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<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">a</font>, <font color="Maroon" title="c3">b</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="normform.html#K7" title="NORMFORM:func.7">DISJOINT_PAIRS</a> <font color="Maroon" title="c1">X</font>;<br/>





<b>assume </b><a NAME="E1:35"/>
not <font color="Maroon" title="c2">a</font> <a href="normform.html#K1" title="NORMFORM:func.1">\/</a> <font color="Maroon" title="c3">b</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="E2:35"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="normform.html#K1" title="NORMFORM:func.1">\/</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a>  <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="normform.html#K1" title="NORMFORM:func.1">\/</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> 
 
;<br/>
<a NAME="E3:35"/><b>then </b><i><font color="Green" title="E21">A1</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="normform.html#K1" title="NORMFORM:func.1">\/</a> <font color="Maroon" title="c3">b</font></span>)</span> <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"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="normform.html#K1" title="NORMFORM:func.1">\/</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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/>
<b>consider </b><font color="Maroon" title="c4">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="normform.html#K1" title="NORMFORM:func.1">\/</a> <font color="Maroon" title="c3">b</font></span>)</span> <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"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="normform.html#K1" title="NORMFORM:func.1">\/</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span>;<br/>
<a NAME="E5:35"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="normform.html#K1" title="NORMFORM:func.1">\/</a> <font color="Maroon" title="c3">b</font></span>)</span> <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"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="normform.html#K1" title="NORMFORM:func.1">\/</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="finsub_1.html#T32" title="FINSUB_1:th.32">FINSUB_1:32</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">p</font> = <font color="Maroon" title="c4">p</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="txt" href="normform.html#E3:35"><i><font color="Green" title="E21">A1</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">p</font>
;<br/>

<a NAME="E7:35"/>
( <font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="normform.html#K1" title="NORMFORM:func.1">\/</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a>  &amp; <font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="normform.html#K1" title="NORMFORM:func.1">\/</a> <font color="Maroon" title="c3">b</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#E3:35"><i><font color="Green" title="E21">A1</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E8:35"/><b>then </b>
( <font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</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">b</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span> &amp; <font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</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">b</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="E9:35"/><b>then </b>
( ( <font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">a</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a>  or <font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">b</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a>  ) &amp; ( <font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">b</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  or <font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">a</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  ) )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E10:35"/>
( ( <font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">a</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a>  &amp; <font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">b</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  ) or ( <font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">b</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a>  &amp; <font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">a</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  ) )
 <b>by </b><i><a class="ref" href="normform.html#T46" target="_self" title="NORMFORM:th.46">Th46</a></i>;<br/>


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