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

theorem Th09:
  for X being non empty real-membered set st
  (for r being Real st r in X holds upper_bound X = r) holds
  ex r being Real st X = {r}
  proof
    let X be non empty real-membered set;
    assume
A1: for r be Real st r in X holds upper_bound X = r;
    ex s be object st (for x be object st x in X holds s = x)
    proof
      set s = upper_bound X;
      take s;
      thus thesis by A1;
    end;
    then consider r be object such that
A2: X = {r} by Lm03;
    reconsider r0 = the Element of X as Real;
    take r0;
    thus thesis by A2,TARSKI:def 1;
  end;
