 At this point of the argument (p. 779) EPR make two critical assumptions, although they do not call special attention to them. (For the significance of these assumptions in Einstein’s thinking see Howard 1985 and also section 5 of the entry on  Einstein.)  The first assumption (separability) is that at the time when the systems are separated, maybe quite far apart, each has its own reality. In effect, they assume that each system maintains a separate identity characterized by a real physical state, even though each system is also strictly correlated with the other in respect both to momentum and position. They need this assumption to make sense of another. The second assumption is that of locality. Given that the systems are far apart, locality supposes that “no real change can take place” in one system as a direct consequence of a measurement made on the other system. They gloss this by saying “at the time of measurement the two systems no longer interact.” Note that locality does not require that nothing at all about one system can be disturbed directly by a distant measurement on the other system. Locality only rules out that a distant measurement may directly disturb or change what is counted as “real“ with respect to a system, a reality that separability guarantees. On the basis of these two assumptions they conclude that each system can have definite values (“elements of reality”) for both position and momentum simultaneously. There is no straightforward argument for this in the text. Instead they use these two assumptions to show how one could be led to assign position and momentum eigenstates to one system by making measurements on the other system, from which the simultaneous attribution of elements of reality is supposed to follow. Since this is the central and most controversial part of the paper, it pays to go slowly here in trying to reconstruct an argument on their behalf.