
theorem LM204H:
  for G, F being finite commutative Group, a be Element of G,
  f be Homomorphism of G, F
  st f| (the carrier of gr{a}) is one-to-one
  holds ord(f.a) = ord a
  proof
    let G, F be finite commutative Group, a be Element of G,
    f be Homomorphism of G, F;
    assume
    AS: f| (the carrier of gr{a}) is one-to-one;
    reconsider H = f| (the carrier of gr{a})
    as Homomorphism of gr{a}, F by LM204G;
    a in gr{a} by GR_CY_2:2;
    then reconsider a0 = a as Element of gr{a};
    f.a = H.a0 by FUNCT_1:49;
    hence ord(f.a) = ord(a0) by AS, LM204F
    .= card(gr{a0}) by GR_CY_1:7
    .= card(gr{a}) by GR_CY_2:3
    .= ord a by GR_CY_1:7;
  end;
