Licensing?

I have a question about licensing and I'm not sure where to post it.

I have three computers, two capable of running sins.

When I purchased GCII, I found out that it was perfectly fine for me to install it on any and all of my computers. I was wowed. I thought "this is great, I don't have to buy 2 copies of a game just so I can legally play it if someone is using one of my other computers?"

Also, I'm lucky enough that my GF will play games with me, unluckily I have to have 2 copies of each game, Warcraft III, Command & Conquer Generals, etc.

To the point, my question is: Do I have to purchase two copies of Sins to install on two of my computers and play a LAN game? Or can I legally, install the same copy on both of my computers to play with my GF.

My computers:
Intel Celeron D 2.66GHz
1 GB cheap ram matched
Nvidia 6200 w/ 256 MB ram
Windows XP Pro
The game runs fantastic

Second one unfortunately is a Dell
Intel Celeron 2.6Ghz
512 MB cheap ram unmatched
ATI Radeon 9250 w/ 128 MB ram
Windows XP Home
The game runs a bit slow/laggy, and crashes alot.