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...and all x, y ∈ XN , x ≽ 0 (R) y ⇐⇒ W (ceR (x)) ≥ W (ceR (y)) . 7 Proof: Let R ′ ∈ D. Define W : RN → R so that W (d) ≥ W (c) if and only if d ≽ 0 (R ′ ) c. Such a W exists as ≽0 (R ′ ) is continuous (=-=Debreu, 1964-=-). W is strictly monotonic by the Pareto property. Now, let R ∈ D be arbitrary. Let x, y ∈ XN . Then by Pareto and rationality, x ≽0 (R) y if and only if ceR (x) ≽0 (R) ceR (y). By independence ceR (x...