reserve G for strict Group,
  a,b,x,y,z for Element of G,
  H,K for strict Subgroup of G,
  p for Element of NAT,
  A for Subset of G;

theorem Th10:
  for a st a is not being_of_order_0
  for m being Integer holds a |^ m = a |^(m mod ord a)
proof
  let a such that
A1: a is not being_of_order_0;
  let m be Integer;
  ord a <> 0 by A1,GROUP_1:def 11;
  then m mod ord a = m - (m div ord a) * ord a by INT_1:def 10;
  then a |^(m mod ord a) = a |^ (m + - 1 * (m div ord a) * ord a)
    .= (a|^m) * a |^((-(m div ord a)) * ord a) by GROUP_1:33
    .= (a|^m) * 1_G by Th9
    .= (a|^m) by GROUP_1:def 4;
  hence thesis;
end;
