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
  M c= union F implies M = union (F|M)
proof
  assume
A1: M c= union F;
  for x being object holds x in M iff x in union(F|M)
  proof let x be object;
    hereby
      assume
A2:   x in M;
      then consider A being set such that
A3:   x in A and
A4:   A in F by A1,TARSKI:def 4;
      reconsider A9=A as Subset of T by A4;
      A /\ M c= M by XBOOLE_1:17;
      then A /\ M c= [#](T|M) by PRE_TOPC:def 5;
      then reconsider B=A9 /\ M as Subset of T|M;
A5:   B in F|M by A4,Def3;
      x in A /\ M by A2,A3,XBOOLE_0:def 4;
      hence x in union(F|M) by A5,TARSKI:def 4;
    end;
    assume x in union(F|M);
    then consider A being set such that
A6: x in A and
A7: A in F|M by TARSKI:def 4;
    reconsider B = A as Subset of T|M by A7;
    ex R being Subset of T st R in F & R /\ M = B by A7,Def3;
    hence thesis by A6,XBOOLE_0:def 4;
  end;
  hence thesis by TARSKI:2;
end;
