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 Th8:
  for T being 1-sorted, F being Subset-Family of T holds
  COMPLEMENT(F) is finite iff F is finite
proof
  let T be 1-sorted, F be Subset-Family of T;
  thus COMPLEMENT(F) is finite implies F is finite by Lm1;
  assume F is finite;
  then COMPLEMENT COMPLEMENT F is finite;
  hence thesis by Lm1;
end;
