
theorem LM204E:
  for G, F being finite commutative Group, a be Element of G,
  f be Homomorphism of G, F
  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;
    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;
    Segm card (the carrier of gr{f.a}) c= Segm card(the carrier of gr{a})
    by P1, CARD_1:67;
    hence thesis by P2, P3, NAT_1:39;
  end;
