Dimension of cohomology group of sheaves associated to a point: $dimH^0(L(−P))=dimH^0(L)−1$

Question

Could anybody help me with this theorem?

Let $L$ be a line bundle on a smooth projective curve with $H^0(L)$ positive dimensional, then for a general point P, $dimH^0(L(−P))=dimH^0(L)−1$.

I don't know how to prove this. Does anybody know an idea or a reference? In Hartshornes proof of the Riemann-Roch-Theorem for curves there is a similar equality for the Euler-characteristic:

$\chi(L(P))=\chi(L)+1$