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. There exist at least four points. Armed with the axioms and theorems we've now proven, we can use a direct proof to construct four different points. Finally, we want to address the question of whether indirect proofs, like the one concerning the irrationality of the square-root of 2, are also examples of deductive arguments. Specifically, the number 2 must appear an odd number of times on the right, and an even number of times on the left.