Vanishing Ideal of Zero Locus of Ideal is Radical

Theorem
Let $k$ be a field.

Let $n \ge 1$ be a natural number.

Let $k \sqbrk {X_1, \ldots, X_n}$ be the polynomial ring in $n$ variables over $k$.

Let $\mathfrak a \subseteq k \sqbrk {X_1, \ldots, X_n}$ be an ideal.

Then:
 * $\map I {\map V {\mathfrak a} } = \map \Rad {\mathfrak a}$

where:
 * $\map V \cdot$ denotes the zero locus
 * $\map I \cdot$ denotes the vanishing ideal
 * $\map \Rad \cdot$ denotes the radical

Proof
This is exactly Hilbert's Nullstellensatz.