I was given the question: If $p,q \in B(\mathcal{H})$ are projections such that $q\mathcal{H}\subset p\mathcal{H}$, show that $p-q$ is a projection with $p\mathcal{H}= q\mathcal{H}\oplus (p-q)\mathcal{H}$.
Is my proof below rigorous enough?
If $q\mathcal{H}\subset p\mathcal{H}$, then $p$ acts as identity on $q\mathcal{H}$, and $$ (p-q)^2 = p^2 - 2pq + q^2 = p - 2q + q = p -q $$ and $$ (p-q)^* = p^* - q^* = p -q $$ since they are projections. Thus $(p-q)$ is a projection. If $v \in \mathcal{H}$ then we can write $v = qv + (1-q)v$ and then applying $p$ across the equation we get $$ pv = pqv + p(1-q)v = qv + (p-pq)v = qv + (p-q)v $$ to show the decomposition we want.