  We take advantage of the internal algebraic structure of the Bockstein
spectral sequence converging to ER(n)^*(pt) to prove that for spaces Z that are
part of Landweber flat real pairs with respect to E(n), the cohomology ring
ER(n)^*(Z) can be obtained from E(n)^*(Z) by base change. In particular, our
results allow us to compute the Real Johnson-Wilson cohomology of the
Eilenberg-MacLane spaces Z = K(Z, 2m+1), K(Z/2^q, 2m), K(Z/2, m) for all
natural numbers $m$ and $q$, as well as connective covers of BO: BO, BSO,
BSpin, and BO<8> (the last for n<3 only).
