<?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>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">Y1</font>, <font color="Olive">Y2</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">Y1</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y2</font> &amp; <font color="Olive">Y2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">Y1</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">X</font> );<br/>
<b>consider </b><font color="Maroon">Z1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:4"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z1</font> iff ( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">X</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">Y</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">Y1</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">Y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">Y1</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">X</font> );<br/>
<b>consider </b><font color="Maroon">Z2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:4"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z2</font> iff ( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">X</font></span>)</span> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">Y</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>1</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">X</font>;<br/>
<b>consider </b><font color="Maroon">Z3</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:4"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z3</font> iff ( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">X</font></span>)</span></span>)</span> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">Y</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<b>set </b><font color="Maroon">V</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z3</font>;<br/>
<i><font color="Green">E44</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z3</font> = 
<span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><font color="Maroon">Z1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z2</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z3</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>
<br/>.= 
<font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Z1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z3</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>
;<br/>
<b>assume </b><a NAME="E8:4"/>
<font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<a NAME="E9:4"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T15">XBOOLE_1:15</a></i>;<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="E11:4"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z3</font>
 <b>and </b><br/><a NAME="E12:4"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">Y</font> <a href="xboole_0.html#R1">misses</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z3</font>
 <b>by </b><i/>;<br/>
<b>assume </b><a NAME="E13:4"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>   holds <br/>(  not <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> or  ex <font color="Olive">Y1</font>, <font color="Olive">Y2</font>, <font color="Olive">Y3</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">Y1</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y2</font> &amp; <font color="Olive">Y2</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y3</font> &amp; <font color="Olive">Y3</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> &amp; not <font color="Olive">Y1</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">X</font> ) )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:4_2"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><b>then consider </b><font color="Maroon">Y1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">Y2</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">Y3</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:4_2"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">Y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y2</font> &amp; <font color="Maroon">Y2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y3</font> &amp; <font color="Maroon">Y3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; not <font color="Maroon">Y1</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">X</font> )
 <b>by </b><i/>;<br/><a NAME="E4:4_2"/>
( <font color="Maroon">Y3</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">X</font> &amp; <font color="Maroon">Y1</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">X</font> )
 <b>by </b><i>, , <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><a NAME="E5:4_2"/><b>then </b>
<font color="Maroon">Y3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z1</font>
 <b>by </b><i>, </i>;<br/><a NAME="E6:4_2"/><b>then </b>
<font color="Maroon">Y3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z1</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E7:4_2"/><b>then </b>
<font color="Maroon">Y3</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z2</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E8:4_2"/><b>then </b>
<font color="Maroon">Y3</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z3</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="E9:4_2"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div>
<a NAME="E15:4"/><b>then </b>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">Z1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z3</font>
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E16:4"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z1</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">Z2</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z3</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:4_3"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z1</font>
 ;<br/><b>then consider </b><font color="Maroon">Y1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">Y2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:4_3"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">Y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y2</font> &amp; <font color="Maroon">Y2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">Y1</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">X</font> )
 <b>by </b><i/>;<br/><a NAME="E4:4_3"/>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">X</font>
 <b>by </b><i>, </i>;<br/><a NAME="E5:4_3"/><b>then </b>
<font color="Maroon">Y2</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">X</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><a NAME="E6:4_3"/><b>then </b>
<font color="Maroon">Y2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z2</font>
 <b>by </b><i>, </i>;<br/><a NAME="E7:4_3"/><b>then </b>
<font color="Maroon">Y2</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z2</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E8:4_3"/><b>then </b>
<font color="Maroon">Y</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z2</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><b>hence </b><a NAME="E9:4_3"/>
contradiction
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T70">XBOOLE_1:70</a></i>;<br/></div><b>end;</b></div>
<a NAME="E18:4"/><b>then </b><i><font color="Green">E57</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z2</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z3</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:4_4"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z2</font>
 ;<br/><b>then consider </b><font color="Maroon">Y1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:4_4"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">Y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">Y1</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">X</font> )
 <b>by </b><i/>;<br/><a NAME="E4:4_4"/>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">X</font></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E5:4_4"/><b>then </b>
<font color="Maroon">Y1</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">X</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><a NAME="E6:4_4"/><b>then </b>
<font color="Maroon">Y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z3</font>
 <b>by </b><i>, </i>;<br/><a NAME="E7:4_4"/><b>then </b>
<font color="Maroon">Y1</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z3</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="E8:4_4"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div>
<a NAME="E20:4"/><b>then </b>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z3</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E21:4"/><b>then </b>
<font color="Maroon">Y</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">X</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E22:4"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xboole_1.html#T70">XBOOLE_1:70</a></i>;<br/>


</div>
