Every polygon of n vertices can be triangulated. The induction step involves drawing a diagonal in a polygon with n vertices, thus partitioning the polygon into two polygons. As in Example A, above, we look for substitution rules to prove the induction step. A simple example, but one which requires some careful algebra. We begin with severeal axioms, and deduce theorems one by one, sometimes directly and sometimes indirectly.
Every line is a set of points.
There exist at least two points. Every line contains at least 1 point. This may seem a trivial thing to want to prove, but because it is not an axiom it must be proven in order for us to use it in future proofs. Every line contains at least two points. We prove this indirectly as well, again showing that a line that contains only one point would contradict A5.