Aumann's proof smuggles in a substantive assumption that players' beliefs are derived from a common prior, which is a stronger condition than common knowledge of rationality alone.

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Aumann(game theory and epistemology): Robert Aumann is an Israeli-American mathematician and economist who created influential models for understanding how people share knowledge and make agreements, particularly focusing on what groups can know together.
Common prior(as the shared foundation Aumann's theorem requires): Starting beliefs or assumptions that two people share before they gather any new information.
Derives from / derived from(describing the relationship between beliefs and their source): Comes from or originates from something else; in this context, it means the players' beliefs are based on or come out of a common prior.
Substantive assumption(as used in epistemology and logic): A real, meaningful claim that requires proof or evidence—not just a minor detail, but something that actually matters to your argument.

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