What does the bet prove

I don't think there is definitive answer to this. The story invites us to consider whether the lawyer is right when he claims that any life is better than none at all. No doubt, the lawyer survives his time in solitude, but when he emerges he no longer desires to interact with society. I suppose the bet proves that a life of solitude has certain benefits if one has access to books. Did it drive the lawyer to insanity? Perhaps, but he gained the wisdom to ultimately realize that the bet was meaningless to begin with.