By theorem of orthogonal projections we have $P^T=P=P^2$ and $Q^T=Q=Q^2$. By definition we need to show that $(P-Q)^T(P-Q) = I = (P-Q)(P-Q)^T$. Note that $P+Q = I \iff P = I - Q$. Observe \begin{equation} \begin{split} (P-Q)^T(P-Q) &= (P^T-Q^T)(P-Q) \\ &= P^TP - P^TQ - Q^TP + Q^2 \\ &= P - PQ - QP + Q \\ &= (I-Q) - (Q-Q^2) - (Q-Q^2) + Q\\ &= I. \end{split} \end{equation} And \begin{equation} \begin{split} (P-Q)(P-Q)^T &= (P-Q)(P^T-Q^T) \\ &= PP^T -PQ^T -QP^T + QQ^T \\ &= P^2 - PQ - QP + Q^2 \\ &= P - PQ-QP - Q \\ &= (I-Q) - (I-Q)Q-Q(I-Q)+Q \\ &= I-Q +Q \\ &= I. \end{split} \end{equation} Hence $P-Q$ is an orthogonal matrix.
Is this a valid prove? Have I made correct use of the carrying out of the transposes?