String is Substring of Itself

Theorem
Let $S$ be a string.

Then $S$ is a substring of itself.

Proof
By definition, a string $T$ is a substring of $S$ in $\AA$ :
 * $S = S_1 T S_2$

where:
 * $S_1$ and $S_2$ are strings in $\AA$ (possibly null)
 * $S_1 T S_2$ is the concatenation of $S_1$, $T$ and $S_2$.

Let $S_1$ and $S_2$ both be the null string.

Then it follows that:
 * $S = T$

Hence the result.