Not sure how you can prove you have saved money in terms of Air Passenger Duty by purchasing the flights in advance of the forthcoming rate increases.
APD is a tax on the airlines and, although they are not legally obliged to, many choose to pass on this charge to the passenger in the form of a business charge. For convenience this is usually done at the time of the ticket sale, although the tax itself is not due until the aircraft takes off. Airlines have been aware of the rate increase since April so could have been been applying an extra charge to customers purchasing flights after 1st November for the past six months.