# Height of truncated system of parameter

Let $$R$$ be a Noetherian local ring of dimension $$d$$, and $$a_1,\dots,a_d$$ is a system of parameters. I am wondering whether the following statement is true:

$$\mathrm{ht}(a_1,\dots,a_i)=i$$ for all $$i$$, $$1 \le i \le d$$.

I am thinking about this because by definition, $$\mathrm{ht}(a_1,\dots,a_d)=d$$ thus the statement holds for $$i=d$$. And if $$R$$ is Cohen-Macaulay, then $$a_1,\dots,a_i$$ is a regular sequence, thus $$\dim R/(a_1,\dots,a_i)=d-i$$ and $$\dim R/I + \operatorname{ht} I = \dim R$$ holds for any ideal in a CM local ring, thus the statement holds for any $$I$$. So I am curious about whether counter example exists in a non CM noetherian local ring. (Or even local Noetherian domain.)

This is true if $$R$$ is catenary and equidimensional. However, for example if $$R$$ is not equidimensional, then there exists a minimal prime $$\mathfrak{p}$$ of $$R$$ such that dim$$(R/\mathfrak{p})<\text{dim}(R)$$. Therefore any $$x\in \mathfrak{p}$$ which not in the union of the other minimal primes is a parameter element of height $$0$$.