
theorem Th52:
  for T, S, R being non empty TopSpace st S is_Retract_of T & S, R
  are_homeomorphic holds R is_Retract_of T
proof
  let T, S, R be non empty TopSpace;
  given f being continuous Function of S,T, g being continuous Function of T,S
  such that
A1: g*f = id S;
  given h being Function of S,R such that
A2: h is being_homeomorphism;
  h" is continuous by A2,TOPS_2:def 5;
  then reconsider f9 = f*(h") as continuous Function of R,T;
  h is continuous by A2,TOPS_2:def 5;
  then reconsider g9 = h*g as continuous Function of T,R;
  take f9,g9;
  thus thesis by A1,A2,Th51;
end;
