
theorem Th9:
  for I be non empty set, G be Group,
      F be component-commutative Subgroup-Family of I,G,
      UF be Subset of G
  holds
    for i be Element of I, x,y be finite-support Function of I,gr UF
    st y = x +* (i,1_F.i) & x in product F
    holds Product x = Product(y) * x.i = x.i * Product(y)
  proof
    let I be non empty set,
        G be Group,
        F be component-commutative Subgroup-Family of I,G,
        UF be Subset of G;
    let i be Element of I,
        x,y be finite-support Function of I,gr UF;
    assume that
    A1: y = x +* (i,1_F.i) and
    A2: x in product F;
    reconsider x0 = x, y0 = y as finite-support Function of I,G by Th4;
    A3: Product x = Product x0 by Th6;
    A4: Product y = Product y0 by Th6;
    A5: Product x0 = Product(y0) * x0.i = x0.i * Product(y0) by A1,A2,Th8; then
    Product x = Product(y) * x.i by A3,A4,GROUP_2:43;
    hence thesis by A3,A4,A5,GROUP_2:43;
  end;
