Equivalence of Definitions of Normal Subset/3 and 4 imply 2

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

Let $N \subseteq G$.

Suppose that $N$ is a normal subset of $G$ by Definition 3 and Definition 4.

Then $N$ is a normal subset of $G$ by Definition:Normal Subset/Definition 2.

Proof
By Equivalence of Normal Subset Definitions/3 iff 4, $N$ being a normal subset of $G$ by Definition 3 and Definition 4 implies that the following hold:
 * $(1)\quad \forall g \in G: g \circ N \circ g^{-1} \subseteq N$
 * $(2)\quad \forall g \in G: g^{-1} \circ N \circ g \subseteq N$


 * $(3)\quad \forall g \in G: N \subseteq g \circ N \circ g^{-1}$
 * $(4)\quad \forall g \in G: N \subseteq g^{-1} \circ N \circ g$

By $(1)$ and $(3)$ and Equality of Sets:
 * $\forall g \in G: g \circ N \circ g^{-1} = N$

By $(2)$ and $(4)$ and Equality of Sets:
 * $\forall g \in G: g^{-1} \circ N \circ g = N$