reserve L for Lattice,
  p,q,r for Element of L,
  p9,q9,r9 for Element of L.:,
  x, y for set;
reserve I,J for Ideal of L,
  F for Filter of L;
reserve D for non empty Subset of L,
  D9 for non empty Subset of L.:;
reserve D1,D2 for non empty Subset of L,
  D19,D29 for non empty Subset of L.:;

theorem
  (D1 c= D2 implies (.D1.> c= (.D2.>) & (.(.D.>.> c= (.D.>
proof
  D2 c= (.D2.> by Def9;
  then D1 c= D2 implies D1 c= (.D2.>;
  hence thesis by Def9;
end;
