Normal Subgroup of Symmetric Group on More than 4 Letters is Alternating Group

Theorem
Let $n \in \N$ be a natural number such that $n > 4$.

Let $S_n$ denote the symmetric group on $n$ letters.

Let $A_n$ denote the alternating group on $n$ letters.

$A_n$ is the only proper non-trivial normal subgroup of $S_n$.

Proof
From Alternating Group is Normal Subgroup of Symmetric Group, $A_n$ is seen to be normal in $S_n$.

It remains to be shown that $A_n$ is the only such normal subgroup of $S_n$.