What is $\text{trdeg}(F( X_i Y_j \mid 1 \leq i \leq n, 1 \leq j \leq m) / F)$?

Question

We have indeterminate variables $X_i$ and $Y_j$ for $1 \leq i \leq n$ and $1 \leq j \leq m$. It is known $\text{trdeg}(L/F) = \text{trdeg}(L/K) + \text{trdeg}(K/F)$ for every field extension $F \subseteq K \subseteq L$. By putting $L = F(X_1,\ldots,X_n, Y_1, \ldots, Y_m)$ we can see that the transcendence degree of $F( X_i Y_j \mid 1 \leq i \leq n, 1 \leq j \leq m) / F$ is less or equal to $n+m$. But what is it exactly and how to prove it?