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<b>consider </b><font color="Maroon" title="c3">P</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="lattice4.html#M2" title="LATTICE4:mode.2">ClosedSubset</a> of <font color="Maroon" title="c1">L</font>, <font color="Maroon" title="c4">o1</font>, <font color="Maroon" title="c5">o2</font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c3">P</font><b> such that </b><br/><a NAME="E2:99_1_1"/><i><font color="Green" title="E39">A1</font></i>: 
( <font color="Maroon" title="c4">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">L</font>) <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c5">o2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c1">L</font>) <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c3">P</font> &amp;  <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">l</font>,the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c2">l</font>,the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <font color="Maroon" title="c2">l</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">o1</font>,<font color="Maroon" title="c5">o2</font> #) )
 <b>by </b><i><a class="ref" href="filter_2.html#D15" target="_self" title="FILTER_2:def.15">Def15</a></i>;<br/>
<a NAME="E3:99_1_1"/>
<font color="Maroon" title="c2">l</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a>  is    <a href="nat_lat.html#M2" title="NAT_LAT:mode.2">Sublattice</a> of <font color="Maroon" title="c1">L</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="99_1_1_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon" title="c3">P</font> <a href="filter_2.html#K10" title="FILTER_2:func.10">.:</a> 
; <i><font color="Red">:: according to </font></i><a class="ref" href="filter_2.html#D15" target="_self" title="FILTER_2:def.15">FILTER_2:def 15</a><br/>

<b>thus </b><a NAME="E1:99_1_1_1"/>
 ex <font color="Olive" title="b1">o1</font>, <font color="Olive" title="b2">o2</font> being  <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c3">P</font> <a href="filter_2.html#K10" title="FILTER_2:func.10">.:</a>  st <br/>( <font color="Olive" title="b1">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a> </span>)</span> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="filter_2.html#K10" title="FILTER_2:func.10">.:</a> </span>)</span> &amp; <font color="Olive" title="b2">o2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a> </span>)</span> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="filter_2.html#K10" title="FILTER_2:func.10">.:</a> </span>)</span> &amp;  <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">l</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a> </span>)</span>,the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">l</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a> </span>)</span>,the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">l</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a> </span>)</span> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="filter_2.html#K10" title="FILTER_2:func.10">.:</a> </span>)</span>,<font color="Olive" title="b1">o1</font>,<font color="Olive" title="b2">o2</font> #) )
 <b>by </b><i><a class="txt" href="filter_2.html#E2:99_1_1"><i><font color="Green" title="E39">A1</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:99_1_1"/>
<font color="Maroon" title="c2">l</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a>  is  <a href="lattices.html#V3" title="LATTICES:attr.3">strict</a>  <a href="nat_lat.html#M2" title="NAT_LAT:mode.2">Sublattice</a> of <font color="Maroon" title="c1">L</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a> 
 ;<br/>


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