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
  F is closed implies F|M is closed
proof
  assume
A1: F is closed;
  let Q be Subset of T|M;
  assume Q in F|M;
  then consider R being Subset of T such that
A2: R in F and
A3: R /\ M = Q by Def3;
  reconsider R as Subset of T;
A4: Q = R /\ [#](T|M) by A3,PRE_TOPC:def 5;
  R is closed by A1,A2;
  hence thesis by A4,PRE_TOPC:13;
end;
