Union of two Zero Loci is Zero Locus

Theorem
Let $k$ be a field.

Let $n \in \N_{>0}$.

Let $A = k \sqbrk {X_1, \ldots, X_n}$ be the ring of polynomials in $n$ variables over $k$.

Let $S,T \subseteq A$.

Then:
 * $\ds {\map V S} \cup \map V T = \map V {S \cdot T}$

where:
 * $\map V \cdot$ denotes the zero locus
 * $S \cdot T := \set { fg : f \in S, g \in T }$

Proof
Let $x \in {\map V S} \cup \map V T$.

That is, either $x \in \map V S$ or $x \in \map V T$.

Then, if $x \in \map V S$:
 * $\forall \paren {f,g} \in S \times T : \map f x \map g x = 0 \map g x = 0$

else:
 * $\forall \paren {f,g} \in S \times T : \map f x \map g x = \map f x 0 = 0$

Therefore:
 * $x \in \map V {S \cdot T}$

On the other hand: