Let \(\left(B_{t}ight)_{t \geqslant 0}\) be a \(\mathrm{BM}^{1},\left(Y_{t}ight)_{t \geqslant 0}\) and adapted process such that \(Y_{t} Y_{s}=Y_{t}\) for all \(s \leqslant t\) and \(\left(X_{t}ight)_{t \geqslant 0} \in \mathcal{L}_{T}^{2}\). Show that \[Y_{t} \int_{0}^{t} X_{S} d B_{s}=Y_{t} \int_{0}^{t} Y_{s} X_{s} d B_{s}\]