So those are just the starting point for some basic constructions. the proofs of the transfer theorems, again, they come just immediately from the definitions and from generating function, counting, the way that we talked about earlier. so A+ again, they separate into the disjoint sets and that immediately gives the result. for star, it's a convolution of, of the type that we've seen before but let's take a look. so to do all different relabellings, so we take one alpha from A and beta from B, and to do all different relabellings it's alpha plus beta choose alpha just like with the example I did with the cycles. and then so that's a number of ways you can relabel. and then the size of the, an object composed of an alpha and a beta is Z to the alpha plus beta. 