
theorem
  for R, S being antisymmetric reflexive transitive with_suprema non
empty RelStr for x being set st x in the carrier of S holds x is Element of R
  [*] S
proof
  let R, S be antisymmetric reflexive transitive with_suprema non empty
  RelStr;
  let x be set;
  assume x in the carrier of S;
  then x in (the carrier of R) \/ (the carrier of S) by XBOOLE_0:def 3;
  hence thesis by Def2;
end;
