One can find all over the internet that it is well-known (and obvious) that given a fiber bundle $F \to E \to B$, the equality $\chi(E) = \chi(F)\chi(B)$ holds ($\chi$ is the Euler characteristic). This is supposed to be true without any serious (= other than the Euler characteristics of $B$ and $F$ being defined) restrictions on $B$, $F$ and $E$. The thing is, I cannot find any reference for this statement.
What I know (and what I've found proof of) is that if $F \to E \to B$ is an orientable fibration, then this is indeed true. But how do I get rid of the orientability assumption?
I'd really appreciate any reference or a sketch of proof.