reserve G for Group,
  a,b for Element of G,
  m, n for Nat,
  p for Prime;

theorem Th6:
  ord (a |^ b) = ord a
proof
  (a |^ b) |^ ord a = (a |^ ord a) |^ b by GROUP_3:27
                     .= (1_G) |^ b by GROUP_1:41
                     .= 1_G by GROUP_3:17; then
A1: ord (a |^ b) divides ord a by GROUP_1:44;
  (a |^ ord (a |^ b)) |^ b = (a |^ b) |^ ord (a |^ b) by GROUP_3:27
                          .= 1_G by GROUP_1:41;
  then ord a divides ord (a |^ b) by GROUP_1:44,GROUP_3:18;
  hence thesis by A1,NAT_D:5;
end;
