If a string $s$ contains $t$ as an internal substring, then $s$ can be written as $s_1 t s_2$, where $s_1$ and $s_2$ are nonempty. A t-suffix contains all of $s_2$ and none of $s_1$, so that its corresponding prefix contains all of $s_1$ and none of $s_2$. The corresponding cut will therefore fall somewhere in $t$.