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We settle the fusion product decomposition theorem for higher-level affineDemazure modules for the cases  $E^{(1)}_{6, 7, 8}, F^{(1)}_4$ and $E^{(2)}_{6}$, thus completing the maintheorems of Chari et al. (J. Algebra, 2016) and Kus et al. (Represent. Theory,2016). We obtain a new combinatorial proof for the key fact, that was used inChari et al. (op cit.), to prove this decomposition theorem. We give a casefree uniform proof for this key fact.

A note on the fusion product decomposition of Demazure modules