
theorem Th2:
  for X,Y be AbGroup holds
  ex I be Homomorphism of [:X,Y:],product <*X,Y*> st I is bijective &
  (for x be Element of X, y be Element of Y holds I.(x,y) = <*x,y*>)
proof
  let X,Y be AbGroup;
  set CarrX = the carrier of X;
  set CarrY = the carrier of Y;
  consider I be Function of [:CarrX,CarrY:],product <*CarrX,CarrY*> such that
A1: I is one-to-one & I is onto &
for x,y be object st x in CarrX & y in CarrY
  holds I.(x,y) = <*x,y*> by PRVECT_3:5;
  len carr <*X,Y*> = len <*X,Y*> by PRVECT_1:def 11;
  then
A2: len carr <*X,Y*> = 2 by FINSEQ_1:44;
  then
A3: dom carr <*X,Y*> = {1,2} by FINSEQ_1:2,def 3;
  len <*X,Y*> = 2 by FINSEQ_1:44;
  then
A4: dom <*X,Y*> = {1,2} by FINSEQ_1:2,def 3;
A5: <*X,Y*>.1 = X & <*X,Y*>.2 = Y;
  1 in {1,2} & 2 in {1,2} by TARSKI:def 2;
  then (carr <*X,Y*>).1 = CarrX & (carr <*X,Y*>).2 = CarrY
  by A4,A5,PRVECT_1:def 11;
  then
A6: carr <*X,Y*> = <* CarrX,CarrY *> by A2,FINSEQ_1:44;
  then reconsider I as Function of [:X,Y:],product <*X,Y*>;
  for v,w be Element of [:X,Y:] holds I.(v+w) = I.v + I.w
  proof
    let v,w be Element of [:X,Y:];
    consider x1 be Element of X, y1 be Element of Y such that
A7: v = [x1,y1] by SUBSET_1:43;
    consider x2 be Element of X, y2 be Element of Y such that
A8: w = [x2,y2] by SUBSET_1:43;
    I.v = I.(x1,y1) & I.w = I.(x2,y2) by A7,A8;
    then
A9: I.v = <*x1,y1*> & I.w = <*x2,y2*> by A1;
A10: I.(v+w) =I.(x1+x2,y1+y2) by A7,A8,PRVECT_3:def 1
      .= <* x1+x2,y1+y2 *> by A1;
    reconsider Iv = I.v, Iw = I.w as Element of product carr <*X,Y*>;
    reconsider j1=1, j2=2 as Element of dom (carr <*X,Y*>) by A3,TARSKI:def 2;
A12: (addop <*X,Y*>).j1 = the addF of (<*X,Y*>.j1) by PRVECT_1:def 12;
A13: ([:addop <*X,Y*>:].(Iv,Iw)).j1 = ((addop <*X,Y*>).j1).(Iv.j1,Iw.j1)
    by PRVECT_1:def 8
      .= x1+x2 by A12,A9;
A14: (addop <*X,Y*>).j2 = the addF of (<*X,Y*>.j2) by PRVECT_1:def 12;
A15: ([:addop <*X,Y*>:].(Iv,Iw)).j2 = ((addop <*X,Y*>).j2).(Iv.j2,Iw.j2)
    by PRVECT_1:def 8
      .= y1+y2 by A14,A9;
    consider Ivw be Function such that
A16: I.v + I.w = Ivw & dom Ivw = dom carr <*X,Y*> & (for i be object st
    i in dom carr <*X,Y*> holds Ivw.i in carr (<*X,Y*>).i) by CARD_3:def 5;
A17: dom Ivw = Seg 2 by A2,A16,FINSEQ_1:def 3;
    then reconsider Ivw as FinSequence by FINSEQ_1:def 2;
    len Ivw = 2 by A17,FINSEQ_1:def 3;
    hence thesis by A10,A16,A13,A15,FINSEQ_1:44;
  end;
  then reconsider I as Homomorphism of [:X,Y:],product <*X,Y*>
  by VECTSP_1:def 20;
  take I;
  thus thesis by A1,A6;
end;
