A graphic characterization of the lines of an affine space.

A partial line space (S, R=R 1 R 2 ) is called an abstract affine Grassmann space if the following axioms hold: (1) PS∖r P≁r , P┴ r or P~r rR . (2) P,QS∖r such that P≁Q , P~r and Q ~ r rR . (3) Let rR1 and PS∖r with P≁r ; then r(P):={QS: Q~ P and Q ~ r } R1 and Qr(P) Q ~1 r . (4) P~2 Q and Q~2 T P~2 T P,Q,TS . (5) Let P~r and Q~ r ; then P~ Q P ~ i r and Q ~i r . Here P ~Q means that P and Q are collinear, P≁Q that P and Q are not collinear, P≁r that P≁Q Qr , P┴ r that ! Qr with P~Q , and P ~i Q that P~ Q and that the line (P,Q) R i (i=1,2) . The authors prove that every abstract affine Grassmann space is isomorphic to the Grassmann space of the lines of an affine space.