### Subfunction Structure

A. P. Morse
1969 Proceedings of the American Mathematical Society
Of prime interest to us here is Theorem 4 in the proof of which an interesting structure is revealed by (2) , (3), and (4). An easy consequence of Theorem 4 is Theorem 5 which embraces results of specific interest. One such result is Theorem B.4 stated and proved in the Amer. Math. Monthly 74 (1967), 773-776. Another such result1 is a Theorem of Erdös and Szekeres2 succinctly proved by A. Seidenberg in the J. London Math. Soc. 34 (1959), 352. In both these proofs a central trick appears and
more » ... ick appears and then reappears here in Theorem 4 (1) below. We assume that a relation is a set of ordered pairs and, in the usual way, that a function is a special sort of relation. We agree that SR is the set of all real numbers, that % is the set of all those functions whose domains are subsets of 9Î, that w is the set of integers « for which « = 0, that ~S is the complement of 5 with respect to the universe, and that pwr'S = E 1-»es Thus pwr'5 = the number of points in S. We agree that x and y are consecutive in S if and only if x and y are such members of 5 that x<y and there is no zQS for which x < z < y. With a relation P tentatively in mind we agree that upP is the set of /Go for which