Subgroup equals Conjugate iff Normal/Proof 2

Proof
From Subgroup Superset of Conjugate iff Normal, $N$ is normal in $G$ :
 * $\forall g \in G: N \supseteq g \circ N \circ g^{-1}$
 * $\forall g \in G: N \supseteq g^{-1} \circ N \circ g$

From Subgroup Subset of Conjugate iff Normal, $N$ is normal in $G$ :
 * $\forall g \in G: N \subseteq g \circ N \circ g^{-1}$
 * $\forall g \in G: N \subseteq g^{-1} \circ N \circ g$

The result follows by definition of set equality.