
theorem ::p.121 lemma 3.2.(ii)
  for T being non empty TopSpace st T is injective for S being non empty
  SubSpace of T st S is_a_retract_of T holds S is injective
proof
  let T be non empty TopSpace;
  assume
A1: T is injective;
  let S be non empty SubSpace of T;
  set p = incl S;
  assume S is_a_retract_of T;
  then consider r being continuous Function of T,S such that
A2: r is being_a_retraction by BORSUK_1:def 17;
  let X be non empty TopSpace, F be Function of X, S;
  assume
A3: F is continuous;
  reconsider f = p*F as Function of X,T;
  let Y be non empty TopSpace;
  assume
A4: X is SubSpace of Y;
  p is continuous by TMAP_1:87;
  then consider g be Function of Y,T such that
A5: g is continuous and
A6: g|(the carrier of X) = f by A1,A3,A4;
  take G = r*g;
A7: the carrier of S c= the carrier of T by BORSUK_1:1;
A8: the carrier of X c= the carrier of Y by A4,BORSUK_1:1;
A9: for x being object st x in dom F holds F.x = G.x
  proof
    let x be object;
    assume
A10: x in dom F;
    then
A11: x in the carrier of X & g.x = f.x by A6,FUNCT_1:49;
A12: F.x in rng F by A10,FUNCT_1:def 3;
    then F.x in the carrier of S;
    then reconsider y = F.x as Point of T by A7;
A13: f.x = p.y by A10,FUNCT_2:15
      .= F.x by A12,TMAP_1:84;
    F.x = r.y by A2,A12,BORSUK_1:def 16;
    hence thesis by A8,A13,A11,FUNCT_2:15;
  end;
  thus G is continuous by A5;
A14: dom F = the carrier of X by FUNCT_2:def 1;
  dom G /\ the carrier of X = (the carrier of Y) /\ the carrier of X by
FUNCT_2:def 1
    .= the carrier of X by A4,BORSUK_1:1,XBOOLE_1:28;
  hence thesis by A14,A9,FUNCT_1:46;
end;
