Fanoian, Pappian and Desarguesian Affine Spaces

The articles [6], [1], [4], [5], [2], and [3] provide the notation and terminology for this paper. Let O1 be an ordered affine space. Then Λ(O1) is an affine space. Let O1 be an ordered affine plane. Then Λ(O1) is an affine plane. We now state several propositions: (1) There exists a real linear space V and there exist vectors u, v of V such that for all real numbers a, b such that a · u + b · v = 0V holds a = 0 and b = 0. (2) For every ordered affine space O1 and for an arbitrary x holds x is an element of the points of O1 if and only if x is an element of the points of Λ(O1) but x is a subset of the points of O1 if and only if x is a subset of the points of Λ(O1). (3) For every ordered affine space O1 and for all elements a, b, c of the points of O1 and for all elements a , b, c of the points of Λ(O1) such that a = a and b = b and c = c holds L(a, b, c) if and only if L(a, b, c). (4) For every real linear space V and for an arbitrary x holds x is an element of the points of OASpaceV if and only if x is a vector of V . (5) Let V be a real linear space. Then for every ordered affine space O1 such that O1 = OASpaceV for all elements a, b, c, d of the points of O1