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 Th2:
  for T being 1-sorted, F being Subset-Family of T, X being set st
  X c= F holds X is Subset-Family of T
proof
  let T be 1-sorted, F be Subset-Family of T, X be set;
  assume
A1: X c= F;
  X c= bool the carrier of T
  proof
    let y be object;
    assume y in X;
    then y in F by A1;
    hence thesis;
  end;
  hence thesis;
end;
