I am trying to understand a claim in a paper https://arxiv.org/pdf/1303.7427.pdf (Observation 4.2) that the integral over the Jacobi-determinant of a regular homotopy $H:[0,1]\times [0,1]\rightarrow \mathbb{R}^2$ of two curves $P$ and $Q$ satisfies $$\int_{0}^1\int_0^1 |JH(t,s)| dt ds = \int_{\mathbb{R}^2} deg_H(x) dx,$$ where the authors define $deg_H(x)$ to be "the number of connected components in the pre-image of $x$ under $H$". To me this looks like the "coarea formula" $$\int_A |JH(u)| du= \int_{\mathbb{R}^2} \mathcal{H}^0(A\cap H^{-1}(x)) dx,$$ where $\mathcal{H}^0$ is the zeroth order Hausdorff measure, however, the cardinality of a set is generally not the number of connected components of a set. Are these notions equivalent, when $H$ is a regular homotopy? I.e. can we say something like $H^{-1}(x)$ is disconnected when $H$ is regular, that is the intermediate curves can visit the same point more than once, but only at disconnected times $s,t$?
Recall a homotopy between two curves (paths) in the plane $P$ and $Q$ is a continuous map $H:[0,1]\times [0,1]\rightarrow \mathbb{R}^2$ where $H(0,\cdot)=P, H(1,\cdot)=Q$ and $H(\cdot,0)=Q(0)=P(0)$ and $H(\cdot,1)=Q(1)=P(1)$. A homotopy is called regular, when all intermediate curves $H(t,\cdot)$ are regular curves (i.e. the derivative does not vanish).