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<div class="add">

<b>let </b><font color="Maroon">L</font> be   <a href="lattices.html#NM1">Lattice</a>;<br/><b>let </b><font color="Maroon">D1</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>;<br/><b>let </b><font color="Maroon">D2</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>;<br/><b>let </b><font color="Maroon">r</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="filter_2.html#D1" target="_self">FILTER_2:def 1</a><br/>







<b>thus </b><a NAME="E1:70"/>
( <font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D1</font> <a href="filter_2.html#K20" target="_self">"\/"</a> <font color="Maroon">D2</font> implies <font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D2</font> <a href="filter_2.html#K20" target="_self">"\/"</a> <font color="Maroon">D1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:70_1"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D1</font> <a href="filter_2.html#K20" target="_self">"\/"</a> <font color="Maroon">D2</font>
 ;<br/>

<b>then consider </b><font color="Maroon">p</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">q</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:70_1"/><i><font color="Green">E14</font></i>: 
( <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">q</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D1</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D2</font> )
 ;<br/>
<b>thus </b><a NAME="E4:70_1"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D2</font> <a href="filter_2.html#K20" target="_self">"\/"</a> <font color="Maroon">D1</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:70"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D2</font> <a href="filter_2.html#K20" target="_self">"\/"</a> <font color="Maroon">D1</font>
 ;<br/>

<b>then consider </b><font color="Maroon">p</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">q</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E4:70"/><i><font color="Green">E15</font></i>: 
( <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">q</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D2</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D1</font> )
 ;<br/>
<b>thus </b><a NAME="E5:70"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D1</font> <a href="filter_2.html#K20" target="_self">"\/"</a> <font color="Maroon">D2</font>
 <b>by </b><i/>;<br/>


</div>
