Not if they pre-registered their copyrights - and I'd be shocked if they didn't, as you can't even sue anyone without a valid registration. If the infringement happens after the registration then you don't have to calculate and prove damages. Instead, the judge or jury awards damages based on some chosen number from $750 to $30,000 times the number of infringements proven. Sometimes, they literally just pick a total damage award and divide out the number of infringements, like in the billion-dollar Cox lawsuit.
Furthermore, even if they didn't preregister, and they somehow cannot prove actual damages, it's still infringing conduct. They could, say, get a permanent injunction barring Internet Archive from operating their library program. This might actually be what they really want, moreso than money damages.
Furthermore, even if they didn't preregister, and they somehow cannot prove actual damages, it's still infringing conduct. They could, say, get a permanent injunction barring Internet Archive from operating their library program. This might actually be what they really want, moreso than money damages.