
theorem Th23:
  for X being Subset of I[01], X9 being Subset of REAL, x being
  Real st x in X9 & X9 = X holds lower_bound X9 <= x &
   x <= upper_bound X9
proof
  let X be Subset of I[01], X9 be Subset of REAL, x be Real;
  assume that
A1: x in X9 and
A2: X9 = X;
  X9 is bounded_above bounded_below by A2,Th22;
  hence thesis by A1,SEQ_4:def 1,def 2;
end;
