Proof of Fundamental Theorem of Poker?

[de Moivre]

Thinking about it some more, I may have a proof.

One formulation of the theorem is: If you play your hand the way you would play it if you could see your opponent's cards, you gain. I regard "you gain" as meaning "your expected gain increases."

Assume a heads-up game. If the game matrix is A, player 1 has a mixed optimal strategy by the minimax theorem. Any departure from this will reduce the expected payoff for player 1 if player 2 plays optimally, which seems to contradict the theorem.

So maybe the meaning is "your expected gain, conditioned on your opponent's cards, increases." For if we condition on our opponent's cards, our payoff matrix changes and is now B, say. Here it would be optimal for player 1 to use his minimax strategy for matrix B, while using that for matrix A would be suboptimal.

So I think I've found an interpretation for the FTP that makes it correct and provable, but rather simple. But Sklansky did say the theorem is obvious, so maybe I'm on the right track.