Section PolyOverRing.
Variables (S : {pred R}) (ringS : subringPred S) (kS : keyed_pred ringS).
Canonical polyOver_smulrPred := SmulrPred (polyOver_mulr_closed kS).
Canonical polyOver_subringPred := SubringPred (polyOver_mulr_closed kS).

Lemma polyOverXsubC c : ('X - c%:P \in polyOver kS) = (c \in kS).