Exercise 15, § 2

With the notations of the preceding exercises, prove now that if for a certain integer $k \geq 0$, one has $\dim(A[X_1, ..., X_m]) \geq k + m + 1$ for an integer $m$, one has the inequality $\dim(A[X_1, ..., X_k]) \geq 2k + 1$.