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In this context, a coordinate system is given by the Ruler Axiom, which states:

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?

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What have you thought of so far? – mixedmath Dec 5 '12 at 6:13
I said in the original post what I understood. I understand what I need to prove, I just don't know how to formally prove it. – anon87576 Dec 5 '12 at 15:25
I'm not asking for anyone to solve the problem, I just need a hint on how to go about it. The function g is just translating the function f to the right on the number line, so it makes sense that g(A) = the new position of P, g(P), plus the old position of A, f(A). – anon87576 Dec 5 '12 at 15:36

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