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 c= G implies F|M c= G|M
proof
  assume
A1: F c= G;
  let x be object;
  assume
A2: x in F|M;
  then reconsider X=x as Subset of T|M;
  ex R being Subset of T st R in F & R /\ M = X by A2,Def3;
  hence thesis by A1,Def3;
end;
