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(4) $\Rightarrow$ (1): By \autoref{theorem:termwise-differentiation}.
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(4) $\Rightarrow$ (1): By \autoref{theorem:termwise-differentiation}.
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(5) $\Rightarrow$ (3): By the equivalence of the prior points, for any $\phi \in E^*$, $\phi \circ f$ satisfies (3). By the \autoref[Hahn-Banach Theorem]{proposition:hahn-banach-utility}, $f$ also satisfies (3).
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(5) $\Rightarrow$ (3): By the equivalence of the prior points, for any $\phi \in E^*$, $\phi \circ f$ satisfies (3). By the \hyperref[Hahn-Banach Theorem]{proposition:hahn-banach-utility}, $f$ also satisfies (3).
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\end{proof}
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\end{proof}
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