Stochastic integral with respect to stochastic integral

Question

Assume that the continuous stochastic process $(X_t)_{t \geq 0}$ is such that the integral $$ Y_t = \int_0^t X_s \, dM_s $$ with respect to some continuous local martingale $(M_t)_{t \geq 0}$ is well defined and is again a continuous local martingale. Moreover, assume that the stochastic process $(Z_t)_{t \geq 0}$ is such that $$ V_t = \int_0^t Z_s d Y_s $$ is well-defined. How does one use the definition of the stochastic integral to show rigorously that $$ V_t = \int_0^t Z_s X_s \, dM_s. $$ What are the necessary assumptions for this? As a special case, one may assume that $(M_t)_{t \geq 0}$ is a standard Brownian motion.