[euphetar]

Well since the next layer outputs are a linear transformation plus some nonliearity of the previous function, it's a fact that it's a change in representation. But I guess the true question is broader: "Is it proven that the next layer is preparing a feature representation for the next one?".

I don't know if it is mathematically proven, but you can easily see it yourself when making an image classifier with one hidden layer. It has some really good evidence for an assumption, at this point I would call it at least empirical evidence.