Why is this local martingale a martingale?

Question

Let $a(t, \omega)$ be a product measurable ($t$ refering to time and $\omega$ to an elementary event of some probability space) and adapted (to some filtration satisfying the "usual" conditions) process on some finite time interval $[0,T]$. Further, assume that $$\mathbb{E}(\int_0^T a^2(s, \omega) \mathrm{d}s) < \infty$$ Then the process $$x(t, \omega):=\int_0^t a(s,\omega)\mathrm{d}W_s(\omega),$$ wherein $W$ denotes a Wiener process, is a square integrable martingale with continuous trajectories (P a.s.). So far the setup. Now my question: Why is the following process $$y(t,\omega)=\int_0^t x(s,\omega)a(s,\omega)\mathrm{d}W_s(\omega)$$ a martingale? (I can see that the process is a local martingale - but i fail to establish that it is a martingale). I would appreciate some advice/help!