An elementary proof of the fundamental theorem of projective geometry

The following version of the fundamental theorem is proved: Let V, W be vector spaces and g: P(V) \ E --> P(W) a morphism between the associated projective spaces. If the image of g is not contained in a line, then there exists a semilinear map f: V --> W which induces g. The difficulty lies in the fact that the homomorphism of division rings associated to the map f can be nonsurjective. As an application, a short proof of Wigner's theorem is also proposed.