
theorem
  for S, T being non empty TopSpace st T is injective & S is_Retract_of
  T holds S is injective
proof
  let S, T be non empty TopSpace such that
A1: T is injective and
A2: S is_Retract_of T;
  consider h being Function of T,T such that
A3: h is continuous and
A4: h*h = h and
A5: Image h, S are_homeomorphic by A2;
  set F = corestr h;
  for W being Point of T st W in the carrier of Image h holds F.W = W
  proof
    let W be Point of T;
    assume W in the carrier of Image h;
    then W in rng h by WAYBEL18:9;
    then ex x being object st x in dom h & W = h.x by FUNCT_1:def 3;
    hence thesis by A4,FUNCT_1:13;
  end;
  then
A6: F is being_a_retraction by BORSUK_1:def 16;
  corestr h is continuous by A3,WAYBEL18:10;
  then Image h is_a_retract_of T by A6,BORSUK_1:def 17;
  then Image h is injective by A1,WAYBEL18:8;
  hence thesis by A5,Th6;
end;
