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

theorem
  A1,A2 are_homeomorphic & A2,A3 are_homeomorphic
  implies A1,A3 are_homeomorphic
proof
  assume
A1: A1,A2 are_homeomorphic; then
A2: T1|A1,T2|A2 are_homeomorphic by METRIZTS:def 1;
  assume A3: A2,A3 are_homeomorphic; then
A4: T2|A2,T3|A3 are_homeomorphic by METRIZTS:def 1;
  per cases;
  suppose A5: A2 is non empty; then
A6: A1 is non empty & A3 is non empty by A1,A3,Th7;
    T1 is non empty & T2 is non empty & T3 is non empty by A5,A1,A3,Th7;
    hence A1,A3 are_homeomorphic by A2,A4,A6,A5,BORSUK_3:3,METRIZTS:def 1;
  end;
  suppose A2 is empty;
    then
A7: A1 is empty & A3 is empty by A1,A3,Th7;
    reconsider f = {} as Function;
A8: the carrier of T1|A1 = {} & the carrier of T3|A3 = {} by A7;
   dom f = {} & rng f = {};
    then reconsider f as Function of T1|A1,T3|A3 by A8,FUNCT_2:1;
A9: dom f = [#](T1|A1) & rng f = [#](T3|A3) by A8;
    for P1 being Subset of T3|A3 st P1 is closed holds f"P1 is closed;
    then
A10: f is continuous by PRE_TOPC:def 6;
    reconsider g = f as onto one-to-one PartFunc of {},{} by FUNCTOR0:1;
    for P1 being Subset of T1|A1 st P1 is closed holds (f")"P1 is closed by A7;
    then f" is continuous by PRE_TOPC:def 6;
    then f is being_homeomorphism by A9,A10,TOPS_2:def 5;
    hence thesis by METRIZTS:def 1,T_0TOPSP:def 1;
  end;
end;
