
theorem Th60:
  for I being non empty set
  for J being TopSpace-yielding non-Empty ManySortedSet of I
  for f being one-to-one I-valued Function
  for X being Subset-Family of product Carrier J
  st X c= product_prebasis J & dom f = X & f" is non-empty &
    for A being Subset of product Carrier J st A in X
    holds proj(J,f/.A).:A is open
  holds for i being Element of I holds
    proj(J,i).:product(Carrier J +* product_basis_selector(J,f)) is open &
    (not i in rng f implies
      proj(J,i).:product(Carrier J +* product_basis_selector(J,f)) = [#](J.i))
proof
  let I be non empty set;
  let J be TopSpace-yielding non-Empty ManySortedSet of I;
  let f be one-to-one I-valued Function;
  let X be Subset-Family of product Carrier J;
  set g = product_basis_selector(J,f);
  set P = product(Carrier J +* g);
  assume that
    A1: X c= product_prebasis J & dom f = X & f" is non-empty and
    A2: for A being Subset of product Carrier J st A in X
      holds proj(J,f/.A).:A is open;
  let i be Element of I;
  not i in rng f implies
      proj(J,i).:P = [#](J.i) by A1, A2, Th59;
  hence thesis by A1, A2, Th59;
end;
