Sum of Squares on Pairs of Rows and Columns of Moessner's Order 4 Magic Square

Theorem
The sums of the squares of the entries are equal on the following pairs of rows and columns of Moessner's order $4$ magic square:


 * Rows $1$ and $4$
 * Rows $2$ and $3$
 * Columns $1$ and $4$
 * Columns $2$ and $3$.

Proof
Recall Moessner's order $4$ magic square: