  We strengthen the revised GCH theorem by showing, e.g., that for
lambda=cf(lambda)>beth_omega, for all but finitely many regular
kappa<beth_omega, lambda is accessible on cofinality kappa in a weak version of
it holds. In particular, lambda=2^mu=mu^+>beth_omega implies the diamond on
lambda is restricted to cofinality kappa for all but finitely many kappa in Reg
cap beth_omega and we strengthen the results on the middle diamond. Moreover,
we get stronger results on the middle diamond.
