
theorem
  for I be non empty set,
      F be Group-Family of I,
      G be Group,
      x be finite-support Function of I,G
  st support(x) = {}
   & for i be object st i in I holds F.i is Subgroup of G
  holds x = 1_product F
  proof
    let I be non empty set,
        F be Group-Family of I,
        G be Group,
        x be finite-support Function of I,G;
    assume that
    A1: support(x) = {} and
    A2: for i be object st i in I holds F.i is Subgroup of G;
    for i being set st i in I
    ex G being Group-like non empty multMagma st G = F.i & x.i = 1_G
    proof
      let i being set;
      assume
      A3: i in I; then
      A4: x.i= 1_G by A1,Def2;
      reconsider Fi = F.i as Subgroup of G by A2,A3;
      take Fi;
      thus Fi = F.i & x.i = 1_Fi by A4,GROUP_2:44;
    end;
    hence x = 1_product F by GROUP_7:5;
  end;
