 reserve I for non empty set;
 reserve i for Element of I;
 reserve F for Group-Family of I;
 reserve G for Group;
reserve S for Subgroup-Family of F;
reserve f for Homomorphism-Family of G, F;

theorem Th47:
  for f being Function,
      I being non empty set,
      F being Group-Family of I
  st dom f = I & (for i being Element of I holds f.i in F.i)
  holds f in product F
proof
  let f be Function;
  let I be non empty set;
  let F be Group-Family of I;
  assume A1: dom f = I;
  assume A2: for i being Element of I holds f.i in F.i;
  A3: dom (Carrier F) = I by PARTFUN1:def 2;
  for i being object st i in dom (Carrier F) holds f.i in (Carrier F).i
  proof
    let i be object;
    assume i in dom (Carrier F);
    then reconsider ii=i as Element of I;
    f.i in F.ii by A2;
    hence f.i in (Carrier F).i by Th9;
  end;
  then f in product (Carrier F) by A1, A3, CARD_3:9;
  hence f in product F by GROUP_7:def 2;
end;
