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





<b>thus </b><a NAME="E1:112"/><i><font color="Green">E26</font></i>: 
<span class="p1"><a href="zfmisc_1.html#K2" target="_self">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K4">\</a> <font color="Maroon">Y</font></span>)</span>,<font color="Maroon">Z</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">X</font>,<font color="Maroon">Z</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">Y</font>,<font color="Maroon">Z</font></span><a href="zfmisc_1.html#K2" target="_self">:]</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="112_1"><b>proof </b></a><div class="add">

<a NAME="E1:112_1"/><i><font color="Green">E31</font></i>: 
( <span class="p1"><a href="zfmisc_1.html#K2" target="_self">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K4">\</a> <font color="Maroon">Y</font></span>)</span>,<font color="Maroon">Z</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">X</font> <a href="xboole_0.html#K4">\</a> <font color="Maroon">Y</font></span>)</span>,<font color="Maroon">Z</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">X</font>,<font color="Maroon">Z</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">Y</font>,<font color="Maroon">Z</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"><font color="Maroon">X</font>,<font color="Maroon">Z</font></span><a href="zfmisc_1.html#K2" target="_self">:]</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/>
<a NAME="E2:112_1"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">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"><span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K4">\</a> <font color="Maroon">Y</font></span>)</span>,<font color="Maroon">Z</font></span><a href="zfmisc_1.html#K2" target="_self">:]</a></span> iff <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">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">X</font>,<font color="Maroon">Z</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">Y</font>,<font color="Maroon">Z</font></span><a href="zfmisc_1.html#K2" target="_self">:]</a></span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="112_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>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>thus </b><a NAME="E1:112_1_1"/>
( <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> <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">X</font> <a href="xboole_0.html#K4">\</a> <font color="Maroon">Y</font></span>)</span>,<font color="Maroon">Z</font></span><a href="zfmisc_1.html#K2" target="_self">:]</a></span> implies <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> <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">X</font>,<font color="Maroon">Z</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">Y</font>,<font color="Maroon">Z</font></span><a href="zfmisc_1.html#K2" target="_self">:]</a></span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="112_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:112_1_1_1"/>
<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> <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">X</font> <a href="xboole_0.html#K4">\</a> <font color="Maroon">Y</font></span>)</span>,<font color="Maroon">Z</font></span><a href="zfmisc_1.html#K2" target="_self">:]</a></span>
 ;<br/>

<a NAME="E2:112_1_1_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="xboole_0.html#K4">\</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> )
 
<b>by </b><i><a class="txt" href="zfmisc_1.html#E4"><i><font color="Green">Lemma28</font></i></a></i>;<br/>
<a NAME="E3:112_1_1_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E4:112_1_1_1"/><b>then </b>
( <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> <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">X</font>,<font color="Maroon">Z</font></span><a href="zfmisc_1.html#K2" target="_self">:]</a></span> &amp; not <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> <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">Y</font>,<font color="Maroon">Z</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="E5:112_1_1_1"/>
<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> <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">X</font>,<font color="Maroon">Z</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">Y</font>,<font color="Maroon">Z</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>assume </b><a NAME="E2:112_1_1"/>
<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> <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">X</font>,<font color="Maroon">Z</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">Y</font>,<font color="Maroon">Z</font></span><a href="zfmisc_1.html#K2" target="_self">:]</a></span>
 ;<br/>

<a NAME="E3:112_1_1"/><b>then </b>
( <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> <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">X</font>,<font color="Maroon">Z</font></span><a href="zfmisc_1.html#K2" target="_self">:]</a></span> &amp; not <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> <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">Y</font>,<font color="Maroon">Z</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/>
<a NAME="E4:112_1_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> &amp; (  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> or  not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> ) )
 
<b>by </b><i><a class="txt" href="zfmisc_1.html#E4"><i><font color="Green">Lemma28</font></i></a></i>;<br/>
<a NAME="E5:112_1_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="xboole_0.html#K4">\</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E6:112_1_1"/>
<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> <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">X</font> <a href="xboole_0.html#K4">\</a> <font color="Maroon">Y</font></span>)</span>,<font color="Maroon">Z</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/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:112_1"/>
<span class="p1"><a href="zfmisc_1.html#K2" target="_self">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K4">\</a> <font color="Maroon">Y</font></span>)</span>,<font color="Maroon">Z</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">X</font>,<font color="Maroon">Z</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">Y</font>,<font color="Maroon">Z</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#E23"><i><font color="Green">Lemma43</font></i></a></i>;<br/>


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

<a NAME="E2:112"/><i><font color="Green">E32</font></i>: 
( <span class="p1"><a href="zfmisc_1.html#K2" target="_self">[:</a><span class="default"><font color="Maroon">Z</font>,<span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K4">\</a> <font color="Maroon">Y</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">Z</font>,<span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K4">\</a> <font color="Maroon">Y</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"><font color="Maroon">Z</font>,<font color="Maroon">X</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">Z</font>,<font color="Maroon">Y</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"><font color="Maroon">Z</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2" target="_self">:]</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/>
<a NAME="E3:112"/>
for <font color="Olive">y</font>, <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">y</font>,<font color="Olive">x</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">Z</font>,<span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K4">\</a> <font color="Maroon">Y</font></span>)</span></span><a href="zfmisc_1.html#K2" target="_self">:]</a></span> iff <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">y</font>,<font color="Olive">x</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">Z</font>,<font color="Maroon">X</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">Z</font>,<font color="Maroon">Y</font></span><a href="zfmisc_1.html#K2" target="_self">:]</a></span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="112_2"><b>proof </b></a><div class="add">

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



<a NAME="E1:112_2"/>
( ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</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">Z</font>,<span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K4">\</a> <font color="Maroon">Y</font></span>)</span></span><a href="zfmisc_1.html#K2" target="_self">:]</a></span> implies <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> <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">X</font> <a href="xboole_0.html#K4">\</a> <font color="Maroon">Y</font></span>)</span>,<font color="Maroon">Z</font></span><a href="zfmisc_1.html#K2" target="_self">:]</a></span> ) &amp; ( <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> <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">X</font> <a href="xboole_0.html#K4">\</a> <font color="Maroon">Y</font></span>)</span>,<font color="Maroon">Z</font></span><a href="zfmisc_1.html#K2" target="_self">:]</a></span> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</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">Z</font>,<span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K4">\</a> <font color="Maroon">Y</font></span>)</span></span><a href="zfmisc_1.html#K2" target="_self">:]</a></span> ) &amp; ( <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> <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">X</font>,<font color="Maroon">Z</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">Y</font>,<font color="Maroon">Z</font></span><a href="zfmisc_1.html#K2" target="_self">:]</a></span> implies ( <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> <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">X</font>,<font color="Maroon">Z</font></span><a href="zfmisc_1.html#K2" target="_self">:]</a></span> &amp; not <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> <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">Y</font>,<font color="Maroon">Z</font></span><a href="zfmisc_1.html#K2" target="_self">:]</a></span> ) ) &amp; ( <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> <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">X</font>,<font color="Maroon">Z</font></span><a href="zfmisc_1.html#K2" target="_self">:]</a></span> &amp; not <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> <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">Y</font>,<font color="Maroon">Z</font></span><a href="zfmisc_1.html#K2" target="_self">:]</a></span> implies <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> <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">X</font>,<font color="Maroon">Z</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">Y</font>,<font color="Maroon">Z</font></span><a href="zfmisc_1.html#K2" target="_self">:]</a></span> ) &amp; ( <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> <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">X</font>,<font color="Maroon">Z</font></span><a href="zfmisc_1.html#K2" target="_self">:]</a></span> &amp; not <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> <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">Y</font>,<font color="Maroon">Z</font></span><a href="zfmisc_1.html#K2" target="_self">:]</a></span> implies ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</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">Z</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2" target="_self">:]</a></span> &amp; not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</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">Z</font>,<font color="Maroon">Y</font></span><a href="zfmisc_1.html#K2" target="_self">:]</a></span> ) ) &amp; ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</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">Z</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2" target="_self">:]</a></span> &amp; not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</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">Z</font>,<font color="Maroon">Y</font></span><a href="zfmisc_1.html#K2" target="_self">:]</a></span> implies ( <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> <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">X</font>,<font color="Maroon">Z</font></span><a href="zfmisc_1.html#K2" target="_self">:]</a></span> &amp; not <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> <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">Y</font>,<font color="Maroon">Z</font></span><a href="zfmisc_1.html#K2" target="_self">:]</a></span> ) ) )
 
<b>by </b><i><a class="txt" href="zfmisc_1.html#E21"><i><font color="Green">Lemma42</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E2:112_2"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</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">Z</font>,<span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K4">\</a> <font color="Maroon">Y</font></span>)</span></span><a href="zfmisc_1.html#K2" target="_self">:]</a></span> iff <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</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">Z</font>,<font color="Maroon">X</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">Z</font>,<font color="Maroon">Y</font></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#D4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:112"/>
<span class="p1"><a href="zfmisc_1.html#K2" target="_self">[:</a><span class="default"><font color="Maroon">Z</font>,<span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K4">\</a> <font color="Maroon">Y</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">Z</font>,<font color="Maroon">X</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">Z</font>,<font color="Maroon">Y</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#E23"><i><font color="Green">Lemma43</font></i></a></i>;<br/>


</div>
