
theorem GRCY211:
  for k being Element of NAT, G being finite Group, a being Element of G
  holds (ord a) divides k * ord(a |^ k)
  proof
    let k be Element of NAT, G be finite Group, a be Element of G;
    a in gr{a} by GR_CY_2:2; then
    reconsider a0 = a as Element of gr{a};
    A11: gr{a0} = gr{a} by GR_CY_2:3;
    A12: card (gr{a}) = ord a by GR_CY_1:7;
    ord(a |^ k) = card (gr{ a|^ k}) = card (gr{a0 |^ k})
      by GR_CY_1:7, GROUP_4:2, GR_CY_2:3;
    hence thesis by A11, A12, GR_CY_2:11;
  end;
