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 Th25:
  for O being open non empty Subset of TOP-REAL n holds
  ex s being sequence of dense_countable_OpenHypercubes(n) st
    for x being object holds x in O iff
    ex y being object st y in NAT & x in s.y
  proof
    let O be open non empty Subset of TOP-REAL n;
    consider Y being Subset of dense_countable_OpenHypercubes(n) such that
    Y is non empty and
A1: O = union Y and
A2: ex g being Function of NAT,Y st
    for x being object holds x in O iff ex y being object st y in NAT &
    x in g.y by Th24;
    consider g be Function of NAT,Y such that
A3: for x being object holds x in O iff ex y being object st y in NAT &
    x in g.y by A2;
    reconsider g2 = g as sequence of dense_countable_OpenHypercubes(n)
      by A1,ZFMISC_1:2,FUNCT_2:7;
    for x being object holds x in O iff ex y being object st y in NAT &
    x in g2.y by A3;
    hence thesis;
  end;
