A proof would require showing that the same result would occur for all acceptable values of n. You can see this would occur by noting the pattern within and between the sets in the first output and extending this pattern to what it would be were there n elements in each of n sets. Then it's back to the first using the fourth elements. After sorting, each integer from 1 to n2 would appear exactly once, proving the conjecture. Viewing the same phenomena from the other direction, each row's end cell is n2 greater than what it would be were it a 'normal' cell. Every pair of congruent cells is either equivalent or differs in value by n2.
All cell pairs congruent to a row's seed column's cell pair are equivalent to it.
This is repeated at each level of movement away from the initial pair because there will always be an even number of cells between the members of each new pair of cells. The mean value of the range of all of the rows is n2 - n. 1 - 2x, where x is the value of the seed cell. All rows are magical. Therefore, each row must be magical.