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;

theorem
  for O being open non empty Subset of TOP-REAL n holds
  ex s being sequence of dense_countable_OpenHypercubes(n) st O = Union s
  proof
    let O be open non empty Subset of TOP-REAL n;
    consider s being sequence of dense_countable_OpenHypercubes(n) such that
A1: for x being object holds x in O iff ex y being object st y in NAT &
    x in s.y by Th25;
A2: dom s = NAT by FUNCT_2:def 1;
    take s;
    now
      let x be object;
      hereby
        assume x in O;
        then consider y be object such that
A3:     y in NAT and
A4:     x in s.y by A1;
        s.y in rng s by A3,A2,FUNCT_1:def 3;
        hence x in Union s by A4,TARSKI:def 4;
      end;
      assume x in Union s;
      then consider y be set such that
A5:   x in y and
A6:   y in rng s by TARSKI:def 4;
      consider z be object such that
A7:   z in dom s and
A8:   y = s.z by A6,FUNCT_1:def 3;
      thus x in O by A1,A5,A7,A8;
    end;
    hence thesis by TARSKI:2;
  end;
