Any collineation that transforms one range projectively is a projective collineation (Theorem 6.11)
Let a and a' be the corresponding lines that carry the projectively related ranges. Let Y be a variable point on b and O be a fixed point on neither a nor b. Let OY meet a in X. The given collineation transforms O into a fixed point O' (on neither a' nor b'), and OY into a line O'Y' that meets a' in X'. Since X is on a, X is projectively related to X'. Thus, Y is perspective through O with X, X is projectively related to X', and X' is perspective through O with Y'. So, the collineation induces a projectivity between Y and Y', between b and b', as desired.