
theorem
  for I being 1-element set
  for J being TopSpace-yielding non-Empty ManySortedSet of I
  for i being Element of I, P being Subset of product J
  holds P is open iff ex V being Subset of J.i
    st V is open & P = product ({i} --> V)
proof
  let I be 1-element set;
  let J be TopSpace-yielding non-Empty ManySortedSet of I;
  let i be Element of I;
  let P be Subset of product J;
  hereby
    assume P is open;
    then P in the topology of product J by PRE_TOPC:def 2;
    then P in product_prebasis J by Th65;
    hence ex V being Subset of J.i st V is open & P = product ({i} --> V)
      by Th64;
  end;
  A1: P is Subset of product Carrier J by WAYBEL18:def 3;
  assume ex V being Subset of J.i st V is open & P = product ({i} --> V);
  then P in product_prebasis J by A1, Th64;
  then P in the topology of product J by Th65;
  hence P is open by PRE_TOPC:def 2;
end;
