If you registered it before when you first installed it, the serial number should be associated with your account. Click Serial Number on the link bar below the two TS2010 banners.
Don't know about other versions, but at least for the boxed version of TS2010 the invoice number is not the serial number used when registering, and the serial number is not shown on the order history page. It should, however, count as proof of purchase, so maybe Auran could help him out. (Once they catch up on the backlog of tickets that I'm sure have piled up over the last few weeks.)