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Note the following useful property of the function C: (1) If i # j, then Ci(Cj(m,a ) , b ) = Ci(C,(m,b ) , a). This Chapter is based entirely on works of Tarski. Cf. his papers 1217 and [22]. e. all with an exception of at most finite number. 67 VALUES OF FUNCTIONAL FORMS 2. Values of functional forms and the notion of satisfaction for matrix forms. Before we give exact definitions of these basic semantical notions we shall explain briefly their intuitive content. g. p = p' = v, + (a, x a2) or p = p" = (pv,)(v, x v, M a,).

Let us perform here the operation of s u b s t i t u t i o n of 1 for the v a r i a b l e ai. Since i f j, we have by theorem I1 3 h S(i,1, X(i, ai,@))= S(i,X(i, 1, ai),@)= S(i,aj,@), 48 ARITHMETICAL THEOREMS PROVABLE IN (s) and the result of s u b s t i t u t i o n of 1 for ai in (3) is k1 at 3 [S(i,l , @ ) *X(i, M aj,@)]. Let us s u b s t i t u t e here pl for the v a r i a b l e a,.. J,@)]. Using (1) we obtain now by the propositional calculus (4) 1 v'S(i,pl,@) 3 M p. J - 1, whence by lemma 1 /- 1 v' M pl --f a&, pl - 1, -A@) a@).

According to the corollary 6 3 i = 1, 2, . , m - 1 whence we obtain t-t- [D, M D, v . . v D, M VI D, w Di for Om--,]. This gives us v\ D, CD, in virtue of theorem 7 and known tautologies of the propositional calculus. T h e o r e m 9. t- 1 -4 a v 1 M a. Proof. It follows from lemma 5 1 that f- a M 1 v ( a - 1) +1 M a 53 THEOREMS O N INEQUALITIES whence the desired result follows by means of theorem 3 and axioms of the group IV. + Theorem 10. k b C a --f [b 1 4 a v b Proof. It is easy to show that j- b + a --f in virtue of lemma 5 1 +1M -A a M a].

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