Principia mathematica, by Alfred North Whitehead ... and Bertrand Russell.

*43. THE RELATIONS OF A RELATIVE PRODUCT TO ITS FACTORS.

Summary of *43.
The purpose of the present number is to give certain propositions on the
relation which holds between P and Q whenever P = Q R, or whenever
P= R I Q, or whenever P = Q I S, where R and S are fixed. In virtue of
the general definitions of *38, these relations are respectively I R, R, and
(R I) | ( S). Such relations are of great utility both in cardinal and in ordinal
arithmetic; they are also much used in the theory of induction (Part II,
Section E). In place of the notation (R i)i(S), which is cumbrous, we
adopt the more compact notation R B1 S. If X is a class of relations, R I"X
will be the class of relations R P where P eX, I R"X will be the class of
relations P R where P e X, and (R i S)"X will be the class of relations
R P S where P X. These classes of relations are often required in
subsequent work.
In virtue of our definitions, we have
*43.112. F. (R.1 S)'Q= R I Q S
The propositions most used in the present number (except such as merely
embody definitions) are the following:
*43-302.. (P). P e a(R S)
*43-411. F. R'(C"x = (I" R"x
*43'421. F. ' \ R"X = (s'x) B R
The remaining propositions are used seldom, but their uses, when they are
used, are important.
*43-01. Ri]S= (RIS) Df
At a later stage (in *150) we shall introduce a simpler notation for the
special case of R IR. The following propositions are for the most part
immediate consequences of the definitions, and proofs are therefore usually
omitted,