
theorem Th65:
  for I being 1-element set
  for J being TopSpace-yielding non-Empty ManySortedSet of I
  holds the topology of product J = product_prebasis J
proof
  let I be 1-element set;
  let J be TopSpace-yielding non-Empty ManySortedSet of I;
  reconsider T = product J as TopSpace;
  reconsider K = product_prebasis J as Subset-Family of T by WAYBEL18:def 3;
  K is prebasis of T by WAYBEL18:def 3;
  then A1: UniCl FinMeetCl K = the topology of T by YELLOW_9:22, YELLOW_9:23;
  FinMeetCl K = FinMeetCl product_prebasis J by WAYBEL18:def 3;
  then UniCl FinMeetCl K = UniCl FinMeetCl product_prebasis J
    by WAYBEL18:def 3;
  hence thesis by A1, Lm7;
end;
