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
( a is_less_than X & ( for b being Element of L9 st b is_less_than X holds
b [= a ) )
proof
let X be Subset of L9; :: thesis: ex a being Element of L9 st
( a is_less_than X & ( for b being Element of L9 st b is_less_than X holds
b [= a ) )

set a = "/\" (X,L);
reconsider a9 = "/\" (X,L) as Element of L9 by A1;
take a9 ; :: thesis: ( a9 is_less_than X & ( for b being Element of L9 st b is_less_than X holds
b [= a9 ) )

"/\" (X,L) is_less_than X by LATTICE3:34;
hence a9 is_less_than X by Th12; :: thesis: for b being Element of L9 st b is_less_than X holds
b [= a9

let b9 be Element of L9; :: thesis: ( b9 is_less_than X implies b9 [= a9 )
the carrier of L9 c= the carrier of L by NAT_LAT:def 12;
then reconsider b = b9 as Element of L ;
assume b9 is_less_than X ; :: thesis: b9 [= a9
then b is_less_than X by Th12;
then A2: b [= "/\" (X,L) by LATTICE3:39;
b9 "/\" a9 = b "/\" ("/\" (X,L)) by Th11
.= b9 by ;
hence b9 [= a9 by LATTICES:4; :: thesis: verum
end;
hence L9 is complete by VECTSP_8:def 6; :: thesis: verum