
theorem Th10:
  for I be set,
      G be Group,
      F be Group-Family of I,
      a be object
  st a in sum F
   & for i be object st i in I holds F.i is Subgroup of G
  holds a is finite-support Function of I,G
  proof
    let I be set,
        G be Group,
        F be Group-Family of I,
        a be object;
    assume that
    A1: a in sum F and
    A2: for i be object st i in I holds F.i is Subgroup of G;
    a in product F by A1,GROUP_2:40; then
    reconsider b = a as Element of product F;
    A8: dom b = I by Th3;
    for z be object st z in rng b holds z in [#]G
    proof
      let z be object;
      assume z in rng b; then
      consider i be object such that
      A9: i in dom b & z = b.i by FUNCT_1:def 3;
      i in I by A9,Th3;
      then reconsider Z = F.i as Subgroup of G by A2;
      reconsider I as non empty set by A9,Th3;
      reconsider i as Element of I by A9,Th3;
      reconsider F as multMagma-Family of I;
      b.i in F.i by A1,Th5,GROUP_2:40;
      then b.i in Z;
      then b.i in G by GROUP_2:40;
      hence z in [#]G by A9;
    end; then
    rng b c= [#] G; then
    reconsider b as Function of I,G by A8,FUNCT_2:2;
    support(b,F) = support(b) by A2,Th9;
    hence thesis by A1,Def3;
  end;
