Complete the proof of Lemma 4. (i) For , ' ( , with ' and

Complete the proof of Lemma 4.
(i) For ω, ω' ( Ω, with ω ≠ ω' and Z1 (ω) = Z1 (ω'), suppose that Z2(ω) ≠ Z2(ω '). Use the assumptions that A'2 contains the singletons of points in Z2(Ω), and that A2 ( A1, in order to arrive at a contradiction.
(ii) By part (i), Z is well defined, and Z2 = Z(Z1). If also Z2 = Z'(Z1), then show that Z(ω 1) = Z'(ω1) for all ω1 ( Z1(Ω).
(iii) For D ( A'2, we have A1 ( A2 ϶ A = Z-12 (D) = Z1-1'(B) with B = Z-l(D) ( Z1(Ω), and A = Z1-1 (C) for some C ( A'1 Conclude that B = C ( Z1(Ω).