Proving the equality of two projection matrices

Question

$y$ is a $n \times 1$ vector. $X_1$ and $X_2$ are $n \times k1$ and $n \times k2$ matrices. $P_1$ is the projection matrix for $X_1$. It is asked to prove that $P_1*y = P_1*X_2*((P_1*X_2)'*(P_1*X_2))^{-1}*(P_1*X_2)'*P_1*y$. I took the step that $(I - P_1*X_2*((P_1*X_2)'*(P_1*X_2))^{-1}*(P_1*X_2)')*P_1*y = 0$. One piece of information is that $X_1*(X_1'*X_1)^{-1}*X_1'*X_2 = X_1*c$ where $c$ is a constant. It seems that if I can show that the first product is in some way $M_1$, I would be done because $I - P_1 = M_1$ and because $M_1*P_1 = 0$. But I am not sure why this would be true. What would be another approach to prove this?