Infected Computer Question

I have a question.  I discovered last week that there were 4 heuristic virus and two virus/Trojans (2 different versions of the Powelik Trojan/virus) on my computer.  I run W7 Prof and NIS, connected to the net via modem/router (3 computers, mine was the only one infected since I'm logged on from 6 am to 11 pm.

I ran Norton (NIS) several times and eventually figured out (I'm pretty sure) where they came from.  It was a website I connected to at startup.  After I disconnected from them, I didn't get any more virus/Trojans.

Now, if NIS didn't stop these when they came to my computer, can I be sure that NIS removed all these virus?  Is it still necessary to run a check from another computer?  I've been running a virus check every morning since then, and have not had any further problems.  While my computer was infected, I kept getting pop-up notes from NIS telling me there was a problem, but I didn't pay close attention till it said:  System Infected.  Then I realized I had a problem.  Those notes have stopped since I cleared this mess up.

Appreciate the reply.

JC