We prove the main estimates on the nonperturbative coordinate of the renormalisation group map. This is the heart of the proof of the main result. One ingredient in the proof of the main estimates consists of the stability estimates which exploit the decay arising from e−g|φ|4. A central ingredient in the proof consists of the crucial contraction which is responsible for the contractive property of the renormalisation group map. The contractive property was used in an essential way to construct the global renormalisation group flow in Chap. 6.