[b]6.13[/b] [i]Given 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. [/i] 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.