# Euclidean Geometry Proof using the Ruler Axiom

Let $l$ be any line. Then there is a one-to-one correspondence $f: l \rightarrow \mathbb{R}$ such that, for any two points $A$,$B$ on $l$, the distance $|AB| = |f(A) - f(B)|$.
I think I understand the gist of the problem. All that the coordinate system $g$ is doing is moving point $P$ from $0$ to another point. So I need to show that in the coordinate system $g$, $g(A)$ is equal to $f(A)$ plus the new position of $P$ in the coordinate system $g$. I have no idea how to show this formally using a proof. Can anyone help me out?