  In this paper we create pseudo associativity ($\rho$-assoc) and pseudo
distributivity ($\rho$-dist) properties for not fundamental operators NFO
$\downarrow$, $\uparrow$, using two semantic rules, also we build proofs for
this result in Hilbert-Ackermann ($\mathscr{HA}$) axiomatic system, all in the
0-order logic (0-OL) context.
