Prefix of String is Substring

Theorem
Let $S$ be a string, and 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 $S$.