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;

theorem
  for x being object st x in MeasurableRectangle(ProductLeftOpenIntervals(n))
  holds ex y being Subset of REAL n st
           x = y &
           for i being Nat st i in Seg n holds
              ex a,b being Real st
                for t being Element of REAL n st t in y holds t.i in ].a,b.]
  proof
    let x be object;
    assume
A1: x in MeasurableRectangle(ProductLeftOpenIntervals(n));
    MeasurableRectangle(ProductLeftOpenIntervals(n)) is
      Subset-Family of REAL n by Th30;
    then reconsider y = x as Subset of REAL n by A1;
    reconsider x0 = x as set by TARSKI:1;
    consider g be Function such that
A2: x = product g and
A3: g in product ProductLeftOpenIntervals(n) by A1,SRINGS_4:def 4;
    dom ProductLeftOpenIntervals(n) = Seg n by FUNCOP_1:13; then
A4: dom g = Seg n by A3,CARD_3:9;
    take y;
    thus x = y;
    now
      let i be Nat;
      assume
A5:   i in Seg n;
      then i in dom ProductLeftOpenIntervals(n) by FUNCOP_1:13;
      then g.i in (ProductLeftOpenIntervals(n)).i by A3,CARD_3:9;
      then g.i in the set of all ].a,b.] where a,b is Real by A5,FUNCOP_1:7;
      then consider a,b be Real such that
A6:   g.i = ].a,b.];
      hereby
        take a,b;
        now
          let t be Element of REAL n;
          assume t in y;
          then consider j0 be Function such that
A7:       t=j0 and
          dom j0 = Seg n and
A8:       for u be object st u in Seg n holds j0.u in g.u
            by A2,CARD_3:def 5,A4;
          thus t.i in ].a,b.] by A7,A6,A8,A5;
        end;
        hence ex a,b being Real st
        for t be Element of REAL n st t in y holds t.i in ].a,b.];
      end;
      hence ex a,b being Real st
      for t be Element of REAL n st t in y holds t.i in ].a,b.];
    end;
    hence thesis;
  end;
