
theorem ::p.126 exercise 3.13, 1 => 2
  for J being injective non empty TopSpace, Y being non empty TopSpace
  st J is SubSpace of Y holds J is_Retract_of Y
proof
  let J be injective non empty TopSpace, Y be non empty TopSpace;
  assume
A1: J is SubSpace of Y;
  then consider f being Function of Y, J such that
A2: f is continuous and
A3: f|(the carrier of J) = id J by WAYBEL18:def 5;
A4: the carrier of J c= the carrier of Y by A1,BORSUK_1:1;
  then reconsider ff = f as Function of Y, Y by FUNCT_2:7;
  ex ff being Function of Y, Y st ff is continuous & ff*ff = ff & Image ff
  , J are_homeomorphic
  proof
    reconsider M = [#]J as non empty Subset of Y by A1,BORSUK_1:1;
    take ff;
    thus ff is continuous by A1,A2,PRE_TOPC:26;
A5: dom f = the carrier of Y by FUNCT_2:def 1;
A6: for x being object st x in the carrier of Y holds (f*f).x = f.x
    proof
      let x be object;
      assume
A7:   x in the carrier of Y;
      hence (f*f).x = f.(f.x) by A5,FUNCT_1:13
        .= (id J).(f.x) by A3,A7,FUNCT_1:49,FUNCT_2:5
        .= f.x by A7,FUNCT_1:18,FUNCT_2:5;
    end;
    dom (ff*ff) = the carrier of Y by FUNCT_2:def 1;
    hence ff*ff = ff by A5,A6,FUNCT_1:2;
A8: rng f = the carrier of J
    proof
      thus rng f c= the carrier of J;
      let y be object;
      assume
A9:   y in the carrier of J;
      then y in the carrier of Y by A4;
      then
A10:  y in dom f by FUNCT_2:def 1;
      f.y = (f|(the carrier of J)).y by A9,FUNCT_1:49
        .= y by A3,A9,FUNCT_1:18;
      hence thesis by A10,FUNCT_1:def 3;
    end;
    the carrier of Y|M = [#](Y|M) .= the carrier of J by PRE_TOPC:def 5;
    then Image ff = the TopStruct of J by A1,A8,TSEP_1:5;
    hence thesis by YELLOW14:19;
  end;
  hence thesis;
end;
