 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 ThHom:
  for I being non empty set
  for A, B being Group-Family of I
  for f being ManySortedSet of I
  holds f is Homomorphism-Family of A,B
  iff (for i being Element of I holds f.i is Homomorphism of A.i, B.i)
proof
  let I be non empty set;
  let A, B be Group-Family of I;
  let f be ManySortedSet of I;
  thus f is Homomorphism-Family of A,B
  implies (for i being Element of I holds f.i is Homomorphism of A.i, B.i)
  proof
    assume A1: f is Homomorphism-Family of A,B;
    let i be Element of I;
    reconsider ff=f as Homomorphism-Family of A,B by A1;
    ff.i = f.i & ff.i is Homomorphism of A.i, B.i;
    hence thesis;
  end;
  thus (for i being Element of I holds f.i is Homomorphism of A.i, B.i)
  implies f is Homomorphism-Family of A, B
  proof
    assume A1: for i being Element of I holds f.i is Homomorphism of A.i, B.i;
    for i being object st i in I
    holds f.i is Function of (Carrier A).i, (Carrier B).i
    proof
      let i be object;
      assume i in I;
      then reconsider ii=i as Element of I;
      (Carrier A).i = the carrier of A.ii &
      (Carrier B).i = the carrier of B.ii by Th9;
      hence f.i is Function of (Carrier A).i, (Carrier B).i by A1;
    end;
    then f is ManySortedFunction of (Carrier A), (Carrier B) by PBOOLE:def 15;
    hence thesis by A1, Def11;
  end;
end;
