
theorem Th23:
  for X,Y be RealLinearSpace
  holds
    ex f be LinearOperator of X,[:X,(0).Y:]
    st f is bijective
     & for x be Element of X
       holds f.x = [x,0.Y]
  proof
    let X,Y be RealLinearSpace;
    set A = the carrier of X;
    set B = the carrier of [:X,(0).Y:];

    deffunc H1(Element of A) = In([$1,0.Y],B);
    consider f be Function of A,B such that
    A1: for x be Element of A
        holds f.x = H1(x) from FUNCT_2:sch 4;

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

      then
      A3: [x,0.Y] is Point of [:X,(0).Y:] by PRVECT_3:9;
      thus f.x
       = In([x,0.Y],B) by A1
      .= [x,0.Y] by A3,SUBSET_1:def 8;
    end;

    0.Y in {0.Y} by TARSKI:def 1;
    then reconsider ZY = 0.Y as Point of (0).Y by RLSUB_1:def 3;
    for x1,x2 be Element of X
    holds f.(x1 + x2) = f.x1 + f.x2
    proof
      let x1,x2 be Element of X;
      A4: f.x1 = [x1, 0.Y] by A2;
      A5: f.x2 = [x2, 0.Y] by A2;

      f.x1 + f.x2
       = [x1 + x2, ZY + ZY] by A4,A5,PRVECT_3:9
      .= [x1 + x2, 0.Y + 0.Y] by RLSUB_1:13
      .= f.(x1 + x2) by A2;
      hence thesis;
    end;
    then
    A6: f is additive;

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

    take f;
    now
      let x1,x2 be object;
      assume
      A8: x1 in A
        & x2 in A
        & f.x1 = f.x2;
      [x1,0.Y]
       = f.x1 by A2,A8
      .= [x2,0.Y] by A2,A8;
      hence x1 = x2 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 X, y be Point of (0).Y such that
      A10: z = [x,y] by PRVECT_3:9;

      y in the carrier of (0).Y;
      then y in {0.Y} by RLSUB_1:def 3;
      then y = 0.Y by TARSKI:def 1;
      then f.x = 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;
