His security staff, MI5 and the Royal Family (and probably some members of staff in the eaterie concerned) obviously know if Andrew was indeed at Pizza Express and at home thereafter on the night in question.
Similarly, his medical records would easily be able back up his claim of not being able to perspire - and would be perfect evidence to undermine his accuser.
The fact that his legal team is fighting tooth and nail on every front possible rather than going for those two quick wins speaks volumes.