<?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>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finsub_1.html#K5">Fin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="finsub_1.html#K5">Fin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/>





<b>assume </b><a NAME="E1:32"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="normform.html#K7" target="_self">DISJOINT_PAIRS</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="normform.html#K7" target="_self">DISJOINT_PAIRS</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/>

<a NAME="E2:32"/><b>then </b>
( <font color="Maroon">c<sub>2</sub></font> <a href="domain_1.html#K2">`1</a>  <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>2</sub></font> <a href="domain_1.html#K3">`2</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="domain_1.html#K2">`1</a>  <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>3</sub></font> <a href="domain_1.html#K3">`2</a>  )
 
<b>by </b><i><a class="ref" href="normform.html#T42" target="_self">Th42</a></i>;<br/>
<a NAME="E3:32"/><b>then </b><i><font color="Green">E22</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="domain_1.html#K2">`1</a> </span>)</span> <a href="finsub_1.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="domain_1.html#K3">`2</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="domain_1.html#K2">`1</a> </span>)</span> <a href="finsub_1.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="domain_1.html#K3">`2</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E4:32"/><i><font color="Green">E23</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normform.html#K1" target="_self">\/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="domain_1.html#K2">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="domain_1.html#K2">`1</a> </span>)</span> <a href="setwiseo.html#K5">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="domain_1.html#K2">`1</a> </span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normform.html#K1" target="_self">\/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="domain_1.html#K3">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="domain_1.html#K3">`2</a> </span>)</span> <a href="setwiseo.html#K5">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="domain_1.html#K3">`2</a> </span>)</span> )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<a NAME="E5:32"/><i><font color="Green">E24</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="domain_1.html#K2">`1</a> </span>)</span> <a href="setwiseo.html#K5">\/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="domain_1.html#K2">`1</a> </span>)</span></span>)</span> <a href="finsub_1.html#K3">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="domain_1.html#K3">`2</a> </span>)</span> <a href="setwiseo.html#K5">\/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="domain_1.html#K3">`2</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</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="domain_1.html#K2">`1</a> </span>)</span> <a href="setwiseo.html#K5">\/</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="domain_1.html#K2">`1</a> </span>)</span></span>)</span> <a href="finsub_1.html#K3">/\</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="domain_1.html#K3">`2</a> </span>)</span></span>)</span> <a href="setwiseo.html#K5">\/</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="domain_1.html#K2">`1</a> </span>)</span> <a href="setwiseo.html#K5">\/</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="domain_1.html#K2">`1</a> </span>)</span></span>)</span> <a href="finsub_1.html#K3">/\</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="domain_1.html#K3">`2</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>;<br/>
<a NAME="E6:32"/><i><font color="Green">E25</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="domain_1.html#K2">`1</a> </span>)</span> <a href="setwiseo.html#K5">\/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="domain_1.html#K2">`1</a> </span>)</span></span>)</span> <a href="finsub_1.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="domain_1.html#K3">`2</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="domain_1.html#K2">`1</a> </span>)</span> <a href="finsub_1.html#K3">/\</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="domain_1.html#K3">`2</a> </span>)</span></span>)</span> <a href="setwiseo.html#K5">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="domain_1.html#K2">`1</a> </span>)</span> <a href="finsub_1.html#K3">/\</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="domain_1.html#K3">`2</a> </span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="domain_1.html#K2">`1</a> </span>)</span> <a href="setwiseo.html#K5">\/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="domain_1.html#K2">`1</a> </span>)</span></span>)</span> <a href="finsub_1.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="domain_1.html#K3">`2</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="domain_1.html#K2">`1</a> </span>)</span> <a href="finsub_1.html#K3">/\</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="domain_1.html#K3">`2</a> </span>)</span></span>)</span> <a href="setwiseo.html#K5">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="domain_1.html#K2">`1</a> </span>)</span> <a href="finsub_1.html#K3">/\</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="domain_1.html#K3">`2</a> </span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:32_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="normform.html#K1" target="_self">\/</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="normform.html#K7" target="_self">DISJOINT_PAIRS</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><a NAME="E2:32_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normform.html#K1" target="_self">\/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="domain_1.html#K2">`1</a>  <a href="xboole_0.html#R1">misses</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normform.html#K1" target="_self">\/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="domain_1.html#K3">`2</a> 
 <b>by </b><i><a class="ref" href="normform.html#T42" target="_self">Th42</a></i>;<br/><b>hence </b><a NAME="E3:32_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="domain_1.html#K2">`1</a> </span>)</span> <a href="finsub_1.html#K3">/\</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="domain_1.html#K3">`2</a> </span>)</span></span>)</span> <a href="setwiseo.html#K5">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="domain_1.html#K2">`1</a> </span>)</span> <a href="finsub_1.html#K3">/\</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="domain_1.html#K3">`2</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="normform.html#E3:32"><i><font color="Green">E22</font></i></a>, <a class="txt" href="normform.html#E4:32"><i><font color="Green">E23</font></i></a>, <a class="txt" href="normform.html#E5:32"><i><font color="Green">E24</font></i></a>, <a class="txt" href="normform.html#E6:32"><i><font color="Green">E25</font></i></a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E8:32"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="domain_1.html#K2">`1</a> </span>)</span> <a href="finsub_1.html#K3">/\</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="domain_1.html#K3">`2</a> </span>)</span></span>)</span> <a href="setwiseo.html#K5">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="domain_1.html#K2">`1</a> </span>)</span> <a href="finsub_1.html#K3">/\</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="domain_1.html#K3">`2</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<a NAME="E9:32"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normform.html#K1" target="_self">\/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="domain_1.html#K2">`1</a>  <a href="xboole_0.html#R1">misses</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normform.html#K1" target="_self">\/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="domain_1.html#K3">`2</a> 
 
<b>by </b><i><a class="txt" href="normform.html#E3:32"><i><font color="Green">E22</font></i></a>, <a class="txt" href="normform.html#E4:32"><i><font color="Green">E23</font></i></a>, <a class="txt" href="normform.html#E5:32"><i><font color="Green">E24</font></i></a>, <a class="txt" href="normform.html#E6:32"><i><font color="Green">E25</font></i></a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<b>hence </b><a NAME="E10:32"/>
<font color="Maroon">c<sub>2</sub></font> <a href="normform.html#K1" target="_self">\/</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="normform.html#K7" target="_self">DISJOINT_PAIRS</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>


</div>
