^{20} For the reader who is familiar with *GKL* the argument of Rosser, up to M62, can be simplified as follows. In *GKL*, through II D, Ax. I_{2} has been used only in II B 2, II B 4 Satz 4, and in II D 1, 3 ff. to drop out factors of the form *B*^{m}I. Without Ax. I_{2}, however, we can replaced II B 4 Satz 4 by *X*×*I* = *I*×*X*. Then we can derive Rosser's M 53 directly; this requires first his F 12, F 16, then a proof by induction and II B 4 Satz 6 that

whence his M 53 follows as he indicates. (Here I write *B*^{m} instead of *B*_{m} because Rosser has shown it is the *m*th power of *B*). The proof of II D 2 Satz 1 then goes through with obvious modifications (cf. Rosser's F 26). Next by the method of *The universal quantifier in combinatory logic* §3 Theorem 8 (Annals of mathematics vol. 32 (1931), p. 165Google Scholar), we can show without using variables or the theorems of II E, that if *X* is a normal combinator (in the sense of *GKL*, or better *ATC* convention 3) of minimum order *m* and degree *n*, then

for *p*≦*m* and *q*≦*n*, and also

(or, if preferred

(The first of the properties to be established does not satisfy hypothesis (a) of Theorem 8 (I.c.); but if we use Theorem 7 and replace “order *m* and degree *n*” by “minimum order *m* and degree *n*,” then Theorem 8 can be established for *normal* combinators; cf. remark on p. 164, (I.c.).) From these properties it follows that where a factor of the form *B*^{m}I occurs in a regular combinator, it can be absorbed in the factor to the left of it or passed across it, and this process can be continued until all these factors are at the beginning, where they may be combined by M 53. With this modification in II D 1 Satz 1 the transformation to a normal form of Rosser type goes through as in II D 3 ff.