The fundamental theorem of affine geometry is an easy corollary of the corresponding projective theorem 2.26 in Artin's Geometric Algebra. However, a simple direct proof based on Lipman's paper [this Bulletin, 4, 265−278] and his axioms 1 and 2 may be of some interest.

Lipman's [desarguian] affine geometry G determined a left linear vector space L={a, b,…} over a skew field F. We wish to construct 1−1 transformations γ of G onto itself such that γ and γ-1 map straight lines onto straight lines preserving parallelism. Designate any point 0 as the origin of G. Multiplying γ with a suitable translation, we may assume γ0=0. Thus γ will then be equivalent to a 1−1 transformation Γ of L onto itself which preserves linear dependence. Since Γ-1 will have the same properties, Γ must also preserve linear independence.