Length of Concatenation

Theorem
Let $S$ and $T$ be words, and let $ST$ be their concatenation.

Then:


 * $\left\vert{ST}\right\vert = \left\vert{S}\right\vert + \left\vert{T}\right\vert$

where $\left\vert{S}\right\vert$ denotes the length of $S$.

Proof
Because of the unique readability of $ST$, we can determine for each symbol $s$ that is part of $ST$, whether:


 * $s$ is part of $S$
 * $s$ is part of $T$

and furthermore, precisely one of these options occurs.

There are $\left\vert{S}\right\vert$ symbols in $S$, and $\left\vert{T}\right\vert$ symbols in $T$.

In total, then, $ST$ is seen to consist of $\left\vert{S}\right\vert + \left\vert{T}\right\vert$ symbols.