I am going through a paper on an iterative method related to discrete optimal regulator, G. Hewer, "An iterative technique for the computation of the steady state gains for the discrete optimal regulator," in IEEE Transactions on Automatic Control, where Theorem 1 in the paper is given below
Theorem 1: Let $V_k,k=0,1,2..$ be the solution of the equation $$ V_k=\Phi_k'V_k\Phi_k+L_k'RL_k+C'C $$ where $$ L_k=(D'V_{k-1}D+R)^{-1}D'V_{k-1}\Phi,\ \ for\ k=1,2,...\\ \Phi_k=\Phi-DL_k,\ \ k=0,1,2,... $$ and $L_0$ is chosen such that $\Phi_0$ is a stability matrix. Then $$ K\le V_{k+1} \le V_k...,\ k=0,1,... $$ and $\lim_{k\to \infty}V_k=K.\ \square$
All the above variables are matrices of appropriate dimension.
In the proof of the theorem, at the very beginning there is an identity $$ \Phi_0'V_0\Phi_0+L_0'RL_0=\Phi_1'V_0\Phi_1+L_1'RL_1+(L_1-L_0)'(D'V_0D+R)(L_1-L_0)\ \ \ \ \ (7) $$ I could not arrive at (7) exactly. My attempt is as follows: From $\Phi_k=\Phi-DL_k$, construct $$ \Phi_0=\Phi-DL_0\\ \Phi_1=\Phi-DL_1\\ \Phi_0-\Phi_1=D(L_1-L_0)\\ (\Phi_0-\Phi_1)'V_0(\Phi_0-\Phi_1)=(L_1-L_0)'D'V_0D(L_1-L_0) $$ Adding $(L_1-L_0)'R(L_1-L_0)$ on both sides $$ \begin{align} (\Phi_0-\Phi_1)'V_0(\Phi_0-\Phi_1)+(L_1-L_0)'R(L_1-L_0)&=(L_1-L_0)'D'V_0D(L_1-L_0)+(L_1-L_0)'R(L_1-L_0)\\ &=(L_1-L_0)'(D'V_0D+R)(L_1-L_0)\end{align} $$ which is not exactly (7). I could not see where the mistake is and any help in this would be highly appreciated.
Thank you very much.