
theorem Th24:
  for X,Y be RealLinearSpace
  holds
    ex f be LinearOperator of Y,[:(0).X,Y:]
    st f is bijective
     & for y be Element of Y
       holds f.y = [0.X,y]
  proof
    let X,Y be RealLinearSpace;
    set A = the carrier of Y;
    set B = the carrier of [:(0).X,Y:];

    deffunc H1(Element of A) = In([0.X,$1],B);

    consider f be Function of A,B such that
    A1: for y be Element of A holds
        f.y = H1(y) from FUNCT_2:sch 4;

    A2: for y be Element of Y
        holds f.y = [0.X,y]
    proof
      let y be Element of Y;
      0.X in {0.X} by TARSKI:def 1;
      then 0.X in the carrier of (0).X by RLSUB_1:def 3;

      then
      A3: [0.X,y] is Point of [:(0).X,Y:] by PRVECT_3:9;
      thus f.y
       = In([0.X,y],B) by A1
      .= [0.X,y] by A3,SUBSET_1:def 8;
    end;
    0.X in {0.X} by TARSKI:def 1;
    then reconsider ZY = 0.X as Point of (0).X by RLSUB_1:def 3;

    for y1,y2 be Element of Y holds f.(y1 + y2) = f.y1 + f.y2
    proof
      let y1,y2 be Element of Y;
      A4: f.y1 = [0.X,y1] by A2;
      A5: f.y2 = [0.X,y2] by A2;

      f.y1 + f.y2
       = [ZY + ZY, y1 + y2] by A4,A5,PRVECT_3:9
      .= [0.X + 0.X, y1 + y2] by RLSUB_1:13
      .= f.(y1 + y2) by A2;
      hence thesis;
    end;
    then
    A6: f is additive;

    for y be VECTOR of Y
    for r be Real
    holds f.(r * y) = r * f.y
    proof
      let y1 be VECTOR of Y,
          r be Real;
      A7: f.y1 = [0.X,y1] by A2;
      r * f.y1
       = [r * ZY, r * y1] by A7,PRVECT_3:9
      .= [r * 0.X, r * y1] by RLSUB_1:14
      .= [0.X, r * y1];
      hence thesis by A2;
    end;
    then
    reconsider f as LinearOperator of Y,[:(0).X,Y:]
      by A6,LOPBAN_1:def 5;

    take f;
    now
      let y1,y2 be object;
      assume
      A8: y1 in A
        & y2 in A
        & f.y1 = f.y2;

      [0.X,y1]
       = f.y1 by A2,A8
      .= [0.X,y2] by A2,A8;
      hence y1 = y2 by XTUPLE_0:1;
    end;
    then
    A9: f is one-to-one by FUNCT_2:19;

    now
      let z be object;
      assume z in B;
      then consider x be Point of (0).X, y be Point of Y such that
      A10: z = [x,y] by PRVECT_3:9;
      x in the carrier of (0).X;
      then x in {0.X} by RLSUB_1:def 3;
      then x = 0.X by TARSKI:def 1;
      then f.y = z by A2,A10;
      hence z in rng f by FUNCT_2:112;
    end;
    then B c= rng f;
    then f is onto by FUNCT_2:def 3,XBOOLE_0:def 10;
    hence thesis by A2,A9;
  end;
