reserve a, b, r, s for Real;

theorem Th22:
  for X being Subset of REAL st X is bounded_above & not
  upper_bound X in X holds X c= left_open_halfline(upper_bound X)
proof
  let X be Subset of REAL such that
A1: X is bounded_above and
A2: not upper_bound X in X;
  let x be object;
  assume
A3: x in X;
  then reconsider x as Real;
  x <= upper_bound X by A1,A3,SEQ_4:def 1;
  then x < upper_bound X by A2,A3,XXREAL_0:1;
  hence thesis by XXREAL_1:233;
end;
