2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/150801In a recent letter, new representations were proposed for the pair of sequences ($γ,δ$), as defined formally by Bailey in his famous lemma. Here we extend and prove this result, providing pairs ($γ,δ$) labelled by the Lie algebra A$_{N-1}$, two non-negative integers $\ell$ and $k$ and a partition $λ$, 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.Latex2e, 21 pages, 1 Postscript figure. Several typos have been corrected including a serious one, a figure has been added and the discussion has been improved. Version to appear in the Ramanujan JournalQuantum Algebratext