
theorem LM204I:
  for G being finite commutative Group, p being Prime, m be Nat,
  a be Element of G st card(G) = p|^m & a <> 1_G
  holds ex n be Nat st ord a = p|^(n+1)
  proof
    let G be finite commutative Group,
    p be Prime, m be Nat, a be Element of G;
    assume
    A1: card(G) = p|^m & a <> 1_G;
    reconsider Gra = gr{a} as normal strict Subgroup of G by GROUP_3:116;
    consider n1 be Nat such that
    A8: (card Gra) = p|^n1 & n1 <= m by GROUPP_1:2, A1, GROUP_2:148;
    ord a = p|^n1 by A8, GR_CY_1:7;
    then n1 <> 0 by A1, GROUP_1:43, NEWTON:4;
    then 1 <= n1 by NAT_1:14;
    then n1-1 in NAT by INT_1:3, XREAL_1:48;
    then reconsider n = n1-1 as Nat;
    take n;
    thus ord a = p|^(n+1) by A8, GR_CY_1:7;
  end;
