Let be the projection If , then is a subset of and free by the induction hypothesis.

By the induction hypothesis, all of the options are equivalent to nimbers, say .

As induction hypothesis, assume the generalization is true for n 1.

In other words, the induction hypothesis holds for S(c).

As the induction hypothesis, we may assume that .

As this is just the induction hypothesis, we are done.

By the induction hypothesis, the number of ways to do that is 2.

By the induction hypothesis, the number of such subsets is 2.

Because of the induction hypothesis, the products inside the brackets are unambiguous.

We have , where the second equality follows from the induction hypothesis.