When you receive a letter like that it makes you wonder ...
Mike
Sure. The letter is worded to imply Getty has some sort of airtight case. In fact, they sometimes have totally screwed up. The bot might have "seen" an image in a ad banner. The bot might mis-identify. The bot wasn't programmed to see whether the image was hotlinked and so on. The human 'checker' don't necessarily check these things properly.
Just because Getty's letter is over-confident of their case doesn't mean you need to prove anything *to them*. If they take you to court, *they* would need to present their evidence to the court. You would also get to see their evidence in discovery.
Bear in mind: If you ask for evidence, they will say they can't provide you anything until discovery. But the same holds for you. You don't have to tell them anything prior to discovery either.