Assume there is an infinite sequence of row-reductions

By our choice of the reduction in Proposition 14, at any step the lowest possible leading term of is removed. But all leading terms of a standard basis of are leading terms of a standard basis of , too, which determines the Hilbert-function of the module

Let *r*_{M} be the regularity bound of the Hilbert function of *M*. Then the homogeneous degree of all leading terms of a standard basis of *Q* (resp.
of
)
is not greater than *r*_{M}, contradicting our assumption of an
infinite reduction sequence.