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<b>let </b><font color="Maroon" title="c1">BL</font> be  <a href="lattices.html#V17" title="LATTICES:attr.17">Boolean</a> <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</a>;<br/><b>let </b><font color="Maroon" title="c2">A</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">BL</font>;<br/><b>let </b><font color="Maroon" title="c3">c</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">BL</font>;<br/>





<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:50_1"/>
<font color="Maroon" title="c3">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lattice4.html#K4" title="LATTICE4:func.4">SetImp</a> <font color="Maroon" title="c2">A</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c4">p</font>, <font color="Maroon" title="c5">q</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">BL</font><b> such that </b><br/><a NAME="E3:50_1"/><i><font color="Green" title="E42">A1</font></i>: 
( <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c5">q</font> &amp; <font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c5">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> )
 ;<br/><b>take </b><font color="Maroon" title="c6">p</font> = <font color="Maroon" title="c4">p</font>;<br/><b>take </b><font color="Maroon" title="c7">q</font> = <font color="Maroon" title="c5">q</font>;<br/><b>thus </b><a NAME="E4:50_1"/>
( <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a> </span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c7">q</font> &amp; <font color="Maroon" title="c6">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c7">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> )
 <b>by </b><i><a class="txt" href="lattice4.html#E3:50_1"><i><font color="Green" title="E42">A1</font></i></a>, <a class="ref" href="filter_0.html#T55" title="FILTER_0:th.55">FILTER_0:55</a></i>;<br/></div>
<b>end;</b></div>

<b>given </b><font color="Maroon" title="c4">p</font>, <font color="Maroon" title="c5">q</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">BL</font><b> such that </b><a NAME="E3:50"/><i><font color="Green" title="E42">A2</font></i>: 
( <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a> </span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c5">q</font> &amp; <font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c5">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> )
 ;<br/>

<a NAME="E4:50"/>
<font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c5">q</font>
 
<b>by </b><i><a class="txt" href="lattice4.html#E3:50"><i><font color="Green" title="E42">A2</font></i></a>, <a class="ref" href="filter_0.html#T55" title="FILTER_0:th.55">FILTER_0:55</a></i>;<br/>
<b>hence </b><a NAME="E5:50"/>
<font color="Maroon" title="c3">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lattice4.html#K4" title="LATTICE4:func.4">SetImp</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="txt" href="lattice4.html#E3:50"><i><font color="Green" title="E42">A2</font></i></a></i>;<br/>


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