Complementary Projection is Projection

Theorem
Let $H$ be a Hilbert space.

Let $A$ be a projection.

Then the complementary projection $I - A$ is also a projection.

Proof
By Characterization of Projections, $A$ is self-adjoint.

Then $\left({I - A}\right)^* = I^* - A^* = I - A$ from Adjoining is Linear.

So $I - A$ is also self-adjoint.

From Complementary Idempotent is Idempotent, $I - A$ is idempotent.

Hence, applying Characterization of Projections, $I - A$ is a projection.