Composition of Permutations is not Commutative

Theorem
Let $S$ be a set.

Let $\map \Gamma S$ denote the set of permutations on $S$.

Let $\pi, \rho$ be elements of $\map \Gamma S$

Then it is not necessarily the case that:
 * $\pi \circ \rho = \rho \circ \pi$

where $\circ$ denotes composition.

Proof
Proof by Counterexample:

Let $S := \set {1, 2, 3}$.

Let:

Then:

while: