holds for given (ordered) polynomials .
An operation for polynomials is generalized for polynomials with cofactors in the following way:
Hence, whenever a new polynomial with cofactors is the result of a sequence of operations, Equation (1) still holds for the result. For instance, the result of the calculation of an S-polynomial and reducing it w.r.t. an ordered set of polynomials is the normal form of the S-polynomial plus a representation in terms of the initial polynomials.