let L be complete Lattice; :: thesis: for L9 being SubLattice of L st L9 is \/-inheriting holds
L9 is complete
let L9 be SubLattice of L; :: thesis: ( L9 is \/-inheriting implies L9 is complete )
assume A1: L9 is \/-inheriting ; :: thesis: L9 is complete
for X being Subset of L9 ex a being Element of L9 st
( X is_less_than a & ( for b being Element of L9 st X is_less_than b holds
a [= b ) )

proof

let X be Subset of L9; :: thesis: ex a being Element of L9 st
( X is_less_than a & ( for b being Element of L9 st X is_less_than b holds
a [= b ) )
set a = "\/" (X,L);
reconsider a9 = "\/" (X,L) as Element of L9 by A1;
take a9 ; :: thesis: ( X is_less_than a9 & ( for b being Element of L9 st X is_less_than b holds
a9 [= b ) )
X is_less_than "\/" (X,L) by LATTICE3:def 21;
hence X is_less_than a9 by Th13; :: thesis: for b being Element of L9 st X is_less_than b holds
a9 [= b
let b9 be Element of L9; :: thesis: ( X is_less_than b9 implies a9 [= b9 )
the carrier of L9 c= the carrier of L by NAT_LAT:def 12;
then reconsider b = b9 as Element of L ;
assume X is_less_than b9 ; :: thesis: a9 [= b9
then X is_less_than b by Th13;
then A2: "\/" (X,L) [= b by LATTICE3:def 21;
a9 "/\" b9 = ("\/" (X,L)) "/\" b by Th11
.= a9 by A2, LATTICES:4 ;
hence a9 [= b9 by LATTICES:4; :: thesis: verum

end;

hence L9 is complete ; :: thesis: verum