- A Proofs Proof. [Proof of Theorem 1] . Axioms 1 to 4 imply that can be represented by a continuous function Φ : Zn → R that is increasing in |ui − vi|, i = 1, ..., n. Using Axiom 3 part (a) of the result follows from Theorem 5.3 of Fishburn (1970). Now take z0 and z in as specied in Axiom 4. Using (11) it is clear that z ∼ z0 if and only if Æi (ui + δ, ui + δ) − Æi (ui, ui) − Æj (uj + δ, uj + δ) + Æj (uj + δ, uj + δ) = 0 which can only be true if Æi (ui + δ, ui + δ) − Æi (ui, ui) = f (δ) for arbitrary ui and δ. This is a standard Pexider equation and its solution implies (12). Proof. [Proof of Theorem 2] Using the function Φ introduced in the proof of Theorem 1 Axiom 5 implies Φ (z) = Φ (z0 ) Φ (tz) = Φ (tz0 ) and so, since this has to be true for arbitrary z, z0 we have Φ (tz) Φ (z) = Φ (tz0 )
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- X i=1 bi (t) see Aczél (1966), page 142. Therefore we have Hi t ui vi = bi (t) ui + B (t) Hi ui vi , i = 1, ..., n (69) From Eichhorn (1978), Theorem 2.7.3 the solution to (69) is of the form Hi (v) = βivα−1 + γi, α 6= 1 βi log v + γi α = 1 (70) where βi > 0 is an arbitrary positive number. Substituting for Hi () from (70) into (14) for the case where βi is the same for all i gives the result.
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