Before revision:  Ax_{k+1}-b\right\|^2_{2} \leq \left(1 + c_{A}{\|A\|_F^2}\right) \left\|A x_k -b \right\|^2_{2} - 2{\|A\|_F^2} \left\|A^T (Ax _k - b )\right\|^2_{2}.
After revision:  Ax_{k+1}-b\right\|^2_{2} \leq \left(1 + c_{A}{\|A\|_F^2}\right) \left\|A x_k -b \right\|^2_{2} - 2{\|A\|_F^2} \left\|A^T (Ax _k - x )\right\|^2_{2}.