Theorem 1
Any permutation is a product of transpositions.
Remark
As before, this means that f produces the same effect as consecutive application of a series of transpositions.
Proof of Theorem 1
Given a permutation f, first factor it into a product of cycles: f = g1g2 ...
Let g1 = (x1 x2 ... xk). Then it's easy to see that