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 "\/" D2 = D2 "\/" D1
proof
  let r;
  thus r in D1"\/"D2 implies r in D2"\/"D1
  proof
    assume r in D1"\/"D2;
    then ex p,q st r = p"\/"q & p in D1 & q in D2;
    hence thesis;
  end;
  assume r in D2"\/"D1;
  then ex p,q st r = p"\/"q & p in D2 & q in D1;
  hence thesis;
end;
