prove that $M_xM_1 = ?$
The only assertion here that goes beyond self-evident matrix algebra and substitutions, is that 

$$P_XP_1 = P_1$$

Since the columns of $X_1$ are in $X$ it follows that

$$P_X X_1 = X_1$$

because in general it holds that 

$$P_X X = X$$

Namely the projection matrix operates as the identity matrix for the columns of the matrix from which it has been constructed. But then

$$P_X P_1 = P_XX_1(X_1'X_1)^{-1}X_1' = (P_XX_1)(X_1'X_1)^{-1}X_1'$$

$$=X_1(X_1'X_1)^{-1}X_1' = P_1$$