reserve T1,T2,T3 for TopSpace,
  A1 for Subset of T1, A2 for Subset of T2, A3 for Subset of T3;

theorem Th7:
  A1,A2 are_homeomorphic implies (A1 is empty iff A2 is empty)
proof
  assume A1,A2 are_homeomorphic;
  then consider f be Function of T1|A1,T2|A2 such that
A1: f is being_homeomorphism by METRIZTS:def 1,T_0TOPSP:def 1;
  dom f = [#](T1|A1) & rng f = [#](T2|A2) & f is one-to-one &
  f is continuous & f" is continuous by A1,TOPS_2:def 5;
  hence thesis by PRE_TOPC:def 5;
end;
