Suppose A ,,..., A, are subsets of a finite set A, and B ,,..., B, are subsets of a finite set B. For each subset S of N= (1, 2,..., n), let A, = nies Ai and B, = nisS Bi. It is shown that if explicit bijections fs: A, + B, for each S c N are given, an explicit bijection h: A-U,:, Ai 4 B Ui_I Bi can be constructed. The map h is independent of any ordering of the elements of A and B, and of the order in which the subsets Ai and Bi are listed.