  We report the first implementation of the von Neumann instantaneous
measurements of nonlocal variables which becomes possible due to technological
achievements in creating hyperentangled photons. Tests of reliability and of
the nondemolition property of the measurements have been performed with high
precision showing the suitability of the scheme as a basic ingredient of
numerous quantum information protocols. The method allows to demonstrate for
the first time with strong measurements a special feature of pre- and
postselected quantum systems: the failure of the product rule. It has been
verified experimentally that for a particular pre- and postselected pair of
particles a single measurement on particle $A$ yields with certainty
$\sigma_x^A=-1$, a single measurement on particle $B$ yields with certainty
$\sigma_y^B=-1$, and a single nonlocal measurement on particles $A$ and $B$
yields with certainty $\sigma_x^A \sigma_y^B=-1$.
