Coxeter- Theorem 6.41
6.41 Any correlation that transforms one range projectively is a projective correlation Let a and A’ be the corresponding line and point that carry the projectively related range and pencil X and x’. Let Y be a variable point on arbitrary b, and O a fixed point on neither a or b. Let OY meet a in X. The given correlation transforms O into a fixed line o’ (through neither A’ nor B’), and OY into a point o’.y’ which is joined to A’ by a line x’. Since Y is perpective with X through O, which is projectively related to x’, which is perpective with y’ through o’, the correlation induces a projectivity between Y and y’, between b and B’.