<?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/>





<a NAME="E1:107"/><i><font color="Green">E26</font></i>: 
for <font color="Olive">z</font>, <font color="Olive">X1</font>, <font color="Olive">X2</font>, <font color="Olive">Y1</font>, <font color="Olive">Y2</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">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="Olive">X1</font>,<font color="Olive">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="Olive">Y1</font>,<font color="Olive">Y2</font></span><a href="zfmisc_1.html#K2" target="_self">:]</a></span> holds <br/> ex <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="107_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>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/>









<b>assume </b><a NAME="E1:107_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#K2">\/</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>
 ;<br/>

<a NAME="E2:107_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> or <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="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E3:107_1"/>
 ex <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <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="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T9" target="_self">Th9</a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E2:107"/><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#K2">\/</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#K2">\/</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="107_2"><b>proof </b></a><div class="add">

<a NAME="E1:107_2"/><i><font color="Green">E32</font></i>: 
( ( for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">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">X</font>,<font color="Maroon">Z</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">Y</font>,<font color="Maroon">Z</font></span><a href="zfmisc_1.html#K2" target="_self">:]</a></span> holds <br/> ex <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <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> ) &amp; ( for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">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">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Y</font></span>)</span>,<font color="Maroon">Z</font></span><a href="zfmisc_1.html#K2" target="_self">:]</a></span> holds <br/> ex <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <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> ) )
 
<b>by </b><i><a class="txt" href="zfmisc_1.html#E34"><i><font color="Green">Lemma49</font></i></a>, <a class="ref" href="zfmisc_1.html#T9" target="_self">Th9</a></i>;<br/>
<a NAME="E2:107_2"/>
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#K2">\/</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#K2">\/</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="107_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>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>thus </b><a NAME="E1:107_2_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#K2">\/</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#K2">\/</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="107_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:107_2_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#K2">\/</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="E2:107_2_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#K2">\/</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="E3:107_2_1_1"/><b>then </b>
( ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> or <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#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E4:107_2_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> or <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="E5:107_2_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#K2">\/</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#D2">XBOOLE_0:def 2</a></i>;<br/>


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

<b>assume </b><a NAME="E2:107_2_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#K2">\/</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="E3:107_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">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> or <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#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E4:107_2_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> ) or ( <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="txt" href="zfmisc_1.html#E4"><i><font color="Green">Lemma28</font></i></a></i>;<br/>
<a NAME="E5:107_2_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#K2">\/</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#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E6:107_2_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#K2">\/</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="E3:107_2"/>
<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#K2">\/</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#K2">\/</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#E39"><i><font color="Green">Lemma51</font></i></a>, <a class="txt" href="zfmisc_1.html#E25"><i><font color="Green">Lemma44</font></i></a></i>;<br/>


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

<a NAME="E3:107"/><i><font color="Green">E33</font></i>: 
( ( for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">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">Z</font>,<font color="Maroon">X</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">Z</font>,<font color="Maroon">Y</font></span><a href="zfmisc_1.html#K2" target="_self">:]</a></span> holds <br/> ex <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <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> ) &amp; ( for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">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">Z</font>,<span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Y</font></span>)</span></span><a href="zfmisc_1.html#K2" target="_self">:]</a></span> holds <br/> ex <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <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> ) )
 
<b>by </b><i><a class="txt" href="zfmisc_1.html#E34"><i><font color="Green">Lemma49</font></i></a>, <a class="ref" href="zfmisc_1.html#T9" target="_self">Th9</a></i>;<br/>
<a NAME="E4:107"/>
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#K2">\/</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#K2">\/</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="107_3"><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="E1:107_3"/>
( ( <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#K2">\/</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#K2">\/</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#K2">\/</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#K2">\/</a> <font color="Maroon">Y</font></span>)</span></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">X</font>,<font color="Maroon">Z</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">Y</font>,<font color="Maroon">Z</font></span><a href="zfmisc_1.html#K2" target="_self">:]</a></span> or <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> or <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> or <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#K2">\/</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; ( ( 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">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> ) or <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> or <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; ( ( 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">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> ) or <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> or <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#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E2:107_3"/>
( <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#K2">\/</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#K2">\/</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#E36"><i><font color="Green">Lemma50</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="E5:107"/>
<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#K2">\/</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#K2">\/</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="txt" href="zfmisc_1.html#E25"><i><font color="Green">Lemma44</font></i></a></i>;<br/>


</div>
