
theorem Th62:
  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 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 &
      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))
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 A1: P in FinMeetCl product_prebasis J;
  consider X being Subset-Family of product Carrier J,
    f being one-to-one I-valued Function such that
    A2: X c= product_prebasis J & X is finite & P = Intersect X & dom f = X and
    A3: P = product(Carrier J +* product_basis_selector(J,f)) by A1, Lm3;
  take X, f;
  thus X c= product_prebasis J & X is finite & P = Intersect X & dom f = X
    by A2;
  A4: now
    let A be Subset of product Carrier J;
    assume A5: A in X;
    A is empty implies proj(J,f/.A).:A = {}(J.(f/.A));
    hence proj(J,f/.A).:A is open by A2, A5, Th58;
  end;
  f" is non-empty
  proof
    assume f" is non non-empty;
    then {} in rng(f") by RELAT_1:def 9;
    then {} in X by A2, FUNCT_1:33;
    then X is non empty & meet X = {} by SETFAM_1:4;
    hence contradiction by A2, SETFAM_1:def 9;
  end;
  hence thesis by A2, A3, A4, Th60;
end;
