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
  for T being non empty 1-sorted, F being Subset-Family of T st F is
  Cover of T holds F <> {}
proof
  let T be non empty 1-sorted, F be Subset-Family of T;
  set x = the Element of union F;
  assume F is Cover of T;
  then union F = the carrier of T by SETFAM_1:45;
  then ex A being set st x in A & A in F by TARSKI:def 4;
  hence thesis;
end;
