In fact, I am working on Grobner systems, roughly speaking, can be considered as an extension of Grobner bases of polynomials over fields to polynomials with parametric coefficients. Indeed, a Grobner system is a finite set of triples (N,W,G) (so-called branches or segments); each branch contains (N,W) a couple of null (N)
and non-null parametric sets (W) (parametric constraint) and also a set of polynomials (G).
Now, I must consider the element of N as zero in polynomial set G. For this I use NormalForm(g, N, parametric monomial order) command of Maple for any g in G. But, it is possible the elements of N don't appear in any polynomial of G so using NormalForm command is additional work. Therefore I need a function to decide that is there at least one element at G e.g. g s.t. an element of N appears in g. In this case, the output is true otherwise false. For example, let
N = [b-1], G = [a*n*z^3-1, -b*x+3]. Since b-1 don't appear in any element of G so it is not needed to use NormalForm command for this branch (but when I use your function HasSubexpression it's output is true and the Normalform command is called automatically).