Give a two-sentence synopsis of Proof 2 of Theorem 9.15.

Data from 9.15 Theorem

No permutation in S_n can be expressed both as a product of an even number of transpositions and as a product of an odd number of transposition.

Proof 2 of 9.15 Theorem

Let a ∈ S_n and let r = (i, j) be a transposition in S_n. We claim that the number of orbits of σ and of r σ differ by 1.