Premiere Elements has a different forum http://forums.adobe.com/community/premiere_elements
Write your downloaded file(s) to a DVD or USB hard drive... along with a text file containing your serial number
[Moved to Elements forum.]
Adobe allows two installations on the same serial number. So, you should be fine on the second computer.
The problem may come when you try to put the first one back together. Adobe's computers may see it as a third computer.
There are two solutions. Under the help menu is a "Deactivate" choice. It tells Adobe you are done using Premier Elements on that computer. You may have to "Deactive" the second one for a fresh install on the first.
The other solution is to send Adobe customer support an email explaining the loss of the first computer and they can reactivate for you. Nobody can do that here because it is a user forum.
thanks, it's done now.