I was reading the proof of the Proposition (3.4, IV) in Hartshorne's Algebraic Geometry book.
Proposition: Let $D$ be a divisor on a curve $X$. Then:
(a) the complete linear system $\vert D \vert$ has no base points if and only if for every point $P\in X$, $$ dim \vert D-P \vert = dim \vert D \vert - 1;$$ (b) $D$ is very ample if and only if for every points $P, Q\in X$(including the case $P=Q$), $$ dim \vert D-P-Q \vert = dim \vert D \vert - 2.$$
In the proof, I could not understand the line in the proof which says that if $D$ satisfies the condition (b), then we have $$ dim \vert D-P \vert = dim \vert D \vert - 1$$ for every $P\in X.$
(It may be silly question.)
I would be thankful if someone could elaborate this sentence.