A pseudo-complement of a quadrilateral D of order n, n, > 3, is a non-trivial (n+l)-
regular linear space with n - 3n + 3 points and n + n - 3 lines. We prove that if n > 18
and D has at least one line of size n - 1, or if n > 25 , then the set of lines of D consists of
three lines of size n -1, 6(n - 2) lines of size n - 2, and n - 5n + 6 lines of size n - 3.
Furthermore, if n > 21 and D has at least one line of size n - 1, then D is embeddable in a
unique projective plane of order n. These results improve the results of the author