How to show that a projection matrix is invariant w.r.t the choice of a g-inverse

Question

Let X be an $(nxk)$-matrix (not necessarily of full rank) and $P = X(X'X)^-X'$, i should show that P is invariant with respect to the choice of the g-inverse of $X'X$. As a hint I should first verify that $PX=X$. That wasn't a problem since $(X'X)^-(X'X)$ cancels out. I thought about showing the invariance the same way, so that $(X'X)^-(X'X)$ cancels out again, but honestly i got no idea how to do it or even start.