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;
reserve n for Nat,
        S for Subset-Family of REAL;
reserve n       for Nat,
        a,b,c,d for Element of REAL n;
reserve n for non zero Nat;

theorem
  MeasurableRectangle(ProductLeftOpenIntervals(n)) =
    Product(n,the_set_of_all_left_open_real_bounded_intervals)
  proof
    thus MeasurableRectangle(ProductLeftOpenIntervals(n))
      c= Product(n,the_set_of_all_left_open_real_bounded_intervals)
    proof
      let x be object;
      assume x in MeasurableRectangle(ProductLeftOpenIntervals(n));
      then consider y being Subset of REAL n, a, b being Element of REAL n
        such that
A1:   x = y and
A2:   for s being object holds (s in y) iff
          (ex t being Element of REAL n st s = t & for i being Nat st
             i in Seg n holds t.i in ]. a.i, b.i .]) by Th32;
      now
        let t being Element of REAL n;
        hereby
          assume
A3:       t in y;
          hereby
            let i be Nat;
            assume
A4:         i in Seg n;
            consider t0 be Element of REAL n such that
A5:         t = t0 and
A6:         for i be Nat st i in Seg n holds t0.i in ].a.i,b.i.] by A3,A2;
            thus t.i in ].a.i,b.i.] by A6,A4,A5;
          end;
         end;
        assume for i be Nat st i in Seg n holds t.i in ].a.i,b.i.];
        hence t in y by A2;
      end;
      then y is Element of Product(n,
                the_set_of_all_left_open_real_bounded_intervals) by Th44;
      hence thesis by A1;
    end;
    thus Product(n,the_set_of_all_left_open_real_bounded_intervals) c=
    MeasurableRectangle(ProductLeftOpenIntervals(n))
    proof
      let x be object;
      assume x in Product(n,the_set_of_all_left_open_real_bounded_intervals);
      then consider g being Function such that
A7:   x = product g and
A8:   g in product (Seg n -->
                    the_set_of_all_left_open_real_bounded_intervals) by Def2;
      thus x in MeasurableRectangle(ProductLeftOpenIntervals(n))
        by A7,A8,SRINGS_4:def 4;
    end;
  end;
