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;

theorem
  for F being Subset-Family of A holds F is Subset-Family of T
proof
  let F be Subset-Family of A;
  [#] A c= [#]T by PRE_TOPC:def 4;
  then bool the carrier of A c= bool the carrier of T by ZFMISC_1:67;
  hence thesis by XBOOLE_1:1;
end;
