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Parameterization is one of the most powerful features to make specifications and declarative programs modular and reusable, and our best hope for scaling up formal verification efforts. This paper studies order-sorted parameterization at three different levels: (i) its mathematical semantics; (ii) its operational semantics by term rewriting; and (iii) the inductive reasoning principles that can soundly be used to prove properties about such specifications. It shows that achieving the desired properties at each of these three levels is a considerably subtler matter than for many-sorted specifications, but that such properties can be attained under reasonable conditions.