Given a collection of sets X={A1,A2,...,An}. A set {a1,a2,...,an} is called a single representation of X if ai∈Ai for all i. Let ∣S∣=mn, S=A1∪A2∪...∪An=B1∪B2∪...∪Bn with ∣Ai∣=∣Bi∣=m for all i. Prove that S=C1∪C2∪...∪Cn where for every i,Ci is a single represenation for {Aj}j=1nand {Bj}j=1n.