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
  for f be Function of T1,T2 st f is being_homeomorphism holds A1,f.:A1
  are_homeomorphic
proof
  let f be Function of T1,T2 such that
A1: f is being_homeomorphism;
  dom f=[#]T1 by A1;
  then
A2: dom(f|A1) = [#]T1/\A1 by RELAT_1:61
    .= A1 by XBOOLE_1:28
    .= [#](T1|A1) by PRE_TOPC:def 5;
  rng(f|A1) = f.:A1 by RELAT_1:115
    .= [#](T2|(f.:A1)) by PRE_TOPC:def 5;
  then reconsider g=f|A1 as Function of T1|A1,T2|(f.:A1) by A2,FUNCT_2:1;
  g is being_homeomorphism by A1,Th2;
  then T1|A1,T2|(f.:A1)are_homeomorphic by T_0TOPSP:def 1;
  hence thesis;
end;
