In the proof of Theorem III.6.2 (c) in Silverman's The Arithmetic Of Elliptic Curves it says:
Let $x_1, y_1 \in K(E_1)$ and $x_2, y_2 \in K(E_2)$ be Weierstrass coordinates. We start by looking at $E_2$ considered as an elliptic curve defined over the field $K(E_1) = K(x_1, y_1)$. Then another way of saying that $\phi$ is an isogeny is to note that $\phi(x_1, y_1) \in E_2(K(x_1,y_1))$, [...]
Why is that? I'm blind.
(Here, $E_1, E_2$ denote elliptic curves, $\phi \colon E_1 \to E_2$ is an isogeny, $K(E_1), K(E_2)$ denote the function fields of $E_1$ resp. $E_2$ and $E_2(K(x_1,y_1))$ are the points in $K(x_1,y_1)$ on the curve $E_2$.)