My Solution

A pq-string of the form:

$$ x\mathbf{p}y\mathbf{q}z $$

is a theorem if and only if we can get back to $x\mathbf{p}-\mathbf{q}x-$ by removing exactly as many hyphens from $y$ (namely all but one) as we remove from $z$.

The Author’s Solution

A pq-string of the form:

$$ x\mathbf{p}y\mathbf{q}z $$

if and only if the amount of hyphens in $x$ added to the amount of hyphens in $y$ adds up to the amount of hyphens in $z$.

Comments

Hah, duh! I actually managed to write down a description of addition (which I believe is actually also correct?) without noticing!

*imagine me facepalming here*