There is no answer to this question, as there is absolutely no way to provide solid evidence that would leave people without a doubt that one card manufacturer is better than the other.
Think about the statement that you are trying to get them to prove. In order to prove this hypothesis, you would have to literally bench every single card (and by every card, I mean every card ever made, not just every type of card) made by both manufacturers in the exact same enviroment, testing them in literally every single piece of gaming software and other GPU demanding applications, at least three times, in order to reach a conclusion with basis behind it.
There are just too many variables and uncontrolled circumstances that can occur that can alter the answer in even the slightest way, so as much as any of us would like to try, there is absolutely, positively no way to prove that nVidia is superior to ATi, and vice versa.
Keep in my mind there's a whole load of factors you'd have to take into consideration that you may not even think of, power consumption, cost, heat dissipation, longitivity etc.
In the end it all comes down to opinion, and frankly in my opinion, I don't really care if one card will get me 5FPS more in a game than another, as long as it meets my standards at a price I feel is worth the card itself, I really couldn't care less.
Intel C2D E6320 / AMD Athlon X2 3800+
Gigabyte 965P DS3 / DFI nF4 Ultra-D
2GB OCZ Gold PC2-6400 / 2GB OCZ Gold PC4000
eVGA 8800GTS 320MB / eVGA 6800GS 256MB
150GB Raptor / 74GB Raptor
2x500GB / 320GB
OCZ GameXStreme 850w / OCZ StealthXStream 600w