
theorem
  for R,T being non empty TopSpace holds R is_Retract_of T iff ex S
  being non empty SubSpace of T st S is_a_retract_of T & S,R are_homeomorphic
proof
  let R,T be non empty TopSpace;
  hereby
    assume R is_Retract_of T;
    then consider f being Function of T,T such that
A1: f is continuous and
A2: f*f = f and
A3: Image f, R are_homeomorphic by WAYBEL18:def 8;
    reconsider S = Image f as non empty SubSpace of T;
    f = corestr f by WAYBEL18:def 7;
    then reconsider f as continuous Function of T,S by A1,WAYBEL18:10;
    take S;
A4: [#]T = the carrier of T;
    [#]S = the carrier of S;
    then the carrier of S c= the carrier of T by A4,PRE_TOPC:def 4;
    then reconsider rf = rng f as Subset of T by XBOOLE_1:1;
    now
      let x be Point of T;
      assume x in the carrier of S;
      then x in [#](T|rf) by WAYBEL18:def 6;
      then x in rng f by PRE_TOPC:def 5;
      then ex y being object st y in dom f & x = f.y by FUNCT_1:def 3;
      hence f.x = x by A2,FUNCT_1:13;
    end;
    then f is being_a_retraction;
    hence S is_a_retract_of T & S, R are_homeomorphic by A3;
  end;
  given S being non empty SubSpace of T such that
A5: S is_a_retract_of T and
A6: S,R are_homeomorphic;
  thus thesis by A5,A6,Th52,Th55;
end;
