if $D_1\sim D_1'$ and $D_2\sim D_2'$, then we have $D_1+D_2\sim D_1'+D_2'$

Question

I'm studying Fulton's algebraic curves book and I'm trying to prove that if $D_1\sim D_1'$ and $D_2\sim D_2'$, then we have $D_1+D_2\sim D_1'+D_2'$.

If the the first two equivalences work, then we have

$$D_1+D_2=(f_1)+(f_2)+(D_1'+D_2')$$

If I take the rational function $z=f_1f_2$, do I solve the question?

Thanks