$\forall$${\it es}$:ES, $P$, $Q$:(E$\rightarrow\mathbb{P}$), $f_{1}$:(\{$e$:E$\mid$ $P$($e$)\} $\rightarrow$\{$e$:E$\mid$ $Q$($e$)\} ), $f_{2}$:(\{$e$:E$\mid$ $Q$($e$)\} $\rightarrow$E). \\[0ex]($f_{1}$ is locl{-}pre{-}preserving on $P$ \& $f_{2}$ is locl{-}pre{-}preserving on $Q$) \\[0ex]$\Rightarrow$ $f_{2}$ o $f_{1}$ is locl{-}pre{-}preserving on $P$