reserve a, b for Real,
  r for Real,
  rr for Real,
  i, j, n for Nat,
  M for non empty MetrSpace,
  p, q, s for Point of TOP-REAL 2,
  e for Point of Euclid 2,
  w for Point of Euclid n,
  z for Point of M,
  A, B for Subset of TOP-REAL n,
  P for Subset of TOP-REAL 2,
  D for non empty Subset of TOP-REAL 2;
reserve a, b for Real;
reserve a, b for Real;
reserve r for Real;

theorem Th67:
  for D being non empty bounded_above Subset of REAL holds upper_bound D =
  upper_bound Cl D
proof
  let D be non empty bounded_above Subset of REAL;
A1: for q being Real st for p being Real st p in D holds p <=
  q holds upper_bound Cl D <= q
  proof
    let q be Real such that
A2: for p being Real st p in D holds p <= q;
    for p being Real st p in Cl D holds p <= q
    proof
      let p be Real;
      assume p in Cl D;
      then consider s being Real_Sequence such that
A3:   rng s c= D and
A4:   s is convergent and
A5:   lim s = p by MEASURE6:64;
      for n holds s.n <= q
      proof
        let n;
A6:   n in NAT by ORDINAL1:def 12;
        dom s = NAT by FUNCT_2:def 1;
        then s.n in rng s by A6,FUNCT_1:def 3;
        hence thesis by A2,A3;
      end;
      hence thesis by A4,A5,PREPOWER:2;
    end;
    hence thesis by SEQ_4:45;
  end;
A7: upper_bound Cl D >= upper_bound D by MEASURE6:58,SEQ_4:48;
  for p being Real st p in D holds p <= upper_bound Cl D
  proof
    let p be Real;
    assume p in D;
    then upper_bound D >= p by SEQ_4:def 1;
    hence thesis by A7,XXREAL_0:2;
  end;
  hence thesis by A1,SEQ_4:46;
end;
