
theorem GRCY212A:
  for k being Element of NAT, G being finite Group,
      a being Element of G st k gcd (ord a) = 1
  holds ord a = ord(a |^ k)
  proof
    let k be Element of NAT, G be finite Group, a be Element of G;
    assume k gcd (ord a) = 1; then
    A1: gr{a} = gr{(a |^ k)} by GRCY212;
    card (gr{a}) =ord a by GR_CY_1:7;
    hence thesis by A1, GR_CY_1:7;
  end;
