reserve a, b, r, s for Real;

theorem Th13:
  for X being real-bounded interval Subset of REAL st lower_bound X in
  X & upper_bound X in X holds X = [.lower_bound X,upper_bound X.]
proof
  let X be real-bounded interval Subset of REAL such that
A1: lower_bound X in X & upper_bound X in X;
   reconsider X1=X as non empty real-bounded interval Subset of REAL by A1;
   X1 c= [.lower_bound X1,upper_bound X1.] by XXREAL_2:69;
  hence X c= [.lower_bound X,upper_bound X.];
  thus thesis by A1,XXREAL_2:def 12;
end;
