Steep Diagonals / Diabolics ! (Interesting)...

Hello All, I have an interesting problem I would like to propose to the number theory masterminds here. (For reference, this question came from the "For Thought" section of my text book, that is why it may seem a little odd)

Using a picture, we can describe a set of subsets within a square that resemble diagonals but aren't quite the same. We shall call these "steep diagonals". Notice one of them labelled 'd' in the square below, and there are 6 others parallel to it.

$\displaystyle \begin{pmatrix}

x & d & x & x & x & x & x\\

x & x & x & x & d & x & x\\

d & x & x & x & x & x & x\\

x & x & x & d & x & x & x\\

x & x & x & x & x & x & d\\

x & x & d & x & x & x & x\\

x & x & x & x & x & d & x

\end{pmatrix}$

It can be proven under __certain__ conditions that the sums of the positive (or negative) steep diagonals are constant providing that we are dealing with a square full of consecutive integers starting at 0.

My question to you all is: How can this be proven? What are the __certain__ conditions that it is referencing?