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The aim of this paper is to prove a generalization of the famous Theorem A of Quillen for strict ∞-categories. This result is central to the homotopy theory of strict ∞-categories developed by the authors. The proof presented here is of a simplicial nature and uses Steiner's theory of augmented directed complexes. In a subsequent paper, we will prove the same result by purely ∞-categorical methods. [ABSTRACT FROM AUTHOR]