The following is a direct proof of a theorem by Zia-ud-Din.
Let {ν} ≡ {ν1, ν2, …, νp)} be any S-function of weight r + s such that
in the alternant denoted in the theorem by A(αβγ….). Let {μ} be an S-function of weight s, equal to and let {λ} be an S-function of weight r, such that {λ}δ(x1, …,xp) is obtained with coefficient gλμν by diminishing the indices in the alternant {ν}δ(x1, …,xp) according to the theorem.