question about martingale In my lecture notes,I found the following problem:

Let $X$ an $F_{t}$  adapted continuous process and $G_{t}\subset F_{t}$.
  show that
   $$E\left(\left. \int^{t}_{0}X_{s}ds \right|G_{t}\right)-\int^{t}_{0} E(X_{s}|G_{s})ds $$ is a $G_{t}$ martingale.

Does anybody know how to prove this? Some help would be appreciated 

$$
M_{t+h} = E\left( \int^{t+h}_{0}X_{s}ds |G_{t+h}\right)
-\int^{t+h}_{0} E(X_{s}|G_{s})ds\\
M_t = E\left( \int^{t}_{0}X_{s}ds|
G_{t+h} G_{t}\right)-\int^{t}_{0} E(X_{s}|G_{s})ds 
\\
M_{t+h} - M_t =
E\left( \int^{t+h}_{0}X_{s}ds | G_{t+h}\right)
-\int^{t+h}_{t} E(X_{s}|G_{s})ds -
E\left(
 \int^{t}_{0}X_{s}ds | G_{t}\right)\\
E(M_{t+h} - M_t|G_t)=
E\left( \int^{t+h}_{0}X_{s}ds | G_{t}\right)
-\int^{t+h}_{t} E(X_{s}|G_{t})ds -
E\left(
 \int^{t}_{0}X_{s}ds | G_{t}\right) =0
$$using the linearity of the conditional expectation in the last step.
