# Change chain of prime ideals so that $a \in P_1$

A text I am following uses of the following (probably basic) commutative algebraic lemma, omitting its proof.

Lemma: Let $$n\in\mathbb{N}_{>0}$$, and let $$P_0\subsetneq P_1\subsetneq\cdots\subsetneq P_n$$ be a chain of prime ideals in a Noetherian ring $$R$$. Moreover, let $$a\in P_n$$. Then there is a chain of prime ideals $$P_0'\subsetneq P_1'\subsetneq\cdots\subsetneq P_{n-1}'\subsetneq P_n$$ (i. e., in the given chain we may change all prime ideals except the last one) so that $$a\in P_1'$$.

Given its place in the text, there should be a proof of this using nothing more advanced than Krull's Principal Ideal Theorem. However I have not been able to find such a proof.

Any help?