Puzzle - Why does this work

You are presented with this grid of numbers:

2 3 4 5
3 4 5 6
4 5 6 7
5 6 7 8

Select a number and circle it. Then cross out all other numbers in its row or column. For example:

2 3 x 5
3 4 x 6
x x 6 x (with the 6 circled)
5 6 x 8

Then circle another number, and cross out all numbers in the rows and columns that it's in. Then repeat until you have 4 circled numbers and all others have been crossed out. (You cannot circle a crossed out number.) For example your final grid could look like this (unrelated to first example):

2 x x x
x x 5 x
x 5 x x
x x x 8

Then you add up your 4 numbers - the answer will always be 20.

The question : WHY?

You can generalise the grid by calling the top-left number n, and filling out the grid with n, n + 1, n + 2, etc, and the total will always be 4n + 12. So why is it always 4n + 12?
(The 4n part is fairly obvious if you draw out the grid, but why 12?)

Thoughts?

Simple enough. The procedure ensures that each remaining number is in a different row and a different column from any other, so there will be one number in each row and column. From any position on the grid, if you go down one row, or across one column, the numerical value increments by one. If you place your four numbers in the left-hand column, with the top one taking the minimum value n, the total would be [n + (n + 1) + (n + 2) + (n + 3)] = 4.n + 6. To make sure that they are in different columns, you must move any three of the number positions across by 1, 2 or 3 steps in any order, and this adds 1 + 2 + 3 to the total, making 4.n + 12 altogether.

2[n + n + 6] = 4n + 12 ANSWER

Hard one! But I did notice that the roWs going down to the bottom go in an order: 2345, 3456, 4567