<?xml version="1.0"?>
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<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>, <font color="Olive">Y3</font>, <font color="Olive">Y4</font>, <font color="Olive">Y5</font>, <font color="Olive">Y6</font>, <font color="Olive">Y7</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">Y4</font> &amp; <font color="Olive">Y4</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y5</font> &amp; <font color="Olive">Y5</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y6</font> &amp; <font color="Olive">Y6</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y7</font> &amp; <font color="Olive">Y7</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="E2:1"/><i><font color="Green">E51</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>, <font color="Olive">Y2</font>, <font color="Olive">Y3</font>, <font color="Olive">Y4</font>, <font color="Olive">Y5</font>, <font color="Olive">Y6</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">Y4</font> &amp; <font color="Olive">Y4</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y5</font> &amp; <font color="Olive">Y5</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y6</font> &amp; <font color="Olive">Y6</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="E4:1"/><i><font color="Green">E53</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> ex <font color="Olive">Y1</font>, <font color="Olive">Y2</font>, <font color="Olive">Y3</font>, <font color="Olive">Y4</font>, <font color="Olive">Y5</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">Y4</font> &amp; <font color="Olive">Y4</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y5</font> &amp; <font color="Olive">Y5</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">Z3</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:1"/><i><font color="Green">E55</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>defpred </b><font color="Maroon">S<sub>4</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">Y1</font>, <font color="Olive">Y2</font>, <font color="Olive">Y3</font>, <font color="Olive">Y4</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">Y4</font> &amp; <font color="Olive">Y4</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">Z4</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:1"/><i><font color="Green">E57</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">Z4</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> <span class="p3">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">X</font></span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon">S<sub>4</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>5</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b> 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="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">Z5</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E10:1"/><i><font color="Green">E59</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">Z5</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> <span class="p3">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p4">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">X</font></span>)</span></span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon">S<sub>5</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>6</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">Z6</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E12:1"/><i><font color="Green">E61</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">Z6</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> <span class="p3">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p4">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p5">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">X</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon">S<sub>6</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>7</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">Z7</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E14:1"/><i><font color="Green">E63</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">Z7</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> <span class="p3">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p4">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p5">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p0">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">X</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon">S<sub>7</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>8</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">Z8</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E16:1"/><i><font color="Green">E65</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">Z8</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> <span class="p3">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p4">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p5">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p0">(<span class="default"><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></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon">S<sub>8</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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><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></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z3</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z4</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z5</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z6</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z7</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z8</font>;<br/>
<i><font color="Green">E67</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><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></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z3</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z4</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z5</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z6</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z7</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z8</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</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></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z3</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z4</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z5</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z6</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z7</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z8</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <span class="p0">(<span class="default"><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></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z4</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z5</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z6</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z7</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z8</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><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></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z4</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z5</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z6</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z7</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z8</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><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></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z4</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z5</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z6</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z7</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z8</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><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></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z4</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z5</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z6</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z7</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z8</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><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></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z4</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z5</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z6</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z7</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z8</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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><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></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z4</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z5</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z6</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z7</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z8</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="E18:1"/>
<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="E19:1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><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></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z3</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z4</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z5</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z6</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z7</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z8</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="E21:1"/><i><font color="Green">E68</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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><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></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z3</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z4</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z5</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z6</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z7</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z8</font>
 <b>and </b><br/><a NAME="E22:1"/><i><font color="Green">E69</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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><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></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z3</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z4</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z5</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z6</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z7</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z8</font>
 <b>by </b><i><a class="ref" href="mcart_1.html#T1">MCART_1:1</a></i>;<br/>
<b>assume </b><a NAME="E23:1"/><i><font color="Green">E70</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>, <font color="Olive">Y4</font>, <font color="Olive">Y5</font>, <font color="Olive">Y6</font>, <font color="Olive">Y7</font>, <font color="Olive">Y8</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">Y4</font> &amp; <font color="Olive">Y4</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y5</font> &amp; <font color="Olive">Y5</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y6</font> &amp; <font color="Olive">Y6</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y7</font> &amp; <font color="Olive">Y7</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y8</font> &amp; <font color="Olive">Y8</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="1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:1_2"/><i><font color="Green">E71</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> , <font color="Maroon">Y4</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">Y5</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">Y6</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">Y7</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">Y8</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:1_2"/><i><font color="Green">E72</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">Y4</font> &amp; <font color="Maroon">Y4</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y5</font> &amp; <font color="Maroon">Y5</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y6</font> &amp; <font color="Maroon">Y6</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y7</font> &amp; <font color="Maroon">Y7</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y8</font> &amp; <font color="Maroon">Y8</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="E4:1_2"/>
( <font color="Maroon">Y8</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="E5:1_2"/><b>then </b>
<font color="Maroon">Y8</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z1</font>
 <b>by </b><i>, </i>;<br/><a NAME="E6:1_2"/><b>then </b>
<font color="Maroon">Y8</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="E7:1_2"/><b>then </b>
<font color="Maroon">Y8</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="E8:1_2"/><b>then </b>
<font color="Maroon">Y8</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/><a NAME="E9:1_2"/><b>then </b>
<font color="Maroon">Y8</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z4</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E10:1_2"/><b>then </b>
<font color="Maroon">Y8</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<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>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z4</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z5</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E11:1_2"/><b>then </b>
<font color="Maroon">Y</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<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>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z4</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z5</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><a NAME="E12:1_2"/><b>then </b>
<font color="Maroon">Y</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<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>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z4</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z5</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z6</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T70">XBOOLE_1:70</a></i>;<br/><a NAME="E13:1_2"/><b>then </b>
<font color="Maroon">Y</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<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>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z4</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z5</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z6</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z7</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T70">XBOOLE_1:70</a></i>;<br/><b>hence </b><a NAME="E14:1_2"/>
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="E25:1"/><b>then </b>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<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> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z4</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z5</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z6</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z7</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z8</font>
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E26:1"/><b>then </b>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">Z1</font> <a href="xboole_0.html#K2">\/</a> <span class="p0">(<span class="default"><font color="Maroon">Z2</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z3</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z4</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z5</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z6</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z7</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z8</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>;<br/>
<a NAME="E27:1"/><b>then </b>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Z1</font> <a href="xboole_0.html#K2">\/</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">Z2</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z3</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z4</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z5</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z6</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z7</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z8</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>;<br/>
<a NAME="E28:1"/><b>then </b>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">Z1</font> <a href="xboole_0.html#K2">\/</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">Z2</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z3</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z4</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z5</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z6</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z7</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z8</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>;<br/>
<a NAME="E29:1"/><b>then </b>
<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">Z1</font> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">Z2</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z3</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z4</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z5</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z6</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z7</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z8</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>;<br/>
<a NAME="E30:1"/><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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">Z2</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z3</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z4</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z5</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z6</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z7</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z8</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>;<br/>
<a NAME="E31:1"/><b>then </b><i><font color="Green">E73</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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">Z2</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z3</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z4</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z5</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z6</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z7</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z8</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="1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:1_3"/><i><font color="Green">E74</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> , <font color="Maroon">Y3</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">Y4</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">Y5</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">Y6</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">Y7</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:1_3"/><i><font color="Green">E75</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">Y4</font> &amp; <font color="Maroon">Y4</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y5</font> &amp; <font color="Maroon">Y5</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y6</font> &amp; <font color="Maroon">Y6</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y7</font> &amp; <font color="Maroon">Y7</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="E4:1_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="E5:1_3"/><b>then </b>
<font color="Maroon">Y7</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="E6:1_3"/><b>then </b>
<font color="Maroon">Y7</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z2</font>
 <b>by </b><i>, </i>;<br/><a NAME="E7:1_3"/><b>then </b>
<font color="Maroon">Y7</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="E8:1_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/><a NAME="E9:1_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"><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_1.html#T70">XBOOLE_1:70</a></i>;<br/><a NAME="E10:1_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"><span class="p2">(<span class="default"><span class="p3">(<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>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z4</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T70">XBOOLE_1:70</a></i>;<br/><a NAME="E11:1_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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<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>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z4</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z5</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T70">XBOOLE_1:70</a></i>;<br/><a NAME="E12:1_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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<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>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z4</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z5</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z6</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T70">XBOOLE_1:70</a></i>;<br/><a NAME="E13:1_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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<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>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z4</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z5</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z6</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z7</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T70">XBOOLE_1:70</a></i>;<br/><b>hence </b><a NAME="E14:1_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="E33:1"/><b>then </b>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">Z2</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z3</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z4</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z5</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z6</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z7</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z8</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E34:1"/><b>then </b>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Z2</font> <a href="xboole_0.html#K2">\/</a> <span class="p5">(<span class="default"><font color="Maroon">Z3</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z4</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z5</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z6</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z7</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z8</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>;<br/>
<a NAME="E35:1"/><b>then </b>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">Z2</font> <a href="xboole_0.html#K2">\/</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">Z3</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z4</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z5</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z6</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z7</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z8</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>;<br/>
<a NAME="E36:1"/><b>then </b>
<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">Z2</font> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">Z3</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z4</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z5</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z6</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z7</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z8</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>;<br/>
<a NAME="E37:1"/><b>then </b>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">Z2</font> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">Z3</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z4</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z5</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z6</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z7</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z8</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>;<br/>
<a NAME="E38:1"/><b>then </b><i><font color="Green">E76</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">Z3</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z4</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z5</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z6</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z7</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z8</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="1_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:1_4"/><i><font color="Green">E77</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> , <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> , <font color="Maroon">Y4</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">Y5</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">Y6</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:1_4"/><i><font color="Green">E78</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">Y4</font> &amp; <font color="Maroon">Y4</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y5</font> &amp; <font color="Maroon">Y5</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y6</font> &amp; <font color="Maroon">Y6</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="E4:1_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="E5:1_4"/><b>then </b>
<font color="Maroon">Y6</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="E6:1_4"/><b>then </b>
<font color="Maroon">Y6</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z3</font>
 <b>by </b><i>, </i>;<br/><a NAME="E7:1_4"/><b>then </b>
<font color="Maroon">Y6</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/><a NAME="E8:1_4"/><b>then </b>
<font color="Maroon">Y6</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z4</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E9:1_4"/><b>then </b>
<font color="Maroon">Y6</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<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>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z4</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z5</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E10:1_4"/><b>then </b>
<font color="Maroon">Y6</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<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>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z4</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z5</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z6</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E11:1_4"/><b>then </b>
<font color="Maroon">Y6</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<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>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z4</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z5</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z6</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z7</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E12:1_4"/><b>then </b>
<font color="Maroon">Y6</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><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></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z3</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z4</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z5</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z6</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z7</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z8</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="E13:1_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="E40:1"/><b>then </b>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Z3</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z4</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z5</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z6</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z7</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z8</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E41:1"/><b>then </b>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">Z3</font> <a href="xboole_0.html#K2">\/</a> <span class="p4">(<span class="default"><font color="Maroon">Z4</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z5</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z6</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z7</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z8</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>;<br/>
<a NAME="E42:1"/><b>then </b>
<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">Z3</font> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Z4</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z5</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z6</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z7</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z8</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>;<br/>
<a NAME="E43:1"/><b>then </b>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">Z3</font> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Z4</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z5</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z6</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z7</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z8</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>;<br/>
<a NAME="E44:1"/><b>then </b><i><font color="Green">E79</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z3</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Z4</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z5</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z6</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z7</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z8</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="1_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:1_5"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z3</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> , <font color="Maroon">Y4</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">Y5</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:1_5"/><i><font color="Green">E81</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">Y4</font> &amp; <font color="Maroon">Y4</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y5</font> &amp; <font color="Maroon">Y5</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="E4:1_5"/>
<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> <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>, </i>;<br/><a NAME="E5:1_5"/><b>then </b>
<font color="Maroon">Y5</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> <span class="p3">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">X</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><a NAME="E6:1_5"/><b>then </b>
<font color="Maroon">Y5</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z4</font>
 <b>by </b><i>, </i>;<br/><a NAME="E7:1_5"/><b>then </b>
<font color="Maroon">Y5</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z4</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E8:1_5"/><b>then </b>
<font color="Maroon">Y5</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<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>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z4</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z5</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E9:1_5"/><b>then </b>
<font color="Maroon">Y5</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<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>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z4</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z5</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z6</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E10:1_5"/><b>then </b>
<font color="Maroon">Y5</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<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>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z4</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z5</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z6</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z7</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E11:1_5"/><b>then </b>
<font color="Maroon">Y5</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><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></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z3</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z4</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z5</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z6</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z7</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z8</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="E12:1_5"/>
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="E46:1"/><b>then </b>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">Z4</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z5</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z6</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z7</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z8</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E47:1"/><b>then </b>
<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">Z4</font> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><font color="Maroon">Z5</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z6</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z7</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z8</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>;<br/>
<a NAME="E48:1"/><b>then </b>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">Z4</font> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">Z5</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z6</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z7</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z8</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>;<br/>
<a NAME="E49:1"/><b>then </b><i><font color="Green">E82</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z4</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">Z5</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z6</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z7</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z8</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="1_6"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:1_6"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z4</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> , <font color="Maroon">Y4</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:1_6"/><i><font color="Green">E84</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">Y4</font> &amp; <font color="Maroon">Y4</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="E4:1_6"/>
<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> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">X</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E5:1_6"/><b>then </b>
<font color="Maroon">Y4</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> <span class="p3">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p4">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">X</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><a NAME="E6:1_6"/><b>then </b>
<font color="Maroon">Y4</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z5</font>
 <b>by </b><i>, </i>;<br/><a NAME="E7:1_6"/><b>then </b>
<font color="Maroon">Y4</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<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>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z4</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z5</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E8:1_6"/><b>then </b>
<font color="Maroon">Y4</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<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>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z4</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z5</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z6</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E9:1_6"/><b>then </b>
<font color="Maroon">Y4</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<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>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z4</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z5</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z6</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z7</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E10:1_6"/><b>then </b>
<font color="Maroon">Y4</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><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></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z3</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z4</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z5</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z6</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z7</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z8</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="E11:1_6"/>
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="E51:1"/><b>then </b>
<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">Z5</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z6</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z7</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z8</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E52:1"/><b>then </b>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">Z5</font> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><font color="Maroon">Z6</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z7</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z8</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>;<br/>
<a NAME="E53:1"/><b>then </b><i><font color="Green">E85</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z5</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Z6</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z7</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z8</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="1_7"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:1_7"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z5</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="E3:1_7"/><i><font color="Green">E87</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; <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="E4:1_7"/>
<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> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p4">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">X</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E5:1_7"/><b>then </b>
<font color="Maroon">Y3</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> <span class="p3">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p4">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p5">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">X</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><a NAME="E6:1_7"/><b>then </b>
<font color="Maroon">Y3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z6</font>
 <b>by </b><i>, </i>;<br/><a NAME="E7:1_7"/><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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<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>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z4</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z5</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z6</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E8:1_7"/><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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<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>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z4</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z5</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z6</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z7</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E9:1_7"/><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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><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></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z3</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z4</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z5</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z6</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z7</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z8</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="E10:1_7"/>
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="E55:1"/><b>then </b>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">Z6</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z7</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z8</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E56:1"/><b>then </b><i><font color="Green">E88</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z6</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">Z7</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z8</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="1_8"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:1_8"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z6</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="E3:1_8"/><i><font color="Green">E90</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="E4:1_8"/>
<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> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p4">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p5">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">X</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E5:1_8"/><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> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p4">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p5">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p0">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">X</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><a NAME="E6:1_8"/><b>then </b>
<font color="Maroon">Y2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z7</font>
 <b>by </b><i>, </i>;<br/><a NAME="E7:1_8"/><b>then </b>
<font color="Maroon">Y2</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<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>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z4</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z5</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z6</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z7</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E8:1_8"/><b>then </b>
<font color="Maroon">Y2</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><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></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z3</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z4</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z5</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z6</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z7</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z8</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="E9:1_8"/>
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="E58:1"/><b>then </b><i><font color="Green">E91</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z7</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z8</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="1_9"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:1_9"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z7</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="E3:1_9"/><i><font color="Green">E93</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="E4:1_9"/>
<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> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p4">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p5">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p0">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">X</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E5:1_9"/><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> <span class="p3">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p4">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p5">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p0">(<span class="default"><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></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><a NAME="E6:1_9"/><b>then </b>
<font color="Maroon">Y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z8</font>
 <b>by </b><i>, </i>;<br/><a NAME="E7:1_9"/><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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><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></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z3</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z4</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z5</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z6</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z7</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z8</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="E8:1_9"/>
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="E60:1"/><b>then </b>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z8</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E61:1"/><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="E62:1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xboole_1.html#T70">XBOOLE_1:70</a></i>;<br/>


</div>
