  Bell's theorem applies to the normalizable approximations of the original
Einstein-Podolsky-Rosen (EPR) state. The constructions of the proof require
measurements difficult to perform, and dichotomic observables. By noticing the
fact that the four mode squeezed vacuum state produced in type II
down-conversion can be seen both as two copies of approximate EPR states, and
also as a kind of polarization supersinglet, we show a straightforward way to
test violations of the EPR concepts with direct use of their state. The
observables involved are simply photon numbers at outputs of polarizing beam
splitters. Suitable chained Bell inequalities are based on the geometric
concept of distance. For a few settings they are potentially a new tool for
quantum information applications, involving observables of a nondichotomic
nature, and thus of higher informational capacity. In the limit of infinitely
many settings we get a Greenberger-Horne-Zeilinger-type contradiction: EPR
reasoning points to a correlation, while quantum prediction is an
anticorrelation. Violations of the inequalities are fully resistant to
multipair emissions in Bell experiments using parametric down-conversion
sources.
