
theorem
  for X be Group-Sequence, Y be AbGroup holds
  ex I be Homomorphism of [:product X,Y:],product(X^<*Y*>) st I is bijective &
  (for x be Element of product X, y be Element of Y ex x1,y1 be FinSequence
  st x = x1 & <*y*> = y1 & I.(x,y) = x1^y1)
proof
  let X be Group-Sequence;
  let Y be AbGroup;
  consider I be Homomorphism of [:product X,Y:],[:product X,product <*Y*>:]
  such that
A1: I is bijective & (for x be Element of product X, y be Element of Y holds
  I.(x,y) = [x,<*y*>]) by Th9;
  consider J be Homomorphism of [:product X,product <*Y*>:], product (X^<*Y*>)
  such that
A2: J 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 & J.(x,y) = x1^y1) by Th3;
  set K = J*I;
  reconsider K as Homomorphism of [:product X,Y:],product (X^<*Y*>);
  take K;
A3: rng J = the carrier of product (X^<*Y*>) by A2,FUNCT_2:def 3;
  rng I = the carrier of [:product X,product <*Y*>:] by A1,FUNCT_2:def 3;
  then rng(J*I) = J.:(the carrier of [:product X,product <*Y*>:])
  by RELAT_1:127
    .= the carrier of product (X^<*Y*>) by A3,RELSET_1:22;
  then K is onto by FUNCT_2:def 3;
  hence K is bijective by A1,A2;
  thus for x be Element of product X, y be Element of Y holds ex x1,y1 be
  FinSequence st x = x1 & <*y*> = y1 & K.(x,y) = x1^y1
  proof
    let x be Element of product X, y be Element of Y;
A4: I.(x,y) = [x,<*y*>] by A1;
    [x,y] in the carrier of [:product X,Y:];
    then [x,<*y*>] in the carrier of [:product X,product <*Y*>:]
    by A4,FUNCT_2:5;
    then reconsider yy=<*y*> as Element of product <*Y*> by ZFMISC_1:87;
    consider x1,y1 be FinSequence such that
A5: x = x1 & yy = y1 & J.(x,yy) = x1^y1 by A2;
    J.(x,yy) = J.(I.([x,y])) by A4
      .= K.(x,y) by FUNCT_2:15;
    hence thesis by A5;
  end;
end;
