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.
The fundamental theorem of affine geometry in regular L0-modules
