
theorem Th56:
  for I being non empty set
  for J being TopSpace-yielding non-Empty ManySortedSet of I
  holds {} in product_prebasis J
proof
  let I be non empty set;
  let J be TopSpace-yielding non-Empty ManySortedSet of I;
  set P = the empty Subset of product Carrier J;
  ex i being set, T being TopStruct, V being Subset of T
    st i in I & V is open & T = J.i & P = product ((Carrier J) +* (i,V))
  proof
    set i = the Element of I;
    set V = the empty Subset of J.i;
    take i, J.i, V;
    dom Carrier J = I by PARTFUN1:def 2;
    hence thesis by Th36;
  end;
  hence thesis by WAYBEL18:def 2;
end;
