# Energy involved in rolling with slipping

A ball rolls without slipping on a table. It rolls onto a piece of paper. You slide the paper around in an arbitrary (horizontal) manner. (It’s fine if there are abrupt, jerky motions, so that the ball slips with respect to the paper.) After you allow the ball to come off the paper, it will eventually resume rolling without slipping on the table. Show that the final velocity equals the initial velocity.

This was a problem I came across in David Morin's Introduction to Classical Mechanics. I don't understand how the final velocity will be equal to the initial velocity since evidently, there is negative work being done by kinetic friction of the paper when slipping occurs. So, its kinetic energy is being lost in the process and suppose, even if it starts rolling without slipping again (I understand that pure rolling is bound to happen since kinetic friction will reduce translational velocity till the point that the v=rw condition will hold and thereafter only static friction will act on the contact point of the ball), wouldn't it have lesser velocity than its initial velocity? Where is my reasoning going wrong here?

As far as I've thought about it, the only way this is possible is if friction does positive work later or there's some other external force I haven't taken into account which gives the ball kinetic energy. And even if that is the case, how can I prove that this will always happen?