
theorem Th47:
  for S,T being non empty TopSpace, f being Function of S,T
  st f is bijective &
    ex K being Basis of S, L being Basis of T st f.:K = L
  holds f is being_homeomorphism
proof
  let S, T be non empty TopSpace, f be Function of S,T;
  assume A1: f is bijective;
  given K being Basis of S, L being Basis of T such that
    A2: f.:K = L;
  for W being Subset of T st W in L holds f"W is open
  proof
    let W be Subset of T;
    assume W in L;
    then consider V being Subset of S such that
      A3: V in K & W = f.:V by A2, FUNCT_2:def 10;
    dom f = the carrier of S by FUNCT_2:def 1;
    then V = f"W by A1, A3, FUNCT_1:94;
    hence thesis by A3, TOPS_2:def 1;
  end;
  then A4: f is continuous by YELLOW_9:34;
  for V being Subset of S st V in K holds f.:V is open
  proof
    let V be Subset of S;
    assume V in K;
    then f.:V in f.:K by FUNCT_2:def 10;
    hence thesis by A2, TOPS_2:def 1;
  end;
  then f is open by Th45;
  then A5: f" is continuous by A1, TOPREALA:14;
  A6: rng f = the carrier of T by A1, FUNCT_2:def 3
    .= [#]T by STRUCT_0:def 3;
  dom f = the carrier of S by FUNCT_2:def 1
    .= [#]S by STRUCT_0:def 3;
  hence thesis by A1, A5, A4, A6, TOPS_2:def 5;
end;
