In this post from George Lowther's blog (shortly after equation (6)), he claims that, for a cadlag semi martingale $X_t$, $f(X_t)$ is a semi-martingale because $f(t)$ obeys the generalized Ito formula $$f(X_t)=f(X_0)+\int_{0+}^tf'(X_{s-})dX_s+\frac{1}{2}\int_{0+}^tf''(X_{s-})d[X]_t+\sum_{t \leq T}(\Delta f(X_t)-f'(X_{t-})\Delta X_t-\frac{1}{2}f''(X_{t-})(\Delta X_t)^2). $$
Question:
Now I don't understand how the claim follows from the expression of the Ito formula. Clearly, the stochastic integrals and the integration wrt to the quadratic variation are semi martingales and the term $f'(X_{t-})\Delta X_t$ can be seen as $\int_{0+}^tf'(X_{t-})dX_t-\int_{0+}^{t-}f'(X_{t-})dX_t$ and similarly for the $\frac{1}{2}f''(x_{t-})(\Delta X_t)^2$. Since the space of semimartingales is a vector space, the countable sum of these terms are again semi-martingale.(using the fact that the jumps are countable for a caglad process). What I dont understand is how can I see that $\Delta f(X_t)$ is a semi-martingale? Can it be rewritten as some stochastic integral or some tricks like that?