proof of theorem A2

 Theorem A2 The affine group A(2) is generated by parallel projections. Proof We use the Fundamental Theorem of Affine Geometry, which shows that an affine transformation is uniquely determined by the images of three non-collinear points. Suppose that t is an affine transformation which maps the non-collinear points A, B, C to the (non-collinear) points A', B', C', respectively. We will define two parallel projections which have the same effect on A, B and C. We begin by taking 3 copies Π, Π' and Π" of the plane. On Π, we mark the points A, B and C. On Π' and Π", we mark the points A', B' and C'. First, we place Π and Π" so that the point A on Π coincides with A' on Π". By rotating the planes about this point, we can ensure that neither B nor B' lies on the line of intersection. Then u = BB' is not parallel to either plane, so we can define the parallel projection p from Π to Π" along u. This maps A to A', and B to B'. Suppose that it maps C to C". Since A, B, C are not collinear, their images A', B', C" are not collinear. Now place Π" and Π' so that the points A', B' coincide. As neither C' nor C" is collinear with these, they do not lie on the line of intersection. Thus v = C'C" is not parallel to either plane, so we can define the parallel projection q from Π" to Π' along v. This fixes A' and B', and maps C" to C'. Thus qop maps A, B, C to A', B', C'. By Theorem A1, the maps p and q are affine, so the composite is also affine. Since t and qop have the same effect on three points, the Fundamental Theorem shows that they are equal.