Equivalence of Definitions of Normal Subset/2 implies 3

Theorem
Let $\left({G, \circ}\right)$ be a group.

Let $N \subseteq G$.

Let $N$ be a normal subset of $G$ by Definition 2.

Then $N$ is a normal subset of $G$ by Definition 3.

That is, if:
 * $\forall g \in G: g \circ N \circ g^{-1} = N$ or
 * $\forall g \in G: g^{-1} \circ N \circ g = N$

then:
 * $\forall g \in G: g \circ N \circ g^{-1} \subseteq N$ and
 * $\forall g \in G: g^{-1} \circ N \circ g \subseteq N$

Proof
Note that:
 * $\left({\forall g \in G: g \circ N \circ g^{-1} = N}\right) \iff \left({\forall g \in G: g^{-1} \circ N \circ g = N}\right)$

Then the theorem holds by definition of set equality.