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A nonempty set A of real numbers is called a B_3-set if the conditions a_1, a_2, a_3, a_4, a_5, a_6 \in A and a_1 + a_2 + a_3 = a_4 + a_5 + a_6 imply that the sequences (a_1, a_2, a_3) and (a_4, a_5, a_6) are identical up to a permutation. Let


A = \{a(0) = 0 < a(1) < a(2) < \ldots \}, B = \{b(0) = 0 < b(1) < b(2) < \ldots \}

be infinite sequences of real numbers with D(A) = D(B), where, for a set X of real numbers, D(X) denotes the difference set \{|x-y| | x, y \in X \}. Prove that if A is a B_3-set, then A = B.

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