reserve a, b, r, s for Real;

theorem Th18:
  for X being real-bounded Subset of REAL st not lower_bound X in X &
  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 lower_bound X in X 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 A3,SEQ_4:def 1;
  then
A4: x < upper_bound X by A2,A3,XXREAL_0:1;
  lower_bound X <= x by A3,SEQ_4:def 2;
  then lower_bound X < x by A1,A3,XXREAL_0:1;
  hence thesis by A4,XXREAL_1:4;
end;
