
theorem Th63:
  for I being non empty set
  for J being TopSpace-yielding non-Empty ManySortedSet of I
  for P being non empty Subset of product Carrier J
  holds P in FinMeetCl product_prebasis J implies
    ex I0 being finite Subset of I st for i being Element of I holds
      proj(J,i).:P is open & (not i in I0 implies proj(J,i).:P = [#](J.i))
proof
  let I be non empty set;
  let J be TopSpace-yielding non-Empty ManySortedSet of I;
  let P be non empty Subset of product Carrier J;
  assume P in FinMeetCl product_prebasis J;
  then consider X being Subset-Family of product Carrier J,
    f being one-to-one I-valued Function such that
    A1: X c= product_prebasis J & X is finite & P = Intersect X & dom f = X and
    A2: for i being Element of I holds proj(J,i).:P is open &
      (not i in rng f implies proj(J,i).:P = [#](J.i)) by Th62;
  reconsider I0 = rng f as finite Subset of I by A1, FINSET_1:8;
  take I0;
  thus thesis by A2;
end;
