 Product dimension bound for sections on a curve

For nonzero finite-dimensional spaces $V,W$ of sections of two line bundles on a smooth integral curve over an algebraically closed field, their product span in sections of the tensor product satisfies this bound. At one point choose bases with strictly increasing orders of vanishing. Multiplying every first-space basis vector by the first second-space vector, and then the last first-space vector by the remaining second-space vectors, produces distinct increasing orders and hence independent sections. This proves the dimension inequality needed for the Clifford inequality for curves.