Complementary Idempotent is Idempotent

Theorem
Let $\HH$ be a Hilbert space.

Let $I$ be an identity operator on $\HH$.

Let $A$ be an idempotent operator.

Then the complementary idempotent $I - A$ is also idempotent.

Proof
That is, $I - A$ is idempotent.

Also see

 * Complementary Idempotent of Complementary Idempotent is Idempotent