Subgroup equals Conjugate iff Normal/Proof 2

Theorem

$\forall g \in G: g \circ N \circ g^{-1} = N$

$\forall g \in G: g^{-1} \circ N \circ g = N$

Proof

From Subgroup is Superset of Conjugate iff Normal, $N$ is normal in $G$ if and only if:

$\forall g \in G: N \supseteq g \circ N \circ g^{-1}$

$\forall g \in G: N \supseteq g^{-1} \circ N \circ g$

From Subgroup is Subset of Conjugate iff Normal, $N$ is normal in $G$ if and only if:

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

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

The result follows by definition of set equality.

$\blacksquare$