theorem Th9:
  for G being finite Group, a being Element of G holds a|^card G = 1_G
proof
  let G be finite Group, a be Element of G;
  ord a divides card G by Th8;
  then consider t being Nat such that
A1: card G = ord a * t by NAT_D:def 3;
  a|^card G = a |^ ord a |^ t by A1,GROUP_1:35
    .= (1_G) |^ t by GROUP_1:41
    .= 1_G by GROUP_1:31;
  hence thesis;
end;
