Birational geometry of Fano direct products

We prove birational superrigidity of direct products $V=F_1\times…\times F_K$ofprimitiveFanovarietiesofthefollowingtwotypes:either$F_i\subset{\mathbb P}^M$isageneralhypersurfaceofdegree$M$,$M\geq 6$, or $F_i{\mathbb{P}}^M$ is a general double space of index 1, $M\ge 3$. In particular, each structure of a rationally connected fiber space on $V$ is given by a projection onto a direct factor. The proof is based on the connectedness principle of Shokurov and Koll' ar and the technique of hypertangent divisors.