
theorem Th14:
  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 i be Element of I holds F.i is normal Subgroup of gr UF
  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;
    let i be Element of I;
    A2: dom(Carrier F) = I by PARTFUN1:def 2;
    A3: F.i is Subgroup of G by Def1;
    (Carrier F).i in rng Carrier F by A2,FUNCT_1:3; then
    [#](F.i) in rng Carrier F by PENCIL_3:7; then
    [#](F.i) c= union rng Carrier F by ZFMISC_1:74; then
    A4: [#](F.i) c= UF by A1,CARD_3:def 4;
    UF c= [#]gr(UF) by GROUP_4:def 4; then
    [#](F.i) c= [#]gr(UF) by A4; then
    reconsider Fi = F.i as Subgroup of gr UF by A3,GROUP_2:57;
    for a be Element of gr UF holds a * Fi c= Fi * a
    proof
      let a be Element of gr UF;
      for x be object st x in a * Fi holds x in Fi * a
      proof
        let x be object;
        assume x in a * Fi; then
        consider g be Element of gr(UF) such that
        A5: x = a * g & g in Fi by GROUP_2:103;
        reconsider a1 = a as Element of G by GROUP_2:42;
        a1 in gr(UF); then
        consider h be finite-support Function of I,gr UF such that
        A6: h in sum F & a = Product h by A1,Th13;
        reconsider m = h.i as Element of gr UF;
        A7: h in product F by A6,GROUP_2:40;
        A8: h.i in F.i by A6,GROUP_19:5,GROUP_2:40;
        reconsider I0 = I \ {i} as Subset of I;
        1_F.i in gr(UF) by A3,GROUP_2:47; then
        reconsider h0 = h +* (i, 1_F.i)
          as finite-support Function of I,gr(UF) by GROUP_19:26;
        A9: h0 in product F by A6,GROUP_19:24,GROUP_2:40;
        dom h = I by FUNCT_2:def 1; then
        A10: h0.i = 1_F.i by FUNCT_7:31;
        A11: a * g = Product(h0) * m * g by A6,A7,Th9
                  .= Product(h0) * (m * g) by GROUP_1:def 3;
        A12: a * g = (m * g) * Product(h0)
                     by A5,A8,A9,A10,A11,GROUP_2:50,Th10
                  .= (m * g) * 1_gr(UF) * Product(h0) by GROUP_1:def 4
                  .= (m * g) * (m" * m) * Product(h0) by GROUP_1:def 5
                  .= (m * g * m") * m * Product(h0) by GROUP_1:def 3
                  .= (m * g * m") * (m * Product(h0)) by GROUP_1:def 3
                  .= (m * g * m") * (Product(h0) * m) by A7,Th9
                  .= (m * g * m") * a by A6,A7,Th9;
        m" in Fi by A8,GROUP_2:51; then
        g * m" in Fi by A5,GROUP_2:50; then
        m * (g * m") in Fi by A8,GROUP_2:50; then
        m * g * m" in Fi by GROUP_1:def 3;
        hence thesis by A5,A12,GROUP_2:104;
      end;
      hence thesis;
    end;
    hence thesis by GROUP_3:118;
  end;
