| (fac(i - j) * fac(j)) * i0 = fac(i), j ≤ i, 0 < i, 0 ≤ j, 0 ≤ i, fac(i + 1 - j) * fac(j - 1) | fac(i), 0 < j ⊦ fac(i) / (fac(i - j) * fac(j)) * j = fac(i) / (fac(i + 1 - j) * fac(j - 1)) * (i + 1 - j) |
(fac(i - j) * fac(j)) * i0 = fac(i), j ≤ i, 0 < i, 0 ≤ j, 0 ≤ i, fac(i + 1 - j) * fac(j - 1) | (fac(i - j) * fac(j)) * i0, 0 < j ⊦ ((fac(i - j) * fac(j)) * i0) / (fac(i - j) * fac(j)) * j = ((fac(i - j) * fac(j)) * i0) / (fac(i + 1 - j) * fac(j - 1)) * (i + 1 - j)