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<b>let </b><font color="Maroon" title="c1">L</font> be   <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">D1</font>, <font color="Olive" title="b2">D2</font> being 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">L</font> holds  <font color="Olive" title="b1">D1</font> <a href="filter_2.html#K20" title="FILTER_2:func.20">"\/"</a> <font color="Olive" title="b2">D2</font> <a href="filter_2.html#R1" title="FILTER_2:pred.1">=</a> <font color="Olive" title="b2">D2</font> <a href="filter_2.html#K20" title="FILTER_2:func.20">"\/"</a> <font color="Olive" title="b1">D1</font></span><br/><b>let </b><font color="Maroon" title="c2">D1</font>, <font color="Maroon" title="c3">D2</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">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">D1</font> <a href="filter_2.html#K20" title="FILTER_2:func.20">"\/"</a> <font color="Maroon" title="c3">D2</font> <a href="filter_2.html#R1" title="FILTER_2:pred.1">=</a> <font color="Maroon" title="c3">D2</font> <a href="filter_2.html#K20" title="FILTER_2:func.20">"\/"</a> <font color="Maroon" title="c2">D1</font></span><br/><b>let </b><font color="Maroon" title="c4">r</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="filter_2.html#D1" target="_self" title="FILTER_2:def.1">FILTER_2:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D1</font> <a href="filter_2.html#K20" title="FILTER_2:func.20">"\/"</a> <font color="Maroon" title="c3">D2</font> iff <font color="Maroon" title="c4">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D2</font> <a href="filter_2.html#K20" title="FILTER_2:func.20">"\/"</a> <font color="Maroon" title="c2">D1</font> )</span><br/>







<b>thus </b><a NAME="E1:70"/>
( <font color="Maroon" title="c4">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D1</font> <a href="filter_2.html#K20" title="FILTER_2:func.20">"\/"</a> <font color="Maroon" title="c3">D2</font> implies <font color="Maroon" title="c4">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D2</font> <a href="filter_2.html#K20" title="FILTER_2:func.20">"\/"</a> <font color="Maroon" title="c2">D1</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D2</font> <a href="filter_2.html#K20" title="FILTER_2:func.20">"\/"</a> <font color="Maroon" title="c2">D1</font> implies <font color="Maroon" title="c4">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D1</font> <a href="filter_2.html#K20" title="FILTER_2:func.20">"\/"</a> <font color="Maroon" title="c3">D2</font> )</span><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" title="c4">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D1</font> <a href="filter_2.html#K20" title="FILTER_2:func.20">"\/"</a> <font color="Maroon" title="c3">D2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D2</font> <a href="filter_2.html#K20" title="FILTER_2:func.20">"\/"</a> <font color="Maroon" title="c2">D1</font></span><br/>

<b>then consider </b><font color="Maroon" title="c5">p</font>, <font color="Maroon" title="c6">q</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E3:70_1"/><i><font color="Green" title="E27">A1</font></i>: 
( <font color="Maroon" title="c4">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c6">q</font> &amp; <font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D1</font> &amp; <font color="Maroon" title="c6">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D2</font> )
 ;<br/>
<b>thus </b><a NAME="E4:70_1"/>
<font color="Maroon" title="c4">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D2</font> <a href="filter_2.html#K20" title="FILTER_2:func.20">"\/"</a> <font color="Maroon" title="c2">D1</font>
 <b>by </b><i><a class="txt" href="#E3:70_1"><i><font color="Green" title="E27">A1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>assume </b><a NAME="E2:70"/>
<font color="Maroon" title="c4">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D2</font> <a href="filter_2.html#K20" title="FILTER_2:func.20">"\/"</a> <font color="Maroon" title="c2">D1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D1</font> <a href="filter_2.html#K20" title="FILTER_2:func.20">"\/"</a> <font color="Maroon" title="c3">D2</font></span><br/>

<b>then consider </b><font color="Maroon" title="c5">p</font>, <font color="Maroon" title="c6">q</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E4:70"/><i><font color="Green" title="E27">A2</font></i>: 
( <font color="Maroon" title="c4">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c6">q</font> &amp; <font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D2</font> &amp; <font color="Maroon" title="c6">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D1</font> )
 ;<br/>
<b>thus </b><a NAME="E5:70"/>
<font color="Maroon" title="c4">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D1</font> <a href="filter_2.html#K20" title="FILTER_2:func.20">"\/"</a> <font color="Maroon" title="c3">D2</font>
 <b>by </b><i><a class="txt" href="#E4:70"><i><font color="Green" title="E27">A2</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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