<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">F</font> be   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font>, <font color="Maroon">G</font> be   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font>;<br/>



<a NAME="E1:45"/>
for <font color="Olive">P</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> holds <br/> ( <font color="Olive">P</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K7" target="_self">COMPLEMENT</a> <font color="Maroon">F</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K7" target="_self">COMPLEMENT</a> <font color="Maroon">G</font></span>)</span> iff <font color="Olive">P</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">G</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">P</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/>

<a NAME="E1:45_1"/>
( <!--HUMANRECFAILED-->( ( not <font color="Maroon">P</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K7" target="_self">COMPLEMENT</a> <font color="Maroon">F</font> &amp; not <font color="Maroon">P</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K7" target="_self">COMPLEMENT</a> <font color="Maroon">G</font> ) or <font color="Maroon">P</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> or <font color="Maroon">P</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> ) &amp; ( ( not <font color="Maroon">P</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; not <font color="Maroon">P</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> ) or <font color="Maroon">P</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K7" target="_self">COMPLEMENT</a> <font color="Maroon">F</font> or <font color="Maroon">P</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K7" target="_self">COMPLEMENT</a> <font color="Maroon">G</font> ) )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:45_1"/>
( <font color="Maroon">P</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K7" target="_self">COMPLEMENT</a> <font color="Maroon">F</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K7" target="_self">COMPLEMENT</a> <font color="Maroon">G</font></span>)</span> iff <font color="Maroon">P</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">G</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:45"/>
 <a href="setfam_1.html#K7" target="_self">COMPLEMENT</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">G</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K7" target="_self">COMPLEMENT</a> <font color="Maroon">F</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K7" target="_self">COMPLEMENT</a> <font color="Maroon">G</font></span>)</span>
 <b>by </b><i/>;<br/>


</div>
