
theorem Th6:
  for G,H,I being AddGroup, h being Homomorphism of G,H
  for h1 being Homomorphism of H,I holds
  h1 * h is Homomorphism of G,I
proof
  let G,H,I be AddGroup;
  let h be Homomorphism of G,H;
  let h1 be Homomorphism of H,I;
  reconsider f = h1 * h as Function of G, I;
  now
    let a,b be Element of G;
    thus f.(a + b) = h1.(h.(a + b)) by FUNCT_2:15
      .= h1.(h.a + h.b) by VECTSP_1:def 20
      .= (h1.(h.a)) + (h1.(h.b)) by VECTSP_1:def 20
      .= f.a + (h1.(h.b)) by FUNCT_2:15
      .= f.a + f.b by FUNCT_2:15;
  end;
  hence thesis by VECTSP_1:def 20;
end;
