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

theorem
  for f be Function of T1,T2 st f is being_homeomorphism holds
  f"A2,A2 are_homeomorphic
proof
  let f be Function of T1,T2 such that
A1: f is being_homeomorphism;
A2: dom(A2|`f) = f"A2 by Th1
    .= [#](T1|f"A2) by PRE_TOPC:def 5;
  rng f = [#]T2 by A1,TOPS_2:def 5;
  then rng(A2|`f) = A2 by RELAT_1:89
   .= [#](T2|A2) by PRE_TOPC:def 5;
  then reconsider g=A2|`f as Function of T1|f"A2,T2|A2 by A2,FUNCT_2:1;
  g is being_homeomorphism by A1,Th4;
  hence thesis by T_0TOPSP:def 1,METRIZTS:def 1;
end;
