SOooooo I'm not sure how to answer this. I have trouble using the correct language but basically I'm thinking this

If after a measurement of $|\psi>$ according to $O$ we obtain $|\phi>$ this means that one of the subspaces of the measurement is generated by $|\phi>$ with projector $|\phi><\phi|$. This means the operator for the measurment of observable $O$ contains $|\phi><\phi|$. Since the subspaces of any observable are orthogonal when $O$ is applied to $|\phi>$ the only non orthonal projector in the operator is $|\phi><\phi|$.

ie.

(1)where $|i><i|$ is the projection operator of the ith subspace that $O$

(2)this is true because the subspaces defined by an observable are orthogonal ie $|i><\phi|=0$ for all i.

Now for $O$ vs $O^2$ I'm really confused. Unless $O^2$ is a different measurement than two individual $O$ measurements I can't think of how they aren't the same…..thoughts anyone