reserve L for Lattice,
  p,q,r for Element of L,
  p9,q9,r9 for Element of L.:,
  x, y for set;

theorem
  for X being Subset of L st for p,q holds p in X & q in X iff p
  "\/"q in X holds X is ClosedSubset of L
proof
  let X be Subset of L such that
A1: for p,q holds p in X & q in X iff p"\/"q in X;
 for p,q holds p in X & q in X implies p"/\"q in X
  proof
  let p,q;
  (p"/\"q)"\/"q = q by LATTICES:def 8;
  hence thesis by A1;
 end;
 hence thesis by A1,LATTICES:def 24,def 25;
end;
