reserve n   for Nat,
        r,s for Real,
        x,y for Element of REAL n,
        p,q for Point of TOP-REAL n,
        e   for Point of Euclid n;
reserve n for non zero Nat;
reserve n for non zero Nat;
reserve n for Nat,
        X for set,
        S for Subset-Family of X;

theorem
  the_set_of_all_closed_real_bounded_intervals =
    {I where I is Subset of REAL : I is closed_interval}
  proof
A1: the_set_of_all_closed_real_bounded_intervals c=
      {I where I is Subset of REAL : I is closed_interval}
    proof
      let x be object;
      assume x in the_set_of_all_closed_real_bounded_intervals;
      then consider a,b be Real such that
A2:   x = [.a,b.];
      reconsider x1 = x as Subset of REAL by A2;
      x1 is closed_interval by A2,MEASURE5:def 3;
      hence thesis;
    end;
    {I where I is Subset of REAL : I is closed_interval} c=
      the_set_of_all_closed_real_bounded_intervals
    proof
      let x be object;
      assume x in {I where I is Subset of REAL : I is closed_interval};
      then consider I0 be Subset of REAL such that
      A3: x = I0 and
      A4: I0 is closed_interval;
      consider a,b being Real such that
A5:   I0 = [.a,b.] by A4,MEASURE5:def 3;
      thus thesis by A3,A5;
    end;
    hence thesis by A1;
  end;
