NOR is not Associative/Proof 1

Theorem
Let $\downarrow$ signify the NOR operation.

Then there exist propositions $p,q,r$ such that:


 * $p \downarrow \left({q \downarrow r}\right) \not \vdash \left({p \downarrow q}\right) \downarrow r$

That is, NOR is not associative.

Proof
Taking $p = \bot$ and $r = \top$, we have $\vdash \neg p \land r$, discharging the last assumption.

Hence the result.