reserve x, X, Y for set;
reserve L for complete LATTICE,
  a for Element of L;

theorem
  for L being non empty Poset holds x in dom SupMap L iff x is Ideal of L
proof
  let L be non empty Poset;
  hereby
    assume x in dom SupMap L;
    then x in Ids L by Th49;
    then ex I being Ideal of L st x = I;
    hence x is Ideal of L;
  end;
  assume x is Ideal of L;
  then x in the set of all X where X is Ideal of L;
  hence thesis by Th49;
end;
