If A and C are two points, there is a point P such that A, C, and P are non-collinear.
I. Given points A and C, there exists a third point, P [Axiom II].
II. Since there is only one line which contains both A and C [Axiom III], and since the third point, P, cannot also be contained in this line (otherwise contradicting Axiom II), no line contains A, C, and P.
III. By Definition 3, points A, C, and P are non-collinear.
Apparently, Axiom II can be broadly interpreted to "manufacture" a non-collinear point with respect to any given line, which would seem to make this a pretty trivial theorem.