reserve a,b,c,d,e for Real;

theorem Th10:
  for X be non empty bounded_below bounded_above real-membered set
  st upper_bound X = lower_bound X
  holds ex r be Real st X = {r}
  proof
    let X be non empty bounded_below bounded_above real-membered set;
    assume
A1: upper_bound X = lower_bound X;
    for r be Real st r in X holds upper_bound X = r
    proof
      let r be Real;
      assume r in X;
      then upper_bound X <= r & r <= upper_bound X by A1,SEQ_4:def 1,def 2;
      hence thesis by XXREAL_0:1;
    end;
    hence thesis by Th09;
  end;
