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 Th31:
  Q in F implies Q /\ M in F|M
proof
  reconsider QQ = Q /\ M as Subset of T|M by Th29;
A1: Q /\ M = QQ;
  assume Q in F;
  hence thesis by A1,Def3;
end;
