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<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7" target="_self">DISJOINT_PAIRS</a> <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">b</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7" target="_self">DISJOINT_PAIRS</a> <font color="Maroon">X</font>;<br/>





<b>assume </b><a NAME="E1:35"/>
not <font color="Maroon">a</font> <a href="normform.html#K1" target="_self">\/</a> <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="normform.html#K7" target="_self">DISJOINT_PAIRS</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E2:35"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="normform.html#K1" target="_self">\/</a> <font color="Maroon">b</font></span>)</span> <a href="domain_1.html#K2">`1</a>  <a href="xboole_0.html#NR5" target="_self">meets</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="normform.html#K1" target="_self">\/</a> <font color="Maroon">b</font></span>)</span> <a href="domain_1.html#K3">`2</a> 
 
;<br/>
<a NAME="E3:35"/><b>then </b><i><font color="Green">E29</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="normform.html#K1" target="_self">\/</a> <font color="Maroon">b</font></span>)</span> <a href="domain_1.html#K2">`1</a> </span>)</span> <a href="finsub_1.html#K3">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="normform.html#K1" target="_self">\/</a> <font color="Maroon">b</font></span>)</span> <a href="domain_1.html#K3">`2</a> </span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</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/>
<b>consider </b><font color="Maroon">p</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="normform.html#K1" target="_self">\/</a> <font color="Maroon">b</font></span>)</span> <a href="domain_1.html#K2">`1</a> </span>)</span> <a href="finsub_1.html#K3">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="normform.html#K1" target="_self">\/</a> <font color="Maroon">b</font></span>)</span> <a href="domain_1.html#K3">`2</a> </span>)</span>;<br/>
<a NAME="E5:35"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="normform.html#K1" target="_self">\/</a> <font color="Maroon">b</font></span>)</span> <a href="domain_1.html#K2">`1</a> </span>)</span> <a href="finsub_1.html#K3">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="normform.html#K1" target="_self">\/</a> <font color="Maroon">b</font></span>)</span> <a href="domain_1.html#K3">`2</a> </span>)</span> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="finsub_1.html#T32">FINSUB_1:32</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">p</font> = <font color="Maroon">p</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="normform.html#T21" target="_self">Th21</a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>take </b>
<font color="Maroon">p</font>
;<br/>

<a NAME="E7:35"/>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="normform.html#K1" target="_self">\/</a> <font color="Maroon">b</font></span>)</span> <a href="domain_1.html#K2">`1</a>  &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="normform.html#K1" target="_self">\/</a> <font color="Maroon">b</font></span>)</span> <a href="domain_1.html#K3">`2</a>  )
 
<b>by </b><i><a class="ref" href="normform.html#T21" target="_self">Th21</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E8:35"/><b>then </b>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">a</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">b</font> <a href="domain_1.html#K2">`1</a> </span>)</span> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">a</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">b</font> <a href="domain_1.html#K3">`2</a> </span>)</span> )
 
<b>by </b><i/>;<br/>
<a NAME="E9:35"/><b>then </b>
( ( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> <a href="domain_1.html#K2">`1</a>  or <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b</font> <a href="domain_1.html#K2">`1</a>  ) &amp; ( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b</font> <a href="domain_1.html#K3">`2</a>  or <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> <a href="domain_1.html#K3">`2</a>  ) )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E10:35"/>
( ( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> <a href="domain_1.html#K2">`1</a>  &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b</font> <a href="domain_1.html#K3">`2</a>  ) or ( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b</font> <a href="domain_1.html#K2">`1</a>  &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> <a href="domain_1.html#K3">`2</a>  ) )
 <b>by </b><i><a class="ref" href="normform.html#T20" target="_self">Th20</a></i>;<br/>


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