I have a question about a step in the proof of Lemma 7.4.8 from Liu's "Algebraic Geometry and Arithmetic Curves" (page 288):
If we assume that $X$ is hiperelliptic and we have our finite separable map $\pi:X \to \mathbb{P}^1_k$ of degree $2$.
A rational point $y_0$ of $\mathbb{P}^1_k$ defines a Cartier divisor and let $D:= \pi^*y_0 \in Div(X)$ it's pullback under $\pi$.
Denote by $O_X(D)$ the corresponding invertible sheaf to $D$ and the take into account that the author uses the notation $$L(D)=H^0(X,O_X(D))$$.
My question is why under giving setting we have an inclusion
$$H^0(\mathbb{P}^1_k, O_{\mathbb{P}^1_k}(y_0)) \subset H^0(X,O_X(D))$$
as stated in the excerpt?