An output feedback type reset system consists of the output feedback connection of two hybrid dynamics (i.e. plant and controller) and the output based reset condition. In the existing results of Lyapunov based stability analysis for reset systems, Lyapunov-like functions which decrease after jump were usually considered. However, if the plant is a continuous-time system, the closed-loop jump dynamics always has nontrivial stationary points. This implies the existing Lyapunov based analysis is not applicable to a certain class of the reset systems which are practically important. To overcome this difficulty, the author proposes a new type of Lyapunov-like function which has a different property for the component of a state in the null space of an output matrix of the system and its complementary subspace. The existence conditions of the proposed Lyapunov-like function and a nonempty interval of reset forbidden period to guarantee the stability of the temporally regularized system are also discussed.