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For 1 < p < 8 and p ? 2 we construct a family of mutually non-equivalent greedy bases in lp having the cardinality of the continuum. In fact, no basis from this family is equivalent to a rearranged subsequence of any other basis thereof. We are able to extend this statement to the spaces Lp and H1. Moreover, the technique used in the proof adapts to the setting of almost greedy bases where similar results are obtained.