Given a (not strictly) decreasing sequence of natural positive numbers $a_1, a_2, \dots, a_n$ prove that
$$ \prod_{i<j} j-i \quad\big|\quad \prod_{i<j} a_i - a_j - i +j $$
I already know a proof through the symmetrical Schur polynomial, but it's complex. I also tried to prove it by seeing the two terms as determinants of VanderMonde matrices, but it lead nowhere.