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