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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>, <font color="Olive">Y8</font>, <font color="Olive">Y9</font>, <font color="Olive">YA</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">Y9</font> &amp; <font color="Olive">Y9</font> <a href="hidden.html#R2">in</a> <font color="Olive">YA</font> &amp; <font color="Olive">YA</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="E2:2"/><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">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>, <font color="Olive">Y7</font>, <font color="Olive">Y8</font>, <font color="Olive">Y9</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">Y9</font> &amp; <font color="Olive">Y9</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="E4:2"/><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">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>, <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="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="E6:2"/><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">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>, <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">Z4</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:2"/><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">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>, <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">Z5</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E10:2"/><i><font color="Green">E67</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>, <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">Z6</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E12:2"/><i><font color="Green">E68</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>, <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">Z7</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E14:2"/><i><font color="Green">E70</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> 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">Z8</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E16:2"/><i><font color="Green">E72</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>defpred </b><font color="Maroon">S<sub>9</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">Z9</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E18:2"/><i><font color="Green">E74</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">Z9</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> <span class="p2">(<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></span>)</span> &amp; <font color="Maroon">S<sub>9</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>10</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">ZA</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E20:2"/><i><font color="Green">E76</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">ZA</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> <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></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon">S<sub>10</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>11</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">ZB</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E22:2"/><i><font color="Green">E78</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">ZB</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> <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></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon">S<sub>11</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"><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></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> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z9</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">ZA</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">ZB</font>;<br/>
<i><font color="Green">E79</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"><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></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> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z9</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">ZA</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">ZB</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"><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"><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></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z9</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">ZA</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">ZB</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"><span class="p0">(<span class="default"><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"><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></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z9</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">ZA</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">ZB</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"><span class="p0">(<span class="default"><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"><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></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z9</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">ZA</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">ZB</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"><span class="p0">(<span class="default"><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"><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></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z9</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">ZA</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">ZB</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"><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> <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></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z9</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">ZA</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">ZB</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"><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> <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></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z9</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">ZA</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">ZB</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"><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> <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></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z9</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">ZA</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">ZB</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"><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> <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> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z9</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">ZA</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">ZB</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"><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> <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> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z9</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">ZA</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">ZB</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"><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> <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> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z9</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">ZA</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">ZB</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="E24:2"/>
<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="E25:2"/><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"><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></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> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z9</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">ZA</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">ZB</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="E27:2"/><i><font color="Green">E80</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"><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></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> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z9</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">ZA</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">ZB</font>
 <b>and </b><br/><a NAME="E28:2"/><i><font color="Green">E81</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"><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></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> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z9</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">ZA</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">ZB</font>
 <b>by </b><i><a class="ref" href="mcart_1.html#T1">MCART_1:1</a></i>;<br/>
<b>assume </b><a NAME="E29:2"/><i><font color="Green">E82</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>, <font color="Olive">Y9</font>, <font color="Olive">YA</font>, <font color="Olive">YB</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">Y9</font> &amp; <font color="Olive">Y9</font> <a href="hidden.html#R2">in</a> <font color="Olive">YA</font> &amp; <font color="Olive">YA</font> <a href="hidden.html#R2">in</a> <font color="Olive">YB</font> &amp; <font color="Olive">YB</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/>

<a NAME="E30:2"/>
( <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"><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></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> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z9</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">ZA</font> or <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ZB</font> )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E31:2"/><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"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z3</font></span>)</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> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z9</font> or <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ZA</font> or <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ZB</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E32:2"/><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"><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> or <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z9</font> or <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ZA</font> or <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ZB</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E33:2"/><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">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> or <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z8</font> or <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z9</font> or <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ZA</font> or <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ZB</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E34:2"/><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">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> or <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z7</font> or <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z8</font> or <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z9</font> or <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ZA</font> or <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ZB</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E35:2"/><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">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> or <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z6</font> or <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z7</font> or <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z8</font> or <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z9</font> or <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ZA</font> or <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ZB</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E36:2"/><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">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> or <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z5</font> or <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z6</font> or <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z7</font> or <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z8</font> or <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z9</font> or <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ZA</font> or <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ZB</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E37:2"/><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">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> or <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z4</font> or <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z5</font> or <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z6</font> or <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z7</font> or <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z8</font> or <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z9</font> or <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ZA</font> or <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ZB</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E38:2"/><b>then </b>
( <font color="Maroon">Y</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> or <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z3</font> or <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z4</font> or <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z5</font> or <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z6</font> or <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z7</font> or <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z8</font> or <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z9</font> or <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ZA</font> or <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ZB</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E39:2"/><b>then </b><i><font color="Green">E83</font></i>: 
( <font color="Maroon">Y</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> or <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z2</font> or <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z3</font> or <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z4</font> or <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z5</font> or <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z6</font> or <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z7</font> or <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z8</font> or <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z9</font> or <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ZA</font> or <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ZB</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="hsNdiv(this)" href="javascript:()" title="2_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_2"/>
( <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> or <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z1</font> or <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z2</font> or <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z3</font> or <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z4</font> or <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z5</font> or <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z6</font> or <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z7</font> or <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z8</font> or <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z9</font> or <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ZA</font> or <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ZB</font> )
 </span><b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1"><b>suppose </b></a><a NAME="E1:2_2_1"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><div class="add"><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> , <font color="Maroon">Y9</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">YA</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">YB</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:2_2_1"/><i><font color="Green">E85</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">Y9</font> &amp; <font color="Maroon">Y9</font> <a href="hidden.html#R2">in</a> <font color="Maroon">YA</font> &amp; <font color="Maroon">YA</font> <a href="hidden.html#R2">in</a> <font color="Maroon">YB</font> &amp; <font color="Maroon">YB</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="E4:2_2_1"/>
( <font color="Maroon">YB</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="E5:2_2_1"/><b>then </b>
<font color="Maroon">YB</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z1</font>
 <b>by </b><i>, </i>;<br/><a NAME="E6:2_2_1"/><b>then </b>
<font color="Maroon">YB</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="E7:2_2_1"/><b>then </b>
<font color="Maroon">YB</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="E8:2_2_1"/><b>then </b>
<font color="Maroon">YB</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="E9:2_2_1"/><b>then </b>
<font color="Maroon">YB</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="E10:2_2_1"/><b>then </b>
<font color="Maroon">YB</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="E11:2_2_1"/><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="E12:2_2_1"/><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="E13:2_2_1"/><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/><a NAME="E14:2_2_1"/><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"><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_1.html#T70">XBOOLE_1:70</a></i>;<br/><a NAME="E15:2_2_1"/><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"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z3</font></span>)</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> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z9</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T70">XBOOLE_1:70</a></i>;<br/><a NAME="E16:2_2_1"/><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"><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></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> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z9</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">ZA</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T70">XBOOLE_1:70</a></i>;<br/><b>hence </b><a NAME="E17:2_2_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T70">XBOOLE_1:70</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_2"><b>suppose </b></a><a NAME="E1:2_2_2"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z1</font>
 ;<br/><div class="add"><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> , <font color="Maroon">Y9</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">YA</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:2_2_2"/><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">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">Y9</font> &amp; <font color="Maroon">Y9</font> <a href="hidden.html#R2">in</a> <font color="Maroon">YA</font> &amp; <font color="Maroon">YA</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="E4:2_2_2"/>
<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="E5:2_2_2"/><b>then </b>
<font color="Maroon">YA</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="E6:2_2_2"/><b>then </b>
<font color="Maroon">YA</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z2</font>
 <b>by </b><i>, </i>;<br/><a NAME="E7:2_2_2"/><b>then </b>
<font color="Maroon">YA</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="E8:2_2_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"><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="E9:2_2_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"><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="E10:2_2_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"><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="E11:2_2_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_1.html#T70">XBOOLE_1:70</a></i>;<br/><a NAME="E12:2_2_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="E13:2_2_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/><a NAME="E14:2_2_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"><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_1.html#T70">XBOOLE_1:70</a></i>;<br/><a NAME="E15:2_2_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"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z3</font></span>)</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> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z9</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T70">XBOOLE_1:70</a></i>;<br/><a NAME="E16:2_2_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"><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></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> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z9</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">ZA</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T70">XBOOLE_1:70</a></i>;<br/><b>hence </b><a NAME="E17:2_2_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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_3"><b>suppose </b></a><a NAME="E1:2_2_3"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z2</font>
 ;<br/><div class="add"><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> , <font color="Maroon">Y9</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:2_2_3"/><i><font color="Green">E89</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">Y9</font> &amp; <font color="Maroon">Y9</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="E4:2_2_3"/>
<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="E5:2_2_3"/><b>then </b>
<font color="Maroon">Y9</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="E6:2_2_3"/><b>then </b>
<font color="Maroon">Y9</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z3</font>
 <b>by </b><i>, </i>;<br/><a NAME="E7:2_2_3"/><b>then </b>
<font color="Maroon">Y9</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="E8:2_2_3"/><b>then </b>
<font color="Maroon">Y9</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="E9:2_2_3"/><b>then </b>
<font color="Maroon">Y9</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="E10:2_2_3"/><b>then </b>
<font color="Maroon">Y9</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="E11:2_2_3"/><b>then </b>
<font color="Maroon">Y9</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="E12:2_2_3"/><b>then </b>
<font color="Maroon">Y9</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/><a NAME="E13:2_2_3"/><b>then </b>
<font color="Maroon">Y9</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"><span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z3</font></span>)</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> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z9</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E14:2_2_3"/><b>then </b>
<font color="Maroon">Y9</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"><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></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> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z9</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">ZA</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E15:2_2_3"/><b>then </b>
<font color="Maroon">Y9</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"><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></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> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z9</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">ZA</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">ZB</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="E16:2_2_3"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_4"><b>suppose </b></a><a NAME="E1:2_2_4"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z3</font>
 ;<br/><div class="add"><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="E3:2_2_4"/><i><font color="Green">E91</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; <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="E4:2_2_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> <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="E5:2_2_4"/><b>then </b>
<font color="Maroon">Y8</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="E6:2_2_4"/><b>then </b>
<font color="Maroon">Y8</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z4</font>
 <b>by </b><i>, </i>;<br/><a NAME="E7:2_2_4"/><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="E8:2_2_4"/><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="E9:2_2_4"/><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"><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="E10:2_2_4"/><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"><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="E11:2_2_4"/><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"><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/><a NAME="E12:2_2_4"/><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"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z3</font></span>)</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> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z9</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E13:2_2_4"/><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"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><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></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> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z9</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">ZA</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E14:2_2_4"/><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"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><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></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> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z9</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">ZA</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">ZB</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="E15:2_2_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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_5"><b>suppose </b></a><a NAME="E1:2_2_5"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z4</font>
 ;<br/><div class="add"><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="E3:2_2_5"/><i><font color="Green">E93</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="E4:2_2_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> <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="E5:2_2_5"/><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> <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="E6:2_2_5"/><b>then </b>
<font color="Maroon">Y7</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z5</font>
 <b>by </b><i>, </i>;<br/><a NAME="E7:2_2_5"/><b>then </b>
<font color="Maroon">Y7</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="E8:2_2_5"/><b>then </b>
<font color="Maroon">Y7</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="E9:2_2_5"/><b>then </b>
<font color="Maroon">Y7</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="E10:2_2_5"/><b>then </b>
<font color="Maroon">Y7</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/><a NAME="E11:2_2_5"/><b>then </b>
<font color="Maroon">Y7</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"><span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z3</font></span>)</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> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z9</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E12:2_2_5"/><b>then </b>
<font color="Maroon">Y7</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"><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></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> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z9</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">ZA</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E13:2_2_5"/><b>then </b>
<font color="Maroon">Y7</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"><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></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> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z9</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">ZA</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">ZB</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="E14:2_2_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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_6"><b>suppose </b></a><a NAME="E1:2_2_6"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z5</font>
 ;<br/><div class="add"><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="E3:2_2_6"/><i><font color="Green">E95</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="E4:2_2_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> <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="E5:2_2_6"/><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> <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="E6:2_2_6"/><b>then </b>
<font color="Maroon">Y6</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z6</font>
 <b>by </b><i>, </i>;<br/><a NAME="E7:2_2_6"/><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="E8:2_2_6"/><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="E9:2_2_6"/><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/><a NAME="E10:2_2_6"/><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"><span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z3</font></span>)</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> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z9</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E11:2_2_6"/><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"><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></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> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z9</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">ZA</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E12:2_2_6"/><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"><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></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> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z9</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">ZA</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">ZB</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="E13:2_2_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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_7"><b>suppose </b></a><a NAME="E1:2_2_7"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z6</font>
 ;<br/><div class="add"><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="E3:2_2_7"/><i><font color="Green">E97</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="E4:2_2_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> <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="E5:2_2_7"/><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> <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="E6:2_2_7"/><b>then </b>
<font color="Maroon">Y5</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z7</font>
 <b>by </b><i>, </i>;<br/><a NAME="E7:2_2_7"/><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="E8:2_2_7"/><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/><a NAME="E9:2_2_7"/><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"><span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z3</font></span>)</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> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z9</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E10:2_2_7"/><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"><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></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> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z9</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">ZA</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E11:2_2_7"/><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"><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></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> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z9</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">ZA</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">ZB</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="E12:2_2_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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8"><b>suppose </b></a><a NAME="E1:2_2_8"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z7</font>
 ;<br/><div class="add"><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="E3:2_2_8"/><i><font color="Green">E99</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="E4:2_2_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> <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="E5:2_2_8"/><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> <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="E6:2_2_8"/><b>then </b>
<font color="Maroon">Y4</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z8</font>
 <b>by </b><i>, </i>;<br/><a NAME="E7:2_2_8"/><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/><a NAME="E8:2_2_8"/><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"><span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z3</font></span>)</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> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z9</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E9:2_2_8"/><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"><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></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> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z9</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">ZA</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E10:2_2_8"/><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"><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></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> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z9</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">ZA</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">ZB</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="E11:2_2_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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_9"><b>suppose </b></a><a NAME="E1:2_2_9"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z8</font>
 ;<br/><div class="add"><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="E3:2_2_9"/><i><font color="Green">E101</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="E4:2_2_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> <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>, </i>;<br/><a NAME="E5:2_2_9"/><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> <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> <span class="p2">(<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></span>)</span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><a NAME="E6:2_2_9"/><b>then </b>
<font color="Maroon">Y3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z9</font>
 <b>by </b><i>, </i>;<br/><a NAME="E7:2_2_9"/><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"><span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z3</font></span>)</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> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z9</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E8:2_2_9"/><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"><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></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> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z9</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">ZA</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E9:2_2_9"/><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"><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></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> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z9</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">ZA</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">ZB</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="E10:2_2_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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_10"><b>suppose </b></a><a NAME="E1:2_2_10"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z9</font>
 ;<br/><div class="add"><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="E3:2_2_10"/><i><font color="Green">E103</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="E4:2_2_10"/>
<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> <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></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E5:2_2_10"/><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> <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></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="E6:2_2_10"/><b>then </b>
<font color="Maroon">Y2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ZA</font>
 <b>by </b><i>, </i>;<br/><a NAME="E7:2_2_10"/><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"><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></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> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z9</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">ZA</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E8:2_2_10"/><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"><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></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> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z9</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">ZA</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">ZB</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="E9:2_2_10"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_11"><b>suppose </b></a><a NAME="E1:2_2_11"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ZA</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">Y1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:2_2_11"/><i><font color="Green">E105</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="E4:2_2_11"/>
<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> <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></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E5:2_2_11"/><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> <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></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="E6:2_2_11"/><b>then </b>
<font color="Maroon">Y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ZB</font>
 <b>by </b><i>, </i>;<br/><a NAME="E7:2_2_11"/><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"><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></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> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z9</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">ZA</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">ZB</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="E8:2_2_11"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_12"><b>suppose </b></a><a NAME="E1:2_2_12"/>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ZB</font>
 ;<br/><div class="add"><a NAME="E2:2_2_12"/><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="E3:2_2_12"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xboole_1.html#T70">XBOOLE_1:70</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
