reserve T, T1, T2 for TopSpace,
  A, B for Subset of T,
  F, G for Subset-Family of T,
  A1 for Subset of T1,
  A2 for Subset of T2,
  TM, TM1, TM2 for metrizable TopSpace,
  Am, Bm for Subset of TM,
  Fm, Gm for Subset-Family of TM,
  C for Cardinal,
  iC for infinite Cardinal;

theorem
  for F be Subset of T|B st A is G_delta & F = A/\B holds F is G_delta
proof
  let F be Subset of T|B;
  assume that
A1: A is G_delta and
A2: F = A/\B;
  consider G be open countable Subset-Family of T such that
A3: A = meet G by A1,TOPGEN_4:def 7;
A4: meet(G|B) c= F
  proof
    let x be object;
    assume
A5: x in meet(G|B);
    then consider g be object such that
A6: g in G|B by SETFAM_1:1,XBOOLE_0:def 1;
    reconsider g as Subset of T|B by A6;
A7: ex h be Subset of T st h in G & h/\B=g by A6,TOPS_2:def 3;
    x in g by A5,A6,SETFAM_1:def 1;
    then
A8: x in B by A7,XBOOLE_0:def 4;
    now
      let Y be set;
      assume Y in G;
      then Y/\B in G|B by TOPS_2:31;
      then x in Y/\B by A5,SETFAM_1:def 1;
      hence x in Y by XBOOLE_0:def 4;
    end;
    then x in A by A3,A7,SETFAM_1:def 1;
    hence thesis by A2,A8,XBOOLE_0:def 4;
  end;
  card(G|B)c=card G & card G c=omega by Th7,CARD_3:def 14;
  then card(G|B)c=omega;
  then
A9: G|B is open & G|B is countable by TOPS_2:37;
  F c=meet(G|B)
  proof
    let x be object;
    assume
A10: x in F;
    then
A11: x in A by A2,XBOOLE_0:def 4;
A12: x in B by A2,A10,XBOOLE_0:def 4;
A13: now
      let f be set;
      assume f in G|B;
      then consider h be Subset of T such that
A14:  h in G and
A15:  h/\B=f by TOPS_2:def 3;
      x in h by A3,A11,A14,SETFAM_1:def 1;
      hence x in f by A12,A15,XBOOLE_0:def 4;
    end;
    ex y be object st y in G by A3,A11,SETFAM_1:1,XBOOLE_0:def 1;
    then G|B is non empty by TOPS_2:31;
    hence thesis by A13,SETFAM_1:def 1;
  end;
  then F=meet(G|B) by A4;
  hence thesis by A9,TOPGEN_4:def 7;
end;
