This Terrain Awareness and Warning System produces a new “Too Low Terrain” predictive alert of “Caution” type when the crew of the aircraft has the possibility of resolving a detected risk of collision with the terrain without interrupting the current maneuver to stabilize at a safety altitude by a leveling-off maneuver, without performing a vertical avoidance maneuver. To do this, it measures the ability of the airplane to avoid the terrain with a sufficient margin without performing a vertical avoidance maneuver, taking into account the location or locations of the penetration or penetrations of the terrain along an alert prober C as well as the capacity of the aircraft to level off knowing the flight conditions.