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In this paper we create pseudo associativity ($\rho$-assoc) and pseudodistributivity ($\rho$-dist) properties for not fundamental operators NFO$\downarrow$, $\uparrow$, using two semantic rules, also we build proofs forthis result in Hilbert-Ackermann ($\mathscr{HA}$) axiomatic system, all in the0-order logic (0-OL) context.

$\rho$-assoc and $\rho$-dist of wfs and f in $\Sigma$ and $\mathscr{L_{HA}}$-theory on 0-OL