The length and area principle for a function on an abstract surface over a domain of a Carnot group

Given a domain of a Carnot group, we say that an abstract surface is defined over this domain if the volume element is induced by some weight function, while the usual vector norm in the horizontal vector bundle is replaced by a more general analog of this norm. Suppose that a continuous function $f$ from a weighted Sobolev space and a smooth function $\varphi$ are defined in a domain of a Carnot group. Assume also that an abstract surface is defined over the domain. In the paper we prove a version of the Lebesgue – Courant lemma (the length and area principle) for the function $f$ in terms of the moduli of families of horizontal curves lying on the level sets of the function $\varphi$.