
theorem Th8:
  for I be set,
      F be Group-Family of I,
      a be Element of product F
  holds
    a in sum F
  iff
    support(a,F) is finite
  proof
    let I be set,
        F be Group-Family of I,
        a be Element of product F;
    thus a in sum F implies support(a,F) is finite;
    assume support(a,F) is finite; then
    reconsider J = support(a,F) as finite Subset of I;
    A2: [#] product F = product Carrier F by GROUP_7:def 2;
    set k = a|J;
    A3: dom a = I by A2,PARTFUN1:def 2; then
    A4: dom k = J by RELAT_1:62; then
    reconsider k as ManySortedSet of J by PARTFUN1:def 2,RELAT_1:def 18;
    set x = 1_product F +* k;
    A5: 1_product F is Element of product Carrier F by GROUP_7:def 2;
    for j being set st j in J
    ex G being Group-like non empty multMagma
    st G = F.j & k.j in the carrier of G & k.j <> 1_G
    proof
      let j be set;
      assume
A6:   j in J;
      then consider G being Group such that
A:    G = F.j & a.j <> 1_G & j in I by Def1;
      take G;
      thus G = F.j by A;     
      a in product F; then
      a.j in G by A,Th5;
      hence k.j in the carrier of G by A6,FUNCT_1:49;
      thus k.j <> 1_G by A,A6,FUNCT_1:49;
    end; then
    reconsider x as Element of sum F by A5,GROUP_7:def 9;
    x in sum F; then
    A10: x in product F by GROUP_2:40; then
    A11: dom x = I by Th3;
    for i be object st i in dom x holds x.i = a.i
    proof
      let i be object;
      assume
      A12: i in dom x;      
      then reconsider G = F.i as Group by A11,Th1;
      per cases;
      suppose
        A15: not i in J; then
        not i in dom k; then
        x.i = (1_product F).i by FUNCT_4:11
           .= 1_G by A11,A12,GROUP_7:6;
        hence x.i = a.i by A11,A12,A15,Def1;
      end;
      suppose
        A16: i in J;
        hence x.i = k.i by A4,FUNCT_4:13
                 .= a.i by A16,FUNCT_1:49;
      end;
    end; then
    x = a by A3,A10,Th3,FUNCT_1:2;
    hence a in sum F;
  end;
