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<div class="add">

<b>let </b><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">y</font>, <font color="Maroon" title="c3">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p2">(<span class="default"><a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c1">X</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>;<br/>





<b>assume </b><a NAME="E1:32"/>
( <font color="Maroon" title="c2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="normform.html#K7" title="NORMFORM:func.7">DISJOINT_PAIRS</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="normform.html#K7" title="NORMFORM:func.7">DISJOINT_PAIRS</a> <font color="Maroon" title="c1">X</font> )
 ;<br/>

<a NAME="E2:32"/><b>then </b>
( <font color="Maroon" title="c2">y</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a>  <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c2">y</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  &amp; <font color="Maroon" title="c3">x</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a>  <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">x</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  )
 
<b>by </b><i><a class="ref" href="normform.html#T42" target="_self" title="NORMFORM:th.42">Th42</a></i>;<br/>
<a NAME="E3:32"/><b>then </b><i><font color="Green" title="E19">A1</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E4:32"/><i><font color="Green" title="E20">A2</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="normform.html#K1" title="NORMFORM:func.1">\/</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="normform.html#K1" title="NORMFORM:func.1">\/</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span> )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>;<br/>
<a NAME="E5:32"/><i><font color="Green" title="E21">A3</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span></span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span></span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span></span>)</span> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span></span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T23" title="XBOOLE_1:th.23">XBOOLE_1:23</a></i>;<br/>
<a NAME="E6:32"/><i><font color="Green" title="E22">A4</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span></span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span></span>)</span> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span></span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span></span>)</span> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T23" title="XBOOLE_1:th.23">XBOOLE_1:23</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:32_1"/>
<font color="Maroon" title="c2">y</font> <a href="normform.html#K1" title="NORMFORM:func.1">\/</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="normform.html#K7" title="NORMFORM:func.7">DISJOINT_PAIRS</a> <font color="Maroon" title="c1">X</font>
 ;<br/><a NAME="E2:32_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="normform.html#K1" title="NORMFORM:func.1">\/</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a>  <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="normform.html#K1" title="NORMFORM:func.1">\/</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> 
 <b>by </b><i><a class="ref" href="normform.html#T42" target="_self" title="NORMFORM:th.42">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" title="c2">y</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span></span>)</span> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="normform.html#E3:32"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="normform.html#E4:32"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="normform.html#E5:32"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="normform.html#E6:32"><i><font color="Green" title="E22">A4</font></i></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">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" title="c2">y</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span></span>)</span> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<a NAME="E9:32"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="normform.html#K1" title="NORMFORM:func.1">\/</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a>  <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="normform.html#K1" title="NORMFORM:func.1">\/</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> 
 
<b>by </b><i><a class="txt" href="normform.html#E3:32"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="normform.html#E4:32"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="normform.html#E5:32"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="normform.html#E6:32"><i><font color="Green" title="E22">A4</font></i></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<b>hence </b><a NAME="E10:32"/>
<font color="Maroon" title="c2">y</font> <a href="normform.html#K1" title="NORMFORM:func.1">\/</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="normform.html#K7" title="NORMFORM:func.7">DISJOINT_PAIRS</a> <font color="Maroon" title="c1">X</font>
 ;<br/>


</div>
