reserve X for TopStruct,
  A for Subset of X;

theorem Th1:
  (A = {}X iff A` = [#]X) & (A = {} iff A` = the carrier of X)
proof
  thus A = {}X iff A` = [#]X
  proof
    thus A = {}X implies A` = [#]X;
    assume A` = [#]X;
    then A`` = {}X by XBOOLE_1:37;
    hence thesis;
  end;
  hence thesis;
end;
