I came across the following claims (from https://aghitza.github.io/publication/translation_velu/, the very beginning of section 1).
Let $E$ be an elliptic curve (in Weierstrass form) over over an algebraic closed field $k$. For each point $P\in E$ and $t\in k(E)$, let $\nu_P(t)$ be the order of $t$ at $P$. Fix a point $Q$ on $E$.
- Claim 1: There exist $f,g \in k(E)$ such that $\nu_Q(f) = -2, \nu_Q(g) = -3, (g^2/f^3)(Q) = 1$, $\nu_P(f) \ge 0$ and $\nu_P(g) \ge 0$ for all $P\neq Q$.
- Claim 2: If any $f, g \in k(E)$ satisfy these conditions, then $k(E) = k(f, g)$.
Regarding claim 1, I know that $x$ and $y$ satisfy these conditions at the infinity point $O$ and $k(E) = k(x, y)$. I suppose we can translate $Q$ to $O$ and the claim 1 follows. But why is the claim 2 true?