Length of Concatenation

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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.

$\blacksquare$