Positive existential definability of parallelism in terms of betweenness in archimedean ordered affine geometry

We prove that one can define the relation ∥, with ab ∥ cd to be read as 'a = 6 or c = d or aft and cd are parallel lines (or coincide)' positively existentially in L_w1win terms of ≠ and the ternary relation B of betweenness, with B(abc) to be read as '6 lies between a and c' in Archimedean ordered affine geometry. We also show that a self-map of an Archimedean ordered translation plane or of a flat affine plane which preserves both B and ¬B must be a surjective affine mapping.