
theorem Th10:
  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 y be finite-support Function of I,gr UF,
        i be Element of I, g be Element of gr UF
    st y in product F & y.i = 1_F.i & g in F.i
    holds Product(y) * g = g * 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 y be finite-support Function of I,gr UF,
        i be Element of I,
        g be Element of gr UF;
    assume that
    A1: y in product F and
    A2: y.i = 1_F.i and
    A3: g in F.i;
    reconsider x = y +* (i,g)
      as finite-support Function of I,gr UF by GROUP_19:26;
    A4: y = x +* (i,1_F.i) by A2,Th7;
    A5: x in product F by A1,A3,GROUP_19:24;
    dom y = I by PARTFUN1:def 2; then
    x.i = g by FUNCT_7:31;
    hence thesis by A4,A5,Th9;
  end;
