">X2</font></span><a href="zfmisc_1.html#K2" target="_self">:]</a></span> <a href="xboole_0.html#K4">\</a> <span class="p1"><a href="zfmisc_1.html#K2" target="_self">[:</a><span class="default"><font color="Maroon">Y1</font>,<font color="Maroon">X2</font></span><a href="zfmisc_1.html#K2" target="_self">:]</a></span> &amp; <span class="p1"><a href="zfmisc_1.html#K2" target="_self">[:</a><span class="default"><font color="Maroon">X1</font>,<span class="p2">(<span class="default"><font color="Maroon">X2</font> <a href="xboole_0.html#K4">\</a> <font color="Maroon">Y2</font></span>)</span></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">X1</font>,<font color="Maroon">X2</font></span><a href="zfmisc_1.html#K2" target="_self">:]</a></span> <a href="xboole_0.html#K4">\</a> <span class="p1"><a href="zfmisc_1.html#K2" target="_self">[:</a><span class="default"><font color="Maroon">X1</font>,<font color="Maroon">Y2</font></span><a href="zfmisc_1.html#K2" target="_self">:]</a></span> &amp; <span class="p1"><a href="zfmisc_1.html#K2" target="_self">[:</a><span class="default"><font color="Maroon">Y1</font>,<font color="Maroon">X2</font></span><a href="zfmisc_1.html#K2" target="_self">:]</a></span> <a href="xboole_0.html#K3">/\</a> <span class="p1"><a href="zfmisc_1.html#K2" target="_self">[:</a><span class="default"><font color="Maroon">X1</font>,<font color="Maroon">Y2</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"><span class="p2">(<span class="default"><font color="Maroon">Y1</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">X2</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">Y2</font></span>)</span></span><a href="zfmisc_1.html#K2" target="_self">:]</a></span> &amp; <font color="Maroon">Y1</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y1</font> &amp; <font color="Maroon">X2</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">Y2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y2</font> )
 
<b>by </b><i><a class="txt" href="zfmisc_1.html#E34"><i><font color="Green">Lemma49</font></i></a>, <a class="txt" href="zfmisc_1.html#E36"><i><font color="Green">Lemma50</font></i></a>, <a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E2:113"/><b>then </b>
( <span class="p1"><a href="zfmisc_1.html#K2" target="_self">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="xboole_0.html#K4">\</a> <font color="Maroon">Y1</font></span>)</span>,<font color="Maroon">X2</font></span><a href="zfmisc_1.html#K2" target="_self">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2" target="_self">[:</a><span class="default"><font color="Maroon">X1</font>,<span class="p2">(<span class="default"><font color="Maroon">X2</font> <a href="xboole_0.html#K4">\</a> <font color="Maroon">Y2</font></span>)</span></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">X1</font>,<font color="Maroon">X2</font></span><a href="zfmisc_1.html#K2" target="_self">:]</a></span> <a href="xboole_0.html#K4">\</a> <span class="p1"><a href="zfmisc_1.html#K2" target="_self">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Y1</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">X2</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">Y2</font></span>)</span></span><a href="zfmisc_1.html#K2" target="_self">:]</a></span> &amp; <span class="p1"><a href="zfmisc_1.html#K2" target="_self">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Y1</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">X2</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">Y2</font></span>)</span></span><a href="zfmisc_1.html#K2" target="_self">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" target="_self">[:</a><span class="default"><font color="Maroon">Y1</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#E30"><i><font color="Green">Lemma47</font></i></a>, <a class="ref" href="xboole_1.html#T54">XBOOLE_1:54</a></i>;<br/>
<a NAME="E3:113"/><b>then </b><i><font color="Green">E26</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">X2</font></span><a href="zfmisc_1.html#K2" target="_self">:]</a></span> <a href="xboole_0.html#K4">\</a> <span class="p1"><a href="zfmisc_1.html#K2" target="_self">[:</a><span class="default"><font color="Maroon">Y1</font>,<font color="Maroon">Y2</font></span><a href="zfmisc_1.html#K2" target="_self">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" target="_self">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="xboole_0.html#K4">\</a> <font color="Maroon">Y1</font></span>)</span>,<font color="Maroon">X2</font></span><a href="zfmisc_1.html#K2" target="_self">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2" target="_self">[:</a><span class="default"><font color="Maroon">X1</font>,<span class="p2">(<span class="default"><font color="Maroon">X2</font> <a href="xboole_0.html#K4">\</a> <font color="Maroon">Y2</font></span>)</span></span><a href="zfmisc_1.html#K2" target="_self">:]</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T34">XBOOLE_1:34</a></i>;<br/>
<a NAME="E4:113"/>
<span class="p1"><a href="zfmisc_1.html#K2" target="_self">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="xboole_0.html#K4">\</a> <font color="Maroon">Y1</font></span>)</span>,<font color="Maroon">X2</font></span><a href="zfmisc_1.html#K2" target="_self">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2" target="_self">[:</a><span class="default"><font color="Maroon">X1</font>,<span class="p2">(<span class="default"><font color="Maroon">X2</font> <a href="xboole_0.html#K4">\</a> <font color="Maroon">Y2</font></span>)</span></span><a href="zfmisc_1.html#K2" target="_self">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" target="_self">[:</a><span class="default"><font color="Maroon">X1</font>,<font color="Maroon">X2</font></span><a href="zfmisc_1.html#K2" target="_self">:]</a></span> <a href="xboole_0.html#K4">\</a> <span class="p1"><a href="zfmisc_1.html#K2" target="_self">[:</a><span class="default"><font color="Maroon">Y1</font>,<font color="Maroon">Y2</font></span><a href="zfmisc_1.html#K2" target="_self">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="113_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</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="E1:113_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" target="_self">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="xboole_0.html#K4">\</a> <font color="Maroon">Y1</font></span>)</span>,<font color="Maroon">X2</font></span><a href="zfmisc_1.html#K2" target="_self">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2" target="_self">[:</a><span class="default"><font color="Maroon">X1</font>,<span class="p2">(<span class="default"><font color="Maroon">X2</font> <a href="xboole_0.html#K4">\</a> <font color="Maroon">Y2</font></span>)</span></span><a href="zfmisc_1.html#K2" target="_self">:]</a></span>
 ;<br/>

<div><i><font color="Green">E32</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="113_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:113_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" target="_self">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="xboole_0.html#K4">\</a> <font color="Maroon">Y1</font></span>)</span>,<font color="Maroon">X2</font></span><a href="zfmisc_1.html#K2" target="_self">:]</a></span>
 ;<br/><b>then consider </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><br/><a NAME="E3:113_1_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font> <a href="xboole_0.html#K4">\</a> <font color="Maroon">Y1</font>
 <b>and </b><br/><a NAME="E4:113_1_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X2</font>
 <b>and </b><br/><a NAME="E5:113_1_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i/>;<br/><a NAME="E6:113_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y1</font> )
 <b>by </b><i><a class="txt" href="zfmisc_1.html#E46"><i><font color="Green">Lemma54</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E7:113_1_1"/><b>then </b>
( <font color="Maroon">z</font> <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">X1</font>,<font color="Maroon">X2</font></span><a href="zfmisc_1.html#K2" target="_self">:]</a></span> &amp; not <font color="Maroon">z</font> <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">Y1</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="E8:113_1_1"/>
<font color="Maroon">z</font> <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">X1</font>,<font color="Maroon">X2</font></span><a href="zfmisc_1.html#K2" target="_self">:]</a></span> <a href="xboole_0.html#K4">\</a> <span class="p1"><a href="zfmisc_1.html#K2" target="_self">[:</a><span class="default"><font color="Maroon">Y1</font>,<font color="Maroon">Y2</font></span><a href="zfmisc_1.html#K2" target="_self">:]</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="113_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:113_1_2"/>
<font color="Maroon">z</font> <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">X1</font>,<span class="p2">(<span class="default"><font color="Maroon">X2</font> <a href="xboole_0.html#K4">\</a> <font color="Maroon">Y2</font></span>)</span></span><a href="zfmisc_1.html#K2" target="_self">:]</a></span>
 ;<br/><b>then consider </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><br/><a NAME="E3:113_1_2"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font>
 <b>and </b><br/><a NAME="E4:113_1_2"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X2</font> <a href="xboole_0.html#K4">\</a> <font color="Maroon">Y2</font>
 <b>and </b><br/><a NAME="E5:113_1_2"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i/>;<br/><a NAME="E6:113_1_2"/>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X2</font> &amp; not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y2</font> )
 <b>by </b><i><a class="txt" href="zfmisc_1.html#E3:113_1_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E7:113_1_2"/><b>then </b>
( <font color="Maroon">z</font> <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">X1</font>,<font color="Maroon">X2</font></span><a href="zfmisc_1.html#K2" target="_self">:]</a></span> &amp; not <font color="Maroon">z</font> <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">Y1</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#E46"><i><font color="Green">Lemma54</font></i></a>, <a class="txt" href="zfmisc_1.html#E4:113_1_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="E8:113_1_2"/>
<font color="Maroon">z</font> <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">X1</font>,<font color="Maroon">X2</font></span><a href="zfmisc_1.html#K2" target="_self">:]</a></span> <a href="xboole_0.html#K4">\</a> <span class="p1"><a href="zfmisc_1.html#K2" target="_self">[:</a><span class="default"><font color="Maroon">Y1</font>,<font color="Maroon">Y2</font></span><a href="zfmisc_1.html#K2" target="_self">:]</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:113_1"/>
<font color="Maroon">z</font> <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">X1</font>,<font color="Maroon">X2</font></span><a href="zfmisc_1.html#K2" target="_self">:]</a></span> <a href="xboole_0.html#K4">\</a> <span class="p1"><a href="zfmisc_1.html#K2" target="_self">[:</a><span class="default"><font color="Maroon">Y1</font>,<font color="Maroon">Y2</font></span><a href="zfmisc_1.html#K2" target="_self">:]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T26" target="_self">Th26</a>, <a class="txt" href="zfmisc_1.html#E45"><i><font color="Green">Lemma53</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:113"/>
<span class="p1"><a href="zfmisc_1.html#K2" target="_self">[:</a><span class="default"><font color="Maroon">X1</font>,<font color="Maroon">X2</font></span><a href="zfmisc_1.html#K2" target="_self">:]</a></span> <a href="xboole_0.html#K4">\</a> <span class="p1"><a href="zfmisc_1.html#K2" target="_self">[:</a><span class="default"><font color="Maroon">Y1</font>,<font color="Maroon">Y2</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"><span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="xboole_0.html#K4">\</a> <font color="Maroon">Y1</font></span>)</span>,<font color="Maroon">X2</font></span><a href="zfmisc_1.html#K2" target="_self">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2" target="_self">[:</a><span class="default"><font color="Maroon">X1</font>,<span class="p2">(<span class="default"><font color="Maroon">X2</font> <a href="xboole_0.html#K4">\</a> <font color="Maroon">Y2</font></span>)</span></span><a href="zfmisc_1.html#K2" target="_self">:]</a></span>
 <b>by </b><i><a class="txt" href="zfmisc_1.html#E39"><i><font color="Green">Lemma51</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div>
