
theorem Th2:
  for I be non empty set,
      G be Group,
      F be Group-Family of I,
      x be Function of I,G
  st x in product F
  holds x is Function of I,Union(Carrier F)
  proof
    let I be non empty set,
        G be Group,
        F be Group-Family of I,
        x be Function of I,G;
    assume
    A1: x in product F;
    A2: dom(Carrier F) = I by PARTFUN1:def 2;
    for z be object st z in rng x holds z in Union(Carrier F)
    proof
      let z be object;
      assume z in rng x; then
      consider i be object such that
      A3: i in I & z = x.i by FUNCT_2:11;
      reconsider i as Element of I by A3;
      x.i in F.i by A1,GROUP_19:5; then
      z in [#](F.i) by A3; then
      A4: z in (Carrier F).i by PENCIL_3:7;
      (Carrier F).i in rng(Carrier F) by A2,FUNCT_1:3; then
      z in union rng(Carrier F) by A4,TARSKI:def 4;
      hence z in Union(Carrier F) by CARD_3:def 4;
    end; then
    rng x c= Union(Carrier F);
    hence x is Function of I,Union (Carrier F) by FUNCT_2:6;
  end;
