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

theorem
  A is F_sigma & B is F_sigma implies A \/ B is F_sigma
proof
  assume that
A1: A is F_sigma and
A2: B is F_sigma;
  consider F being closed countable Subset-Family of T such that
A3: A = union F by A1;
  consider G being closed countable Subset-Family of T such that
A4: B = union G by A2;
  reconsider H = UNION (F,G) as Subset-Family of T;
  per cases;
  suppose
A5: A <> {} & B <> {};
A6: H is closed by Th22;
    card H c= card [:F,G:] & [:F,G:] is countable by Th26,CARD_4:7;
    then
A7: H is countable by WAYBEL12:1;
    A \/ B = union H by A3,A4,A5,Th28,ZFMISC_1:2;
    hence thesis by A6,A7;
  end;
  suppose
    A = {};
    hence thesis by A2;
  end;
  suppose
    B = {};
    hence thesis by A1;
  end;
end;
