Length of Concatenation

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

Then:


 * $\size {S T}\ = \size S + \size T$

where $\size S$ denotes the length of $S$.

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


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

and furthermore, precisely one of these options occurs.

There are $\size S$ symbols in $S$, and $\size T$ symbols in $T$.

In total, then, $S T$ is seen to consist of $\size S + \size T$ symbols.