$\forall$${\it es}$:ES, $P_{1}$, $P_{2}$:(E$\rightarrow\mathbb{P}$), $Q_{1}$, $R_{1}$, $Q_{2}$, $R_{2}$:(E$\rightarrow$E$\rightarrow\mathbb{P}$), $f$:(\{$e$:E$\mid$ $P_{1}$($e$)\} $\rightarrow$E).
\\[0ex]$P_{2}$ $\Rightarrow$ $P_{1}$
\\[0ex]$\Rightarrow$ $Q_{2}$ =$>$ $Q_{1}$
\\[0ex]$\Rightarrow$ $R_{1}$ =$>$ $R_{2}$
\\[0ex]$\Rightarrow$ \{$f$ is $Q_{1}${-}$R_{1}${-}pre{-}preserving on $P_{1}$ $\Rightarrow$ $f$ is $Q_{2}${-}$R_{2}${-}pre{-}preserving on $P_{2}$\}