Birational geometry of Fano direct products

We prove the birational superrigidity of direct products $V=F_1\times\dots\times F_K$ of primitive Fano varieties of the following two types: either $F_i\subset\mathbb P^M$ is a general hypersurface of degree $M$, $M\geqslant 6$, or $F_i\stackrel{\sigma}{\to}{\mathbb P}^M$ is a general double space of index 1, $M\geqslant 3$. In particular, every structure of a rationally connected fibre space on $V$ is given by the projection onto a direct factor. The proof is based on the connectedness principle of Shokurov and Kollár and the technique of hypertangent divisors.