If $A$ is infinite , $F,G\in A[X_i|i\in I]$, $F\neq 0$ and if $A^I \setminus V_A(F)\subseteq V_A(G)$,then $G=0$

This question was asked in my assignment and I am stuck on it.

Question:If $A$ is infinite integral domain , $F,G\in A[X_i|i\in I]$, $F\neq 0$ and if $A^I \setminus V_A(F)\subseteq V_A(G)$,then show that $G=0$.

Attempt: Let $D(F)= A^I \setminus V_A(F)=${$a\in A^I : F(a)\neq0$} . Given $D(F)\subseteq V_A(G)$. Let $G\neq 0$ ( on the contrary) Since $F$ and $G$ are polynomials ; only finitely many variables occur in them. So, there exists a finite subset $J\subset I$ st both lie in $A[x_j |j \in J]$ . Let $m=J$.

Trying to prove: A non-zero polynomial in finitely many variables over an infinite integral domain doesn't vanishes on all of $A^m$.

Proving using induction on m:

For $m=1,$ a non zero polynomial has only finitely many roots.

For $m>1$, write $H= \sum_{r=0}^d H_r(X_1,...,X_{m-1}){X_r}^m$ with $H_d \neq 0$.

But I am not able to move forward from here and would need guidance.

Please help me.