
theorem
  for X,Y be Group-Sequence holds
  ex I be Homomorphism of product <* product X,product Y *>,product (X^Y)
  st I is bijective &
  (for x be Element of product X, y be Element of product Y holds
  ex x1,y1 be FinSequence st x = x1 & y = y1 & I.<*x,y*> = x1^y1)
proof
  let X,Y be Group-Sequence;
  set PX = product X;
  set PY = product Y;
  set PXY = product(X^Y);
  consider K be Homomorphism of [:PX,PY:],PXY such that
A1: K is bijective & (for x be Element of PX, y be Element of PY
  holds ex x1,y1 be FinSequence st x=x1 & y=y1 & K.(x,y) = x1^y1) by Th3;
  consider J be Homomorphism of [:PX,PY:],product<*PX,PY*> such that
A2: J is bijective & (for x be Element of PX, y be Element of PY
  holds J.(x,y) = <*x,y*>) by Th2;
  reconsider JB = J" as Homomorphism of product <*PX,PY*>,[:PX,PY:] by A2,Th7;
  dom J = the carrier of [:PX,PY:] by FUNCT_2:def 1;
  then rng JB = the carrier of [:PX,PY:] by A2,FUNCT_1:33;
  then
A3: JB is onto by FUNCT_2:def 3;
  reconsider I= K*JB as Homomorphism of product <*PX,PY*>,PXY;
  take I;
  I is onto by A1,A3,FUNCT_2:27;
  hence I is bijective by A1,A2;
  thus for x be Element of PX, y be Element of PY holds ex x1,y1 be FinSequence
  st x = x1 & y = y1 & I.<*x,y*> = x1^y1
  proof
    let x be Element of PX, y be Element of PY;
    consider x1,y1 be FinSequence such that
A4: x=x1 & y=y1 & K.(x,y) = x1^y1 by A1;
A5: J.(x,y) = <*x,y*> by A2;
    [x,y] in the carrier of [:PX,PY:];
    then
A6: [x,y] in dom J by FUNCT_2:def 1;
    I.<*x,y*> = K.(JB.(J.[x,y])) by A5,FUNCT_2:15;
    then I.<*x,y*> = x1^y1 by A4,A6,A2,FUNCT_1:34;
    hence thesis by A4;
  end;
end;
