 The proof utilizes the coarea inequality, the isoperimetric inequality, the cone inequality, and the deformation theorem of Herbert Federer.One of the key ideas of the proof is the introduction of filling invariants, namely the filling radius and the filling volume of "X".Namely, Gromov proved a sharp inequality relating the systole and the filling radius,

valid for all essential manifolds "X"; as well as an inequality

valid for all closed manifolds "X".