reserve T, T1, T2 for TopSpace,
  A, B for Subset of T,
  F, G for Subset-Family of T,
  A1 for Subset of T1,
  A2 for Subset of T2,
  TM, TM1, TM2 for metrizable TopSpace,
  Am, Bm for Subset of TM,
  Fm, Gm for Subset-Family of TM,
  C for Cardinal,
  iC for infinite Cardinal;

theorem
  T1,T2 are_homeomorphic iff [#]T1,[#]T2 are_homeomorphic
proof
A1: T1|([#]T1)=the TopStruct of T1 & T2|([#]T2)=the TopStruct of T2 by
TSEP_1:93;
  per cases;
  suppose
A2: T2 is non empty;
    thus T1,T2 are_homeomorphic implies [#]T1,[#]T2 are_homeomorphic
         by A1,A2,TOPREALA:15;
    assume [#]T1,[#]T2 are_homeomorphic;
    then the TopStruct of T1,the TopStruct of T2 are_homeomorphic by A1;
    hence thesis by A2,TOPREALA:15;
  end;
  suppose
A3: T2 is empty;
    hereby
      assume T1,T2 are_homeomorphic;
      then T1 is empty iff T2 is empty by YELLOW14:18;
      then T1|([#]T1),T2|([#]T2) are_homeomorphic by A3,Lm1;
      hence [#]T1,[#]T2 are_homeomorphic;
    end;
    assume [#]T1,[#]T2 are_homeomorphic;
    then T1|([#]T1),T2|([#]T2) are_homeomorphic;
    then T1 is empty by A3,YELLOW14:18;
    hence thesis by A3,Lm1;
  end;
end;
