Result
R1396: Image elements of orthogonal projection and coprojection are orthogonal shows that $P x, P^{\perp} x$ is an orthogonal collection in $N$. We can write
\begin{equation}
x
= x + P x - P x
= P x + x - P x
= P x + (I - P) x
= P x + P^{\perp} x
\end{equation}
Thus, applying
R26: Pythagorean theorem the the collection $P x, P^{\perp} x$ yields
\begin{equation}
\Vert x \Vert^2
= \left\Vert P x + P^{\perp} x \right\Vert^2
= \left\Vert P x \right\Vert^2 + \left\Vert P^{\perp} x \right\Vert^2
\end{equation}
$\square$
