Degree of line bundles is additive on curves

Question

In Hartshorne (among other places), it is used that on a nonsingular curve $C$, the degree of line bundles is additive. That is,

$$\mbox{deg}_C(L_1 \otimes L_2) = \mbox{deg}_C(L_1)+ \mbox{deg}_C(L_2).$$

I can't see why, nor find a reference for this. Am I missing something obvious?

Thanks.

Edit: definition of degree I'm using is taking the associated Weil divisor $\sum_{P\in C}\eta_P P$, then taking the degree to be $\sum \eta_P$.