<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></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> ] means ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>4</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> );<br/>
<b>consider </b><font color="Maroon">c<sub>2</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:6"/><i><font color="Green">E5</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> iff ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></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> ] means ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> );<br/>
<b>consider </b><font color="Maroon">c<sub>3</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:6"/><i><font color="Green">E6</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> iff ( <font color="Olive">b<sub>1</sub></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">c<sub>1</sub></font></span>)</span> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></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> ] means ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> );<br/>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:6"/><i><font color="Green">E7</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> iff ( <font color="Olive">b<sub>1</sub></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">c<sub>1</sub></font></span>)</span></span>)</span> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></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> ] means ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> );<br/>
<b>consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:6"/><i><font color="Green">E8</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> iff ( <font color="Olive">b<sub>1</sub></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">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></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> ] means <font color="Maroon">a<sub>1</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E10:6"/><i><font color="Green">E9</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> iff ( <font color="Olive">b<sub>1</sub></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">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon">S<sub>5</sub></font>[<font color="Olive">b<sub>1</sub></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">c<sub>7</sub></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"><font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>6</sub></font>;<br/>
<i><font color="Green">E10</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"><font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>6</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>6</sub></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">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>6</sub></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">c<sub>1</sub></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">c<sub>2</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>6</sub></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="E12:6"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<a NAME="E13:6"/><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"><font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="mcart_1.html#E11:6"><i><font color="Green">E10</font></i></a>, <a class="ref" href="xboole_1.html#T15">XBOOLE_1:15</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>8</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E15:6"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>8</sub></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">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>6</sub></font>
 <b>and </b><br/><a NAME="E16:6"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>8</sub></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"><font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="mcart_1.html#T1" target="_self">Th1</a></i>;<br/>
<b>assume </b><a NAME="E17:6"/><i><font color="Green">E13</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> or ex <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font>, <font color="Olive">b<sub>5</sub></font>, <font color="Olive">b<sub>6</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>4</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>5</sub></font> &amp; <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>6</sub></font> &amp; <font color="Olive">b<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; not <font color="Olive">b<sub>2</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> ) )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:6_2"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>13</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:6_2"/><i><font color="Green">E15</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; not <font color="Maroon">c<sub>9</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="mcart_1.html#E17:6"><i><font color="Green">E13</font></i></a></i>;<br/><a NAME="E4:6_2"/>
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="mcart_1.html#E1:6_2"><i><font color="Green">E14</font></i></a>, <a class="txt" href="mcart_1.html#E3:6_2"><i><font color="Green">E15</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><a NAME="E5:6_2"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="mcart_1.html#E2:6"><i><font color="Green">E5</font></i></a>, <a class="txt" href="mcart_1.html#E3:6_2"><i><font color="Green">E15</font></i></a></i>;<br/><a NAME="E6:6_2"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E7:6_2"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E8:6_2"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E9:6_2"/><b>then </b>
<font color="Maroon">c<sub>8</sub></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">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="mcart_1.html#E3:6_2"><i><font color="Green">E15</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><a NAME="E10:6_2"/><b>then </b>
<font color="Maroon">c<sub>8</sub></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">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T70">XBOOLE_1:70</a></i>;<br/><b>hence </b><a NAME="E11:6_2"/>
contradiction
 <b>by </b><i><a class="txt" href="mcart_1.html#E16:6"><i><font color="Green">E12</font></i></a>, <a class="ref" href="xboole_1.html#T70">XBOOLE_1:70</a></i>;<br/></div><b>end;</b></div>
<a NAME="E19:6"/><b>then </b>
<font color="Maroon">c<sub>8</sub></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">c<sub>2</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="mcart_1.html#E11:6"><i><font color="Green">E10</font></i></a>, <a class="txt" href="mcart_1.html#E15:6"><i><font color="Green">E11</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E20:6"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>;<br/>
<a NAME="E21:6"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>;<br/>
<a NAME="E22:6"/><b>then </b><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:6_3"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:6_3"/><i><font color="Green">E16</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="mcart_1.html#E2:6"><i><font color="Green">E5</font></i></a></i>;<br/><a NAME="E4:6_3"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="mcart_1.html#E2:6"><i><font color="Green">E5</font></i></a>, <a class="txt" href="mcart_1.html#E1:6_3"><i><font color="Green">E15</font></i></a></i>;<br/><a NAME="E5:6_3"/><b>then </b>
<font color="Maroon">c<sub>12</sub></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">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="mcart_1.html#E3:6_3"><i><font color="Green">E16</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><a NAME="E6:6_3"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="mcart_1.html#E4:6"><i><font color="Green">E6</font></i></a>, <a class="txt" href="mcart_1.html#E3:6_3"><i><font color="Green">E16</font></i></a></i>;<br/><a NAME="E7:6_3"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E8:6_3"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="mcart_1.html#E3:6_3"><i><font color="Green">E16</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><a NAME="E9:6_3"/><b>then </b>
<font color="Maroon">c<sub>8</sub></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">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T70">XBOOLE_1:70</a></i>;<br/><a NAME="E10:6_3"/><b>then </b>
<font color="Maroon">c<sub>8</sub></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">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T70">XBOOLE_1:70</a></i>;<br/><b>hence </b><a NAME="E11:6_3"/>
contradiction
 <b>by </b><i><a class="txt" href="mcart_1.html#E16:6"><i><font color="Green">E12</font></i></a>, <a class="ref" href="xboole_1.html#T70">XBOOLE_1:70</a></i>;<br/></div><b>end;</b></div>
<a NAME="E24:6"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="mcart_1.html#E22:6"><i><font color="Green">E14</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E25:6"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>;<br/>
<a NAME="E26:6"/><b>then </b><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:6_4"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:6_4"/><i><font color="Green">E17</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="mcart_1.html#E4:6"><i><font color="Green">E6</font></i></a></i>;<br/><a NAME="E4:6_4"/>
<font color="Maroon">c<sub>8</sub></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">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="mcart_1.html#E4:6"><i><font color="Green">E6</font></i></a>, <a class="txt" href="mcart_1.html#E1:6_4"><i><font color="Green">E16</font></i></a></i>;<br/><a NAME="E5:6_4"/><b>then </b>
<font color="Maroon">c<sub>11</sub></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">c<sub>1</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="mcart_1.html#E3:6_4"><i><font color="Green">E17</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><a NAME="E6:6_4"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="mcart_1.html#E6:6"><i><font color="Green">E7</font></i></a>, <a class="txt" href="mcart_1.html#E3:6_4"><i><font color="Green">E17</font></i></a></i>;<br/><a NAME="E7:6_4"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E8:6_4"/><b>then </b>
<font color="Maroon">c<sub>11</sub></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">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E9:6_4"/><b>then </b>
<font color="Maroon">c<sub>11</sub></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">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>6</sub></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="E10:6_4"/>
contradiction
 <b>by </b><i><a class="txt" href="mcart_1.html#E16:6"><i><font color="Green">E12</font></i></a>, <a class="txt" href="mcart_1.html#E3:6_4"><i><font color="Green">E17</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div>
<a NAME="E28:6"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="mcart_1.html#E26:6"><i><font color="Green">E15</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E29:6"/><b>then </b><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:6_5"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:6_5"/><i><font color="Green">E18</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="mcart_1.html#E6:6"><i><font color="Green">E7</font></i></a></i>;<br/><a NAME="E4:6_5"/>
<font color="Maroon">c<sub>8</sub></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">c<sub>1</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="mcart_1.html#E6:6"><i><font color="Green">E7</font></i></a>, <a class="txt" href="mcart_1.html#E1:6_5"><i><font color="Green">E17</font></i></a></i>;<br/><a NAME="E5:6_5"/><b>then </b>
<font color="Maroon">c<sub>10</sub></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">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="mcart_1.html#E3:6_5"><i><font color="Green">E18</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><a NAME="E6:6_5"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="mcart_1.html#E8:6"><i><font color="Green">E8</font></i></a>, <a class="txt" href="mcart_1.html#E3:6_5"><i><font color="Green">E18</font></i></a></i>;<br/><a NAME="E7:6_5"/><b>then </b>
<font color="Maroon">c<sub>10</sub></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">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E8:6_5"/><b>then </b>
<font color="Maroon">c<sub>10</sub></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">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>6</sub></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="E9:6_5"/>
contradiction
 <b>by </b><i><a class="txt" href="mcart_1.html#E16:6"><i><font color="Green">E12</font></i></a>, <a class="txt" href="mcart_1.html#E3:6_5"><i><font color="Green">E18</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div>
<a NAME="E31:6"/><b>then </b><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="mcart_1.html#E29:6"><i><font color="Green">E16</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_6"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:6_6"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:6_6"/><i><font color="Green">E19</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="mcart_1.html#E8:6"><i><font color="Green">E8</font></i></a></i>;<br/><a NAME="E4:6_6"/>
<font color="Maroon">c<sub>8</sub></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">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="mcart_1.html#E8:6"><i><font color="Green">E8</font></i></a>, <a class="txt" href="mcart_1.html#E1:6_6"><i><font color="Green">E18</font></i></a></i>;<br/><a NAME="E5:6_6"/><b>then </b>
<font color="Maroon">c<sub>9</sub></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">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="mcart_1.html#E3:6_6"><i><font color="Green">E19</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><a NAME="E6:6_6"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="mcart_1.html#E10:6"><i><font color="Green">E9</font></i></a>, <a class="txt" href="mcart_1.html#E3:6_6"><i><font color="Green">E19</font></i></a></i>;<br/><a NAME="E7:6_6"/><b>then </b>
<font color="Maroon">c<sub>9</sub></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">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>6</sub></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="E8:6_6"/>
contradiction
 <b>by </b><i><a class="txt" href="mcart_1.html#E16:6"><i><font color="Green">E12</font></i></a>, <a class="txt" href="mcart_1.html#E3:6_6"><i><font color="Green">E19</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div>
<a NAME="E33:6"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="mcart_1.html#E31:6"><i><font color="Green">E17</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E34:6"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="mcart_1.html#E10:6"><i><font color="Green">E9</font></i></a></i>;<br/>
<b>hence </b><a NAME="E35:6"/>
contradiction
 <b>by </b><i><a class="txt" href="mcart_1.html#E11:6"><i><font color="Green">E10</font></i></a>, <a class="txt" href="mcart_1.html#E16:6"><i><font color="Green">E12</font></i></a>, <a class="ref" href="xboole_1.html#T70">XBOOLE_1:70</a></i>;<br/>


</div>
