6.13Given any two complete quadrilaterals (or quadrangles), with their four sides (or vertices) named in a corresponding order, there is just one projective collineation that will transform the first into the second.
Let DEFPQR and D’E’F’P’Q’R’ be the two given quadrilaterals. Choose an arbitrary line a. There are two sides of the first quadrilateral that meet a in two distinct points. Suppose a is XY with X on DE and Y on DQ. The projectivites (DEF is projectively related to D’E’F’) and (DQR is projectively related to D’Q’R’) determine a line a’=X’Y’, where (DEFX is projectively related to D’E’F’X’) and (DQRY is projectively related to D’Q’R’Y’).
Let a vary in a pencil so that X is perspective with Y. By our construction for a’, we now have X’ is related projectively to X, which is perspective with Y, which is projectively related to Y’. Since D is the invariant point of the perspectivity between X and Y, D’ must be an invariant point of the projectivity between X’ and Y’. Hence, by 4.22, this projectivity is again a perspectivity. Thus a’, like a, varies in a pencil. The projectivity between X and X’ suffices to make it a projective collineation, because we have a line-to-line and point-to-point transformation preserving incidence.
Finally, there is no other projective collineation transforming DEFPQR into D’E’F’P’Q’R’; for, if another transformed a into a1, the inverse of the latter would take a1 to a, the original collineation takes a to a’, and altogether we would have a projective collineation leaving D’E’F’P’Q’R’ invariant and taking a1 to a’. By 6.12, this collineation can only be the identity. So, the projective collineation is unique.