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<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:28"/>
<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/>

<b>then 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="E3:28"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E4:28"/><i><font color="Green">E19</font></i>: 
for <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>  holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> or  not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>3</sub></font> or  not <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> or <font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="ref" href="mcart_1.html#T4" target="_self">Th4</a></i>;<br/>
<b>take </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font>;<br/>

<b>thus </b><a NAME="E5:28"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="mcart_1.html#E3:28"><i><font color="Green">E18</font></i></a></i>;<br/>

<b>given </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E6:28"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>and </b><br/><a NAME="E7:28"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="mcart_1.html#K3" target="_self">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="mcart_1.html#K3" target="_self">]</a></span>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K2">}</a></span>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> = <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K2">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K2">}</a></span>;<br/>
<b>set </b><font color="Maroon">c<sub>9</sub></font> = <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K2">{</a><span class="default"><span class="p3"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K2">}</a></span>,<span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K2">}</a></span>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K2">}</a></span>;<br/>
<a NAME="E8:28"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K2">}</a></span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K2">}</a></span> )
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E9:28"/><b>then </b><i><font color="Green">E22</font></i>: 
not <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K2">}</a></span> <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#E6:28"><i><font color="Green">E20</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<font color="Maroon">c<sub>2</sub></font> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="mcart_1.html#E7:28"><i><font color="Green">E21</font></i></a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K2">{</a><span class="default"><span class="p3"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K2">}</a></span>,<span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K2">}</a></span>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="tarski.html#D5">TARSKI:def 5</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K2">{</a><span class="default"><span class="p3"><a href="tarski.html#K2">{</a><span class="default"><span class="p4"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K2">}</a></span>,<span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K2">}</a></span>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K2">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K2">{</a><span class="default"><span class="p4"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K2">}</a></span>,<span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K2">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K2">}</a></span>
<b>by </b><i><a class="ref" href="tarski.html#D5">TARSKI:def 5</a></i>
;<br/>
<a NAME="E11:28"/><b>then </b>
( <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K2">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K2">}</a></span> &amp; <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K2">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K2">{</a><span class="default"><span class="p3"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K2">}</a></span>,<span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K2">}</a></span>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K2">}</a></span> &amp; <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K2">{</a><span class="default"><span class="p3"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K2">}</a></span>,<span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K2">}</a></span>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<b>hence </b><a NAME="E12:28"/>
contradiction
 <b>by </b><i><a class="txt" href="mcart_1.html#E4:28"><i><font color="Green">E19</font></i></a>, <a class="txt" href="mcart_1.html#E9:28"><i><font color="Green">E22</font></i></a></i>;<br/>


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