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 F_sigma & F = A/\B holds F is F_sigma
proof
  let F be Subset of T|B;
  assume that
A1: A is F_sigma and
A2: F=A/\B;
  consider G be closed countable Subset-Family of T such that
A3: A=union G by A1,TOPGEN_4:def 6;
A4: union(G|B)c=F
  proof
    let x be object;
    assume x in union(G|B);
    then consider g be set such that
A5: x in g and
A6: g in G|B by TARSKI:def 4;
    consider h be Subset of T such that
A7: h in G and
A8: h/\B=g by A6,TOPS_2:def 3;
    x in h by A5,A8,XBOOLE_0:def 4;
    then
A9: x in A by A3,A7,TARSKI:def 4;
    x in B by A5,A8,XBOOLE_0:def 4;
    hence thesis by A2,A9,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
A10: G|B is closed & G|B is countable by TOPS_2:38;
  A/\B/\B = A/\(B/\B) by XBOOLE_1:16
    .= F by A2;
  then F c=union(G|B) by A3,TOPS_2:32,XBOOLE_1:17;
  then F=union(G|B) by A4;
  hence thesis by A10,TOPGEN_4:def 6;
end;
