Of course... that antivirus that was already installed on that system is something like demo anyway... so, once it expires, you are on your own... or go and pay for the licence, and you even don't know how good is that product.
This way, you can try avast! home edition. It's completely free of charge, that means you don't have to pay anything for it... and one day, when you realize how good avast! really is, you'll want to pay for the Professional version... but then, you are paying for something that you really want to have installed on your system. That's much better than something that you even don't know what it's name is...
Yes, you MUST completely uninstall that antivirus that was previously installed on that system, but please, don't do anything on your own. Post back in here, what's the name of that antivirus, and we'll tell you how to uninstall it properly... sometimes it's just not enough to go to add/remove programs and just uninstall it. Some nasty heavy antiviruses like to leave many traces in windows registry, so it could give you additional problems... you think that antivirus is uninstalled, but it's not completely uninstalled...
Take care and stick around. Post back if you experience any other problems.
Cheers !