reserve a, b, r, s for Real;

theorem Th16:
  for X being real-bounded Subset of REAL st not upper_bound X in X
  holds X c= [.lower_bound X,upper_bound X.[
proof
  let X be real-bounded Subset of REAL such that
A1: not upper_bound X in X;
  let x be object;
  assume
A2: x in X;
  then reconsider x as Real;
  x <= upper_bound X by A2,SEQ_4:def 1;
  then
A3: x < upper_bound X by A1,A2,XXREAL_0:1;
  lower_bound X <= x by A2,SEQ_4:def 2;
  hence thesis by A3,XXREAL_1:3;
end;
