reserve x, y for set,
  T for TopStruct,
  GX for TopSpace,
  P, Q, M, N for Subset of T,
  F, G for Subset-Family of T,
  W, Z for Subset-Family of GX,
  A for SubSpace of T;
reserve S for non empty TopStruct,
  f for Function of T, S,
  H for Subset-Family of S;

theorem
  for T, S being TopStruct, f being Function of T, S, H being
  Subset-Family of S st f is continuous & H is closed holds for F being
  Subset-Family of T st F=("f).:H holds F is closed
proof
  let T, S be TopStruct, f be Function of T, S, H be Subset-Family of S;
  assume that
A1: f is continuous and
A2: H is closed;
  let F be Subset-Family of T such that
A3: F=("f).:H;
  let X be Subset of T;
  assume X in F;
  then consider y being object such that
A4: y in dom "f and
A5: y in H and
A6: X=("f).y by A3,FUNCT_1:def 6;
  reconsider Y=y as Subset of S by A5;
A7: X = f"Y by A4,A6,FUNCT_3:21;
  Y is closed by A2,A5;
  hence thesis by A1,A7;
end;
