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

theorem Th21:
  for D being non empty Subset of L holds D is Ideal of L iff (for
p,q st p in D & q in D holds p "\/" q in D) & for p,q st p in D & q [= p holds
  q in D
proof
  let D be non empty Subset of L;
  thus D is Ideal of L implies (for p,q st p in D & q in D holds p "\/" q in D
  ) & for p,q st p in D & q [= p holds q in D
  by Lm1;
  assume
A1: ( for p,q st p in D & q in D holds p"\/"q in D)& for p,q st p in D &
  q [= p holds q in D;
   for p,q holds p in D & q in D iff p"\/"q in D by A1,LATTICES:5;
  hence thesis by Lm1;
end;
