In this paper, another step on relaxation for Ta-kagi-Sugeno systems' stability analysis is addressed. Inspired from non-quadratic Lyapunov functions (NQLF), regarding to quadratic ones, a multiple polynomial Lyapunov function (MPLF) is proposed as an extension to polynomial Lyapunov function approaches. Following the latter post-LMI challenge, the obtained stability conditions are written in terms of a sum-of-squares (SOS) optimization problem. The proposed MPLF includes the well-studied NQLF ones as a special case. Moreover, the proposed SOS based stability conditions don't require unknown parameters in advance, as well as guarantee, when a solution exists, global stability. Therefore, these drawbacks of classical LMI based non-quadratic approaches are overcame.