Pasch's axiom and projective spaces

Let @p be a generalized projective geometry and i @e Z^+ such that some i-dimensional subspace of @p contains finitely many (i - 1) dimensional subspaces. We give a characterization of the incidence structure formed by the (i - 1)-dimensional and i-dimensional subspaces of @p (where the incidence relation is set inclusion). The specialization of this characterization to projective geometries follows. Let @q be a connected incidence structure with more than one point and line, and let two lines have at most one point in common. If (i) both @q and the dual of @q satisfy Pasch's Axiom, (ii) for all points x and lines m of @q not containing x, the number of points of m collinear with x is not 1 or 2, (iii) some line has finitely many points, then @q is the incidence structure having the (i - 1)-dimensional and i-dimensional subspaces (for some finite i) of some projective geometry of finite order as its points and lines respectively.