Composition of Symmetries is Symmetry

Theorem
Let $P$ be a geometric figure.

Let $S_P$ be the set of all symmetries of $P$.

Let $\circ$ denote composition of mappings.

Let $\phi$ and $\psi$ be symmetries of $P$.

Then $\phi \circ \psi$ is also a symmetry of $P$.

Proof
By definition of composition of mappings:
 * $\phi \circ \psi$ is a mapping.

We have by definition of symmetry that:
 * $\map \phi P$ is congruent to $P$

and:
 * $\map \psi {\map \phi P}$ is congruent to $\map \phi P$

Therefore:
 * $\phi \circ \psi$ is congruent to $P$

Thus $\phi \circ \psi$ is a symmetry of $P$.