Example of stopped processes violating the equality

Question

I am reading Protter's Stochastic Integration and Differential Equations and trying to understand the following definition better:

Here, $$ X_{t}^{T-}:=X_{t} 1_{\{0 \leq t<T\}}+X_{T-}1_{\{t \geq T\}} $$ and $X_{-T(\omega)}(\omega):=\lim_{s\to T(\omega),\,s<T(\omega)}X_s(\omega)$.

Since the constant $t$ is a stopping time, the requirement (i) is stronger than just saying "for any $t>0$, $X^t=Y^t$ implies..."

But I cannot think of a good example of processes $X$, $Y$ and a stopping time such that $X^{T-}\neq Y^{T-}$ but $X^t=Y^t$. What would be one?

Or maybe this is not a good question and I should think of $F$ that violates this condition...(which I don't have).