 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 ThRHom:
  for G being Group
  for I being non empty set
  for F being Group-Family of I
  for f being ManySortedSet of I
  holds f is Homomorphism-Family of G,F
  iff (for i being Element of I holds f.i is Homomorphism of G, F.i)
proof
  let G be Group;
  let I be non empty set;
  let F be Group-Family of I;
  let f be ManySortedSet of I;
  thus f is Homomorphism-Family of G,F
  implies (for i being Element of I holds f.i is Homomorphism of G, F.i)
  by Def10;
  assume A1: for i being Element of I
  holds f.i is Homomorphism of G, F.i;
  for i being object st i in dom f holds f.i is Function by A1;
  then f is Function-yielding by FUNCOP_1:def 6;
  hence f is Homomorphism-Family of G,F by A1, Def10;
end;
