I want to prove that:
$X$, $Y$ are irreducible varieties $\Rightarrow$ $dim(X \times Y) = dim(X)+ dim(Y)$.
$dim X$ := max {n | $\exists\quad \emptyset\neq X_0\subset X_1\subset...\subset X_n = X$ sequence of irreducible closed sets}
I found this property in Mumford's Red Book. The author wrote there that the proof it's easy, but I have no idea how to start. I saw a proof on this site, but it involves tensor products and I can't understand it.