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$\forall$${\it es}$:ES, ${\it Cmd}$:Type, ${\it In}$:AbsInterface(${\it Cmd}$), ${\it isupdate}$:(${\it Cmd}$$\rightarrow\mathbb{B}$), ${\it Sys}$, ${\it Out}$:AbsInterface(Top).
\\[0ex](E(${\it In}$) $\subseteq$r E(${\it Sys}$))
\\[0ex]$\Rightarrow$ \=($\forall$$f$:sys{-}antecedent(${\it es}$;${\it Sys}$).\+
\\[0ex]($\forall$$e$:E(${\it In}$). $f$($e$) = $e$ $\in$ E)
\\[0ex]$\Rightarrow$ \=($\forall$${\it chain}$:(E(${\it Sys}$)$\rightarrow$(Id List)).\+
\\[0ex]chain{-}consistent($f$;${\it chain}$)
\\[0ex]$\Rightarrow$ \=($\forall$$x$, $y$:E(${\it Sys}$).\+
\\[0ex]$x$ is $f$$\ast$($y$)
\\[0ex]$\Rightarrow$ \=($\forall$$z$:E(${\it Sys}$).\+
\\[0ex]$y$ $\leq$loc $z$ 
\\[0ex]$\Rightarrow$ loc($x$) $<<$= loc($f$($z$))
\\[0ex]$\Rightarrow$ ($\exists$$e$:E(${\it Sys}$). (loc($e$) = loc($f$($z$)) $\in$ Id \& $e$ is $f$$\ast$($y$) \& $x$ is $f$$\ast$($e$)))))))
\-\-\-\-
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