(T --+ a ) a n d (a --+ T)follows by modus ponens from T --+ (a --+ T). Thus f ) a l s o holds. It remains only to show d). Suppose a --+ /3,/3 --+ a, # --+ p, p --+ it are given as hypotheses.

Hence, in this case, M1 can be considered (by modulo of an isomorphism) as a subsystem (a submodel)of M 2 . , f(an))] An useful thing is the representation theorem for homomorphic images by means of the quotient system by the kernel congruence relation, which we rehearse now. For any set A, an equivalence relation on A is a binary relation R such that (i) For every a C A, R(a, a) (reflexivity); (ii) For any a, b e A, R(a, b) =v R(b, a) (symmetricity); (iii) For any a, b, c E A, R(a, b)~R(b, c) ::~ R(a, c) (transitivity).

C H A P T E R 1. ~, := ( c ~ . ]i E I). ,n} then sometimes we will write A/t1 x ... h4,~ instead of 1-Ii~x A4i. Ifall A/Ii - A/I we will also use the brief denotation A/I I for YIiet A4i. Finally if # is a cardinal number we denote by Ad u the product 1-Iiei A/[i, where ,~di = M , I := {i l i <_ #}. , xn) - Vi~t A j ~ j oi,j , where all Oi,j are certain atomic formulas. , xn) is the matrix of (I). e. (~ is equivalent to 0 in CPC. Theorem We say a formula (I) has the prenex form if (~= Q l x l .