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<b>consider </b><font color="Maroon">c<sub>3</sub></font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="lattice4.html#M2">ClosedSubset</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> being   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E2:99_1_1"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>1</sub></font>) <a href="realset1.html#K1">||</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>1</sub></font>) <a href="realset1.html#K1">||</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="lattices.html#G3">LattStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="lattices.html#U1">L_meet</a> of <font color="Maroon">c<sub>2</sub></font> #) <a href="hidden.html#R1">=</a>  <a href="lattices.html#G3">LattStr</a>(# <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> #) )
 <b>by </b><i><a class="ref" href="filter_2.html#D15" target="_self">Def15</a></i>;<br/>
<a NAME="E3:99_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="lattice2.html#K1">.:</a>  is    <a href="nat_lat.html#M2">Sublattice</a> of <font color="Maroon">c<sub>1</sub></font> <a href="lattice2.html#K1">.:</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">c<sub>3</sub></font> <a href="filter_2.html#K10" target="_self">.:</a> 
; <i><font color="Red">:: according to </font></i><a class="ref" href="filter_2.html#D15" target="_self">FILTER_2:def 15</a><br/>

<b>thus </b><a NAME="E1:99_1_1_1"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">c<sub>3</sub></font> <a href="filter_2.html#K10" target="_self">.:</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="lattices.html#U2">L_join</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="lattice2.html#K1">.:</a> </span>)</span> <a href="realset1.html#K1">||</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="filter_2.html#K10" target="_self">.:</a> </span>)</span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> the <a href="lattices.html#U1">L_meet</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="lattice2.html#K1">.:</a> </span>)</span> <a href="realset1.html#K1">||</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="filter_2.html#K10" target="_self">.:</a> </span>)</span> &amp;  <a href="lattices.html#G3">LattStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice2.html#K1">.:</a> </span>)</span>,the <a href="lattices.html#U2">L_join</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice2.html#K1">.:</a> </span>)</span>,the <a href="lattices.html#U1">L_meet</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice2.html#K1">.:</a> </span>)</span> #) <a href="hidden.html#R1">=</a>  <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="filter_2.html#K10" target="_self">.:</a> </span>)</span>,<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> #) )
 <b>by </b><i><a class="txt" href="filter_2.html#E2:99_1_1"><i><font color="Green">E39</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:99_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="lattice2.html#K1">.:</a>  is  <a href="lattices.html#V3">strict</a>  <a href="nat_lat.html#M2">Sublattice</a> of <font color="Maroon">c<sub>1</sub></font> <a href="lattice2.html#K1">.:</a> 
 ;<br/>


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