reserve T for non empty TopSpace,
  A, B for Subset of T,
  F, G for Subset-Family of T;

theorem
  A is G_delta & B is G_delta implies A \/ B is G_delta
proof
  assume that
A1: A is G_delta and
A2: B is G_delta;
  consider F being open countable Subset-Family of T such that
A3: A = meet F by A1;
  consider G being open countable Subset-Family of T such that
A4: B = meet G by A2;
  reconsider H = UNION (F,G) as Subset-Family of T;
  per cases;
  suppose
A5: F <> {} & G <> {};
A6: meet UNION(F,G) c= meet F \/ meet G by Th32;
    meet F \/ meet G c= meet UNION(F,G) by A5,SETFAM_1:29;
    then
A7: A \/ B = meet H by A3,A4,A6;
    card H c= card [:F,G:] & [:F,G:] is countable by Th26,CARD_4:7;
    then
A8: H is countable by WAYBEL12:1;
    H is open by Th24;
    hence thesis by A7,A8;
  end;
  suppose
    F = {} or G = {};
    then A = {} or B = {} by A3,A4,SETFAM_1:def 1;
    hence thesis by A1,A2;
  end;
end;
