
theorem
  for I being non empty set
  for J being TopSpace-yielding non-Empty ManySortedSet of I
  for P being Subset of product Carrier J
  holds P in FinMeetCl product_prebasis J iff
    ex X being Subset-Family of product Carrier J,
      f being one-to-one I-valued Function
    st X c= product_prebasis J & X is finite & P = Intersect X & dom f = X &
      P = product(Carrier J +* product_basis_selector(J,f))
  by Lm3, CANTOR_1:def 3;
