
theorem
  for I,J be non empty set,
      G be Group,
      x be finite-support Function of I,G,
      y be finite-support Function of J,G,
      a be Function of I,J
  st a is bijective & x = y * a
   & for i,j be Element of I holds x.i * x.j = x.j * x.i
  holds Product x = Product y
  proof
    let I,J be non empty set,
        G be Group,
        x be finite-support Function of I,G,
        y be finite-support Function of J,G,
        a be Function of I,J;
    assume that
    A1: a is bijective and
    A2: x = y * a and
    A3: for i,j be Element of I holds x.i * x.j = x.j * x.i;
    reconsider rx = rng x as non empty Subset of G;
    reconsider ry = rng y as non empty Subset of G;
    A4: dom y = J by FUNCT_2:def 1;
    A5: rng a = J by A1,FUNCT_2:def 3;
    A6: gr(rx) = gr(ry) by A2,A4,A5,RELAT_1:28;
    rng x c= [#]gr(rx) by GROUP_4:def 4; then
    reconsider x1 = x as finite-support Function of I,gr(rx) by GROUP_20:5;
    rng y c= [#]gr(ry) by GROUP_4:def 4; then
    reconsider y1 = y as finite-support Function of J,gr(rx) by A6,GROUP_20:5;
    now
      let a,b be Element of G;
      assume that
      A7: a in rx and
      A8: b in rx;
      consider i be object such that
      A9: i in dom x and
      A10: a = x.i by A7,FUNCT_1:def 3;
      consider j be object such that
      A11: j in dom x and
      A12: b = x.j by A8,FUNCT_1:def 3;
      reconsider i as Element of I by A9;
      reconsider j as Element of I by A11;
      x.i * x.j = x.j * x.i by A3;
      hence a * b = b * a by A10,A12;
    end; then
    gr(rx) is commutative by GROUP_19:44; then
    A14: Product x1 = Product y1 by A1,A2,Th16;
    Product x = Product x1 by GROUP_20:6;
    hence thesis by A14,GROUP_20:6;
  end;
