
theorem LM204F:
  for G, F being finite commutative Group, a be Element of G,
  f be Homomorphism of G, F st f is one-to-one
  holds ord(f.a) = ord a
  proof
    let G, F being finite commutative Group, a be Element of G,
    f be Homomorphism of G, F;
    assume
    AS: f is one-to-one;
    P1: the carrier of gr{f.a} = f.: (the carrier of gr{a}) by LM204L;
    P2: card (gr{a}) = ord a by GR_CY_1:7;
    P3: card (gr{f.a}) = ord(f.a) by GR_CY_1:7;
    dom f = the carrier of G by FUNCT_2:def 1;
    then the carrier of gr{a}, the carrier of gr{f.a} are_equipotent
    by P1, AS, CARD_1:33, GROUP_2:def 5;
    hence thesis by P2, P3, CARD_1:5;
  end;
