Why can this joint distribution function be written as this integral of a conditional distribution function? In Section 3.8 of Dependence Modeling with Copulas (Joe), the author starts with the following.

In this section, we show how Sklar's theorem applies to a set of univariate conditional distributions, all conditioned on variables in an index set $S$. Sequential mixtures of conditional distributions lead to the vine pair-copula construction in Section 3.9.
Shorthand notation used here includes: $\boldsymbol{x}_J=\left(x_j: j \in J\right)$ where $J$ is a subset of $\{1, \ldots, d\} ;\left(-\infty, \boldsymbol{x}_J\right)=\prod_{j \in J}\left(-\infty, x_j\right) ;\left(-\infty, \boldsymbol{x}_J\right]=\prod_{j \in J}\left(-\infty, x_j\right]$.


Consider $d$ random variables $X_1, \ldots, X_d$ with multivariate distribution $F$. Let $S$ be a non-empty subset of $\{1, \ldots, d\}$, which will be the conditioning set of variables. Let $T$ be a subset of $S^c$ with cardinality of at least two, which will be the conditioned set of variables. With $M=S \cup T$, we can write
$$
F_M\left(\boldsymbol{x}_M\right)=\int_{\left(-\infty, \boldsymbol{x}_S\right]} F_{T \mid S}\left(\boldsymbol{x}_T \mid \boldsymbol{y}_S\right) \mathrm{d} F_S\left(\boldsymbol{y}_S\right)
$$

I cannot figure out why this equality holds, and any help would be appreciated. Thank you for taking the time to consider my question!
 A: Since we are working with the cumulative distribution we have that
$$\begin{align}
F_M\left(\boldsymbol{x}_M\right) 
& = \mathbb{P}\left(\boldsymbol{X}_M \leq \boldsymbol{x}_M\right) \\
& = \mathbb{P}\left(\boldsymbol{X}_T \leq \boldsymbol{x}_T \mbox{ and } \boldsymbol{X}_S \leq \boldsymbol{x}_S\right) \\
& = \mathbb{P}\left(\boldsymbol{X}_T \leq \boldsymbol{x}_T | \boldsymbol{X}_S \leq \boldsymbol{x}_S\right)\mathbb{P}\left( \boldsymbol{X}_S \leq \boldsymbol{x}_S\right) \\
& = \int_{\left(-\infty, \boldsymbol{x}_S\right]} \mathbb{P}\left(\boldsymbol{X}_T \leq \boldsymbol{x}_T | \boldsymbol{X}_S = \boldsymbol{y}_S\right) \mathrm{d} F_S\left(\boldsymbol{y}_S\right)
\\
& = \int_{\left(-\infty, \boldsymbol{x}_S\right]} F_{T \mid S}\left(\boldsymbol{x}_T \mid \boldsymbol{y}_S\right) \mathrm{d} F_S\left(\boldsymbol{y}_S\right)
\end{align}$$
The passage to the integral is a bit tricky, but compare to the discrete case to convince yourself. Also, see that the integration bound follows naturally from the condition $\boldsymbol{X}_S \leq \boldsymbol{x}_S$.
