Let $T$ be a normal operator on a finite-dimensional complex inner product space. Show that if T is a projection, then it is an orthogonal projection.

Question

My problem is with the second part. If you say $R(T)^\perp = N(T)^{\perp\perp} \ldots$ isn't that the same as $R(T)^\perp = N(T)$ which is what we are trying to prove...and you can't use what you are trying to prove in the proof. So am I misunderstanding this or is it wrong?