Let X and Y be k-smooth sequences of length w. A matching layer of balancers for X and Y is one where each element of X is joined by a balancer to  an element of Y in a one-to-one correspondence....

Let X and Y be k-smooth sequences of length w. A matching layer of balancers for X and Y is one where each element of X is joined by a balancer to  an element of Y in a one-to-one correspondence. Prove that if X and Y are each k-smooth and Z is the result of matching X and Y, then Z is(k+1)-smooth.