Let's say we have $|\langle x|y\rangle|^2=\langle y|x\rangle\langle x|y\rangle$. I have the projection operator $\mathbb{P}=|x\rangle\langle x|$, and a projection operator on the subspace $\mathbb{P}=|x,1\rangle\langle x,1|+|x,2\rangle\langle x,2|$.
If I were to find the probability that a measurement will find the result $x$, then I would write
$$P(x)=\langle y|\mathbb{P}|y\rangle=\langle y|(|x,1\rangle\langle x,1|+|x,2\rangle\langle x,2|)|y\rangle$$
My question is, does this distribute? Meaning, for all cases (and if not all then for what cases) can we write this as
$$P(x)=\langle y|x,1\rangle\langle x,1|y\rangle+\langle y|x,2\rangle\langle x,2|y\rangle$$ $$P(x)=|\langle x,1|y\rangle|^2+|\langle x,2|y\rangle|^2 $$
The answer is yes. In particular, this is a routine consequence of the usual distributive property for linear operators: $$ \langle y|(|x,1\rangle\langle x,1|+|x,2\rangle\langle x,2|)|y\rangle = \\ (\langle y|x,1\rangle\langle x,1|+\langle y|x,2\rangle\langle x,2|)|y\rangle =\\ \langle y|x,1\rangle\langle x,1| y \rangle+\langle y|x,2\rangle\langle x,2|y\rangle $$