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
  W is closed implies meet W is closed
proof
  reconsider C = COMPLEMENT(W) as Subset-Family of GX;
  assume W is closed;
  then COMPLEMENT(W) is open by Th9;
  then
A1: union C is open by Th19;
A2: now
    assume W <> {};
    then union COMPLEMENT(W) = (meet W)` by Th7;
    hence thesis by A1;
  end;
  now
    assume W = {};
    then meet W = {}GX by SETFAM_1:def 1;
    hence thesis;
  end;
  hence thesis by A2;
end;
