Theorem 3

If A and C are two points, there exists a point between A and C.

Jeff's 1979 Notes

I. Given points A and C, line AC also exists [Axiom III].

II. There is a point D not contained in line AC [Axiom II].

III. There is also a point E such that CDE [Axiom IV].

IV. Similarly, there is also a point F such that AEF [Axiom IV again].

V. Since A, E, and C are non-collinear, and both AEF and CDE exist, there is a point B such that FDB and ABC [Axiom IX].  B is a point between A and C.

[I made a note in the margin: "Why are A, E, and C non-collinear?"]