Complementary Idempotent is Idempotent

Theorem
Let $H$ be a Hilbert space.

Let $A$ be an idempotent operator.

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

Proof
In $I - A$, $I$ is the identity operator on $H$.

Hence:

That is, $I - A$ is idempotent.