We present a proof of the fact that given a Hodge isometry Psi between the rational second cohomology of two Kahler K3 surfaces $S_1$ and $S_2$, we can find a finite sequence of K3 surfaces and analytic (2, 2)-classes supported on successive products, such that the isometry Psi is the convolution of these classes. The proof of this fact implies that for projective K3 surfaces $S_1$, $S_2$ the class of Psi is algebraic. This proves a conjecture of I. Shafarevich.