
theorem
  for S,T being non empty TopSpace holds S, T are_homeomorphic iff ex f
  being continuous Function of S,T, g being continuous Function of T,S st f*g =
  id T & g*f = id S
proof
  let S,T be non empty TopSpace;
  hereby
    assume S, T are_homeomorphic;
    then consider f being Function of S,T such that
A1: f is being_homeomorphism;
    reconsider f as continuous Function of S,T by A1,TOPS_2:def 5;
A2: rng f = [#]T by A1,TOPS_2:def 5;
    reconsider g = f" as continuous Function of T,S by A1,TOPS_2:def 5;
    take f,g;
A3: dom f = [#]S by A1,TOPS_2:def 5;
    f is one-to-one by A1,TOPS_2:def 5;
    hence f*g = id T & g*f = id S by A2,A3,TOPS_2:52;
  end;
  given f being continuous Function of S,T, g being continuous Function of T,S
  such that
A4: f*g = id T and
A5: g*f = id S;
A6: f is onto by A4,FUNCT_2:23;
    then
A7: rng f = [#]T by FUNCT_2:def 3;
  take f;
A8: dom f = [#]S by FUNCT_2:def 1;
A9: f is one-to-one by A5,FUNCT_2:23;
  then g = f qua Function" by A5,A7,FUNCT_2:30
    .= f" by A6,A9,TOPS_2:def 4;
  hence thesis by A9,A7,A8,TOPS_2:def 5;
end;
