 Savage showed that whenever these six axioms are satisfied, the comparative belief relation can be represented by a unique probability function. Having done so, he could rely on the vNM representation theorem to show that an agent who satisfies all six  axioms[7]  can be represented as maximising expected utility, relative to a unique probability function that plausibly represents the agent’s beliefs over the states and a cardinal utility function that plausibly represents the agent’s desires for ultimate outcomes (recall the statement of Savage’s theorem  above).[8]  Savage’s own proof is rather complicated, but Kreps (1988) provides a useful illustration of it.