We'll prove the theorem of the title, which states that given a morphism from a complete irreducible variety to a product of r-dimensional projective spaces with image of dimension > r, the inverse image of the diagonal is connected. To this end, we will outline a classical proof of the Bertini Theorem for general linear sections: given a morphism from an irreducible variety to projective space, the inverse image of a low co-dimensional general plane is irreducible.
Some applications of the Fulton-Hansen result will be considered: in particular the irreducibility of intersections of certain subvarieties of projective space, and a nice property of unramified morphisms of good varieties to projective space. Definitions of "certain", "nice", "good" and possibly "unramified" will be provided.