Theorem 2 (Principle of extension of algebraic identities)

— Suppose that A is an infinite integral domain. Let $g_1, \ldots, g_m, f$ be elements of $A[(X_i)_{i \in I}]$ and assume the following hypotheses:

a) $g_1 \neq 0, \ldots, g_m \neq 0$; b) for all $x \in \mathbf{A}^1$ such that $g_1(x) \neq 0, \ldots, g_m(x) \neq 0$, we have $f(x) = 0$. Then $f = 0$.

For if $f \neq 0$, we have $fg_1 \ldots g_m \neq 0$ (IV, p. 9, Prop. 8), hence there exists $x \in \mathbf{A}^1$ such that $f(x)g_1(x) \ldots g_m(x) \neq 0$ (IV, p. 18, Cor. 2), which contradicts the hypothesis.