  We develop further in this work the high order paracontrolled calculus
setting to deal with the analytic part of the study of quasilinear singular
PDEs. A number of continuity results for some operators are proved for that
purpose. Unlike the regularity structures approach of the subject by Gerencser
and Hairer, and Otto, Sauer, Smith and Weber, or Furlan and Gubinelli' study of
the two dimensional quasilinear parabolic Anderson model equation, we do not
use parametrised families of models or paraproducts to set the scene. We use
instead infinite dimensional paracontrolled structures that we introduce here.

Analysis of PDEs;Probability