Tail fields of partially exchangeable arrays

We give an elementary, direct proof that if an array of random variables {(Xij, [alpha], [xi]i, [eta]j); i, j [set membership, variant] } is separately exchangeable, then X = {Xij; i, j [set membership, variant] } and {([alpha], [xi]i, [eta]j); i, j [set membership, variant] } are conditionially independent given the shell [sigma]-field X of X. We show further that if (X, Y) = {(Xij, Yij); i, j [set membership, variant] } is separately exchangeable, then X and X, Y are conditionally independent given X.