Orthogonal Projection Operators and Subspaces

Question

I'm looking for help to confirm my intuition and perhaps some help with more of a concrete proof. My question is:

Let $S_0 ⊆ S_1$ be subspaces, and let $H_0$ and $H_1$ be orthogonal projection operators onto $S_0$ and $S_1$, respectively. Explain why $H_0◦H_1$ = $H_1◦H_0$ = $H_0$.

My answer: given $x$ $∈$ $S_0 ∩ S_1$. Then $H_0H_1x$ = $H_0x$ = $x$

Thus any combination of the operators $H_1$ & $H_0$ brings a vector into the $H_0$ space.

Answer 1

I think you can split the cases into $V\backslash S_1$, $S_1\backslash S_0$ and $S_0$ and directly verify these cases as $S_0\subseteq S_1$.