The minitightness of products

We prove that tm(X×Y)≤tm(X)tm(Y)tm(X×Y)≤tm(X)tm(Y) if the space Y   is locally compact, and that always tm(X×Y)≤tm(X)χ(Y)tm(X×Y)≤tm(X)χ(Y), where tm(Z)tm(Z) is the minitightness (a.k.a. the weak functional tightness) of a space Z.