Given a semi-martingale $X$ on a filtered probability space $(\Omega,\{\mathcal F_t\}_{t\le\infty},P)$, an integrand $A$ and a set $C\in\mathcal F_t$. Show: $$\int_t^T 1_C\cdot A\;d\!X=1_C\cdot\int_t^T A\;d\!X.$$
Is this statement true? How could one prove it? Where can I find the prove?