Set of Isometries in Complex Plane under Composition forms Group

Theorem
Let $S$ be the set of all complex functions $f: \C \to \C$ which preserve distance when embedded in the complex plane.

That is:
 * $\size {\map f a - \map f b} = \size {a - b}$

Let $\struct {S, \circ}$ be the algebraic structure formed from $S$ and the composition operation $\circ$.

Then $\struct {S, \circ}$ is a group.