Singular

Suppose that a finite set of polynomials in C[x,y,z] has a finite number of solutions (i.e. the generated ideal is 0-dimensional).
Suppose also that the Groebner basis with respect to lex order x>y>z is

[f(z), g(y,z), h(y,z), k(x,y,z)]

As well known, the system can be now easily solved: choose a root z0 of f, plug it into g and h and look for a common root (y0) etc.

The question is the following: Is it true that for EVERY root z0 of f there exist y0, z0 such that (x0,y0,z0) satisfy the system?

In all the examples I have seen this is true, but I don't know whether this is true in general or there is a counterexample.

Note that an extension from (y0,z0) to (x0,y0,z0) is not always possible (there is an "Extension theorem" which must be used).
The problem here is to extend from (z0) to (x0,y0,z0) which seems to be always possible. Is it?