Almost directly south of the exit to the mine is a place known as the Condor Fort. For now, it's an optional event, but later on, it will become more important. If you got all your Limit Breaks at the Mythril Mine, you'll probably have a good stockpile of cash, so you might want to try your hand at the little strategy game they're set up for you in here. It's really easy to win if you have the cash. Just set up a fighting perimiter, and keep bringing in reinforcments if you need to. Winning the first battle will earn you a Magic Comb, your first double-AP weapon for Red XIII, and one that can't be bought for quite a while yet. You might want to get it now if you're planning on growing your materia a lot later.