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
  Q c= union F implies Q /\ M c= union(F|M)
proof
  assume
A1: Q c= union F;
  now
    assume M <> {};
    thus Q /\ M c= union(F|M)
    proof
      let x be object;
      assume
A2:   x in Q /\ M;
      then x in Q by XBOOLE_0:def 4;
      then consider Z being set such that
A3:   x in Z and
A4:   Z in F by A1,TARSKI:def 4;
      reconsider ZZ=Z as Subset of T by A4;
      ZZ /\ M in F|M by A4,Th31;
      then reconsider ZP = ZZ /\ M as Subset of T|M;
A5:   ZP in F|M by A4,Th31;
      x in M by A2,XBOOLE_0:def 4;
      then x in ZP by A3,XBOOLE_0:def 4;
      hence thesis by A5,TARSKI:def 4;
    end;
  end;
  hence thesis;
end;
