Let H be a Hilbert Space and P and Q be the projections on closed linear subspaces M and N of H.
Prove that if Q-P is a projection,then its range is N\cap M^\perp.

My attempt:
Let x\in range of Q-P.
Since Q-P is a projection,
\Rightarrow(Q-P)x=x
\Rightarrow Qx-Px=x
I am stuck here as i don't know how to proceed to show that x\in N and x\in M^\perp

Can anyone give me some hints to proceed?
If I am in the wrong way to prove the question,please tell me.