Proving that it was copied is only necessary to show copyright infringement. For example, if you independently write a Book A1 that happens to be very similar

Proving that it was copied is only necessary to show copyright infringement. For example, if you independently write a Book A1 that happens to be very similar to an existing Book B, it is not copyright infringement. If you write Book A2 by taking an existing Book B and changing the name of each character, then it would be copyright infringement. Even if A1 and A2 end up having the exact same words as each other, the origin of those words matters. For a patent to be granted, the implied argument is entirely different. For a patent to be granted, it must be both novel and non-obvious to somebody in the field. If a patent is granted that exactly describes something that already exists, then one of two things must be the case. If the patent description was derived from the existing work, it wasn't novel. If the patent description was derived independently, then it was the obvious solution to a problem. In either case, it fails at least one of the requirements for a patent. Unfortunately, the patent office has frequently interpreted "novel" as "not covered by a previous patent", with the implication that every invention must have been patented at some point. The courts can go back and invalidate a patent, but that would be a defense against an actual lawsuit, and would carry significant risk to initiate.