Magic constant of a normal magic square

Every normal magic square has the magic constant $n(n^{2}+1)/2$.

Let $M$ be an $n \times n$ array whose entries are exactly the integers $1,2,\dots,n^{2}$, each used once, and suppose every row, every column and both main diagonals of $M$ sum to the same number $s$. Then

Equivalently $s = n(n^{2}+1)/2$, the classical magic constant: it is $\tfrac{1}{n}$ of the sum $1+2+\cdots+n^{2} = n^{2}(n^{2}+1)/2$ of all entries.

Formalization Note The division by $2$ is avoided by multiplying through, so the statement is an identity in $\mathbb{N}$. Normality is the conjunction of the entry bounds $1 \le M_{ij} \le n^{2}$ with injectivity of the index-to-entry map.