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

theorem Th5:
  ord (a") = ord a
proof
  a |^ ord a = 1_G by GROUP_1:41;
  then a" |^ ord a = 1_G by Th3; then
A1: ord (a") divides ord a by GROUP_1:44;
  a" |^ ord (a") = 1_G by GROUP_1:41;
  then a |^ ord (a") = 1_G by Th4;
  then ord a divides ord (a") by GROUP_1:44;
  hence thesis by A1,NAT_D:5;
end;
