Prefix of String is Substring

Theorem

Let $S$ be a string.

Let $T$ be a prefix of $S$.

Then $T$ is a substring of $S$.

Proof

By definition of substring, there exists a string $T'$ such that:

$S = TT'$

Hence $S$ is the concatenation of the null string, $T$, and $T'$.

Thus by definition of substring, $T$ is a substring of $S$.

$\blacksquare$