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<span class="kw">consider </span><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#NM2" title="LATTICE4:NM.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><span class="kw"> such that </span><br/><a NAME="E2:90_1_1"/><span class="lab"><font color="Green" title="E36">A1</font></span>: 
( <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> #) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="filter_2.html#D13" target="_self" onmouseover="rs('filter_2/D13')" onmouseout="rh()">Def13</a></span>;<br/>
<a NAME="E3:90_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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="90_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">take </span>
<font color="Maroon" title="c3">P</font> <a href="filter_2.html#K3" title="FILTER_2:func.3">.:</a> 
; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="filter_2.html#D13" target="_self" onmouseover="rs('filter_2/D13')" onmouseout="rh()">FILTER_2:def 13</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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 <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="filter_2.html#K3" title="FILTER_2:func.3">.:</a></span>)</span> 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#K3" title="FILTER_2:func.3">.:</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#K3" title="FILTER_2:func.3">.:</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#K3" title="FILTER_2:func.3">.:</a></span>)</span>,<font color="Olive" title="b1">o1</font>,<font color="Olive" title="b2">o2</font> #) )</span><br/>

<span class="kw">thus </span><a NAME="E1:90_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 <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="filter_2.html#K3" title="FILTER_2:func.3">.:</a></span>)</span> 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#K3" title="FILTER_2:func.3">.:</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#K3" title="FILTER_2:func.3">.:</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#K3" title="FILTER_2:func.3">.:</a></span>)</span>,<font color="Olive" title="b1">o1</font>,<font color="Olive" title="b2">o2</font> #) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:90_1_1"><span class="lab"><font color="Green" title="E36">A1</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:90_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> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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