 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
  for f being Homomorphism-Family of G, F
  ex phi being Homomorphism of G, product F
  st (for i being Element of I holds f.i = (proj (F,i)) * phi)
proof
  let f be Homomorphism-Family of G, F;
  consider phi being Homomorphism of G, product F such that
  A1: for g being Element of G
      holds (for j being Element of I holds (f.j).g = (proj (F,j)) . (phi.g))
  by Th39;
  take phi;
  let i be Element of I;
  for g being Element of G holds ((proj (F,i)) * phi).g = (f.i).g
  proof
    let g be Element of G;
    (f.i).g = (proj (F,i)) . (phi.g) by A1
           .= ((proj (F,i)) * phi).g by FUNCT_2:15;
    hence (f.i).g = ((proj (F,i)) * phi).g;
  end;
  hence f.i = (proj (F,i)) * phi by FUNCT_2:def 8;
end;
