
theorem Th12:
  for I be non empty set,
      G be Group,
      F be Subgroup-Family of I,G,
      h,h0 be finite-support Function of I,G,
      i be Element of I,
      UFi be Subset of G
  st UFi = Union((Carrier F) | (I \ {i}))
   & h0 = h +* (i, 1_F.i)
   & h in product F
  holds Product h0 in gr UFi
  proof
    let I be non empty set,
        G be Group,
        F be Subgroup-Family of I,G,
        h,h0 be finite-support Function of I,G,
        i be Element of I,
        UFi be Subset of G;
    assume that
    A1: UFi = Union((Carrier F) | (I \ {i})) and
    A2: h0 = h +* (i, 1_F.i) and
    A3: h in product F;
    set CFi = (Carrier F) | (I \ {i});
    dom(Carrier F) = I by PARTFUN1:def 2; then
    A4: dom CFi = I \ {i} by RELAT_1:62;
    for y be object st y in rng h0 holds y in [#]gr(UFi)
    proof
      let y be object;
      assume y in rng h0; then
      consider j be Element of I such that
      A5: y = h0.j by FUNCT_2:113;
      per cases;
      suppose
        A6: j <> i; then
        not j in {i} by TARSKI:def 1; then
        A7: j in dom CFi by A4,XBOOLE_0:def 5;
        A8: h0.j = h.j by A2,A6,FUNCT_7:32;
        h.j in F.j by A3,GROUP_19:5; then
        h.j in [#](F.j); then
        h.j in (Carrier F).j by PENCIL_3:7; then
        A9: h0.j in CFi.j by A7,A8,FUNCT_1:47;
        CFi.j c= union rng CFi by A7,FUNCT_1:3,ZFMISC_1:74; then
        A10: CFi.j c= UFi by A1,CARD_3:def 4;
        UFi c= [#]gr(UFi) by GROUP_4:def 4;
        hence thesis by A5,A9,A10;
      end;
      suppose
        A11: j = i;
        dom h = I by A3,GROUP_19:3; then
        A12: h0.j = 1_F.j by A2,A11,FUNCT_7:31;
        F.j is Subgroup of G by Def1; then
        1_F.j = 1_gr(UFi) by GROUP_2:45;
        hence thesis by A5,A12;
      end;
    end; then
    rng h0 c= [#]gr(UFi); then
    reconsider x0 = h0 as finite-support Function of I,gr(UFi) by Th5;
    Product x0 = Product h0 by Th6;
    hence Product h0 in gr(UFi);
  end;
