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 Th9:
  for m being Integer holds a |^ (m * ord a) = 1_G
proof
  let m be Integer;
  per cases;
  suppose a is being_of_order_0;
    then ord a = 0 by GROUP_1:def 11;
    hence thesis by GROUP_1:25;
  end;
  suppose a is not being_of_order_0;
    then
A1: a |^ ord a = 1_G by GROUP_1:def 11;
    (1_G) |^ m = 1_G by GROUP_1:31;
    hence thesis by A1,GROUP_1:35;
  end;
end;
