Doubt in definition of closure under concatenation operation in Recursive Enumerative languages
If $L_1,L_2$ are two languages over a common alphabet, then their concatenation is given by
$$ L_1 L_2 = \{ x_1 x_2 : x_1 \in L_1 \text{ and } x_2 \in L_2 \}. $$
If $L_1,L_2$ are r.e. then so is their concatenation. To see this, you can use the following algorithm. Given an input $x$, consider all possible partitions $x = x_1x_2$. For each partition, run a machine for $L_1$ on $x_1$ and a machine for $L_2$ on $x_2$; do this in parallel for all possible partitions. Accept if for some partition $x = x_1x_2$, you find that both $x_1 \in L_1$ and $x_2 \in L_2$.