
theorem Th13:
  for I be non empty set, G be Group,
      F be component-commutative Subgroup-Family of I,G,
      UF be Subset of G
  st UF = Union Carrier F
  holds
    for g be Element of G st g in gr UF holds
    ex f be finite-support Function of I,gr UF
    st f in sum F & g = Product f
  proof
    let I be non empty set,
        G be Group,
        F be component-commutative Subgroup-Family of I,G,
        UF be Subset of G;
    assume
    A1: UF = Union Carrier F;
    A2: for i be object st i in I holds F.i is Subgroup of G by Def1;
    let g be Element of G;
    assume g in gr UF; then
    consider H be FinSequence of G,
             K be FinSequence of INT such that
    A3: len H = len K & rng H c= UF & Product(H|^K) = g by GROUP_4:28;
    consider f be finite-support Function of I,G such that
    A4: f in product F & g = Product f by A1,A3,Th11;
    f is Function of I,Union Carrier F by A4,Th2; then
    A5: rng f c= UF by A1,RELAT_1:def 19;
    UF c= [#]gr(UF) by GROUP_4:def 4; then
    rng f c= [#]gr(UF) by A5; then
    reconsider f0 = f as finite-support Function of I,gr(UF) by Th5;
    take f0;
    support(f,F) = support(f) by A2,A4,GROUP_19:9;
    hence thesis by A4,GROUP_19:8,Th6;
  end;
