  In a recent letter, new representations were proposed for the pair of
sequences ($\gamma,\delta$), as defined formally by Bailey in his famous lemma.
Here we extend and prove this result, providing pairs ($\gamma,\delta$)
labelled by the Lie algebra A$_{N-1}$, two non-negative integers $\ell$ and $k$
and a partition $\lambda$, whose parts do not exceed $N-1$. Our results give
rise to what we call a ``higher-level'' Bailey lemma. As an application it is
shown how this lemma can be applied to yield general $q$-series identities,
which generalize some well-known results of Andrews and Bressoud.
