© 2019 Elsevier Ltd This paper proposes a new approach to the problem of establishing internal and external stability of interconnection of integral input-to-state stable (iISS) systems. In the literature of iISS and input-to-state stability (ISS), typical Lyapunov functions constructed for interconnected systems are in max-separable or sum-separable from. In contrast to the max construction, solutions to the sum construction have not been given intuitive geometrical interpretations. The max construction is, however, incapable of guaranteeing stability in the presence of iISS components which are not ISS. This paper aims at constructing a Lyapunov function in the non-ISS case through simple geometrical observations. The approach leads to a novel construction mixing the max and sum separability. The new Lyapunov function gives much useful forward invariant sets than the sum-separable ones known in the literature. It is as intuitive as the max-separable function even in the presence of a non-ISS component.