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Finally, we consider the case when w < V. The equivalence needed is (let (I = 0 (I’= 1) (4). (15) serves to simplifY 3 (u), or if not, when (I‘= 1. Othemise, when (I’ > 1 and no conjunct NCI 3 5( u ) occurs in F, one must first replace -I N (u) by a disjunction as in (i3), and then use the distributive laws. The proof of (15) is similar to the proof of whenever there is a conjunct N W,O’ This results in a disjunction each term of which is subject to (15). E. DONER, A. MOSTOWSKI and A. TARSKI 32 proof of Lemma 18 is complete.

A' < 8. nB k (1) Ha(x)[a'l nu, iff C Ha(-). But the right hand side of (1) is equivalent to a' = a , by part ( i ) . 'L is a well-ordering and F If { b :b R f } Z

C~&tld&C London, 1971, vi + 508 pp. Kelley, J. L. TopoeOgy. [551 D. , Princeton, 1955, xiv + 298 PP. Kleene, S. C. [52] Inthoductian & Ahdwtm%em&U. 1952s X + 550 PP- D. , Princeton, Morse, A. P. [651 A Zhtheoky 06 b&. Academic Press, New York, 1965, xxxi + 130 pp. , and Tarski, A. 1491 Arithmetical classes and types of well ordered systems. P r e Z ~ M l y report. &Lee. A ~ u L . Math. , vol. 55 (1949), p . 65. Rabin, M. 0. [69] Decidability of second-order theories and automata on i n f i n i t e trees.