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es{-}kind{-}sends{-}iff(${\it es}$;$k$;$A$;$l$;${\it tg}$;$B$;${\it ds}$;$e$.$f$($e$))
\\[0ex]$\,\equiv$$_{\mbox{\scriptsize def}}$$\;\;$\=(($\forall$$e$:es{-}E(${\it es}$). (es{-}kind(${\it es}$; $e$) = rcv($l$,${\it tg}$) $\in$ Knd) $\Rightarrow$ (es{-}valtype(${\it es}$; $e$) $\subseteq$r $B$))\+
\\[0ex]\& ($\forall$$x$:Id. es{-}vartype(${\it es}$; source($l$); $x$) $\subseteq$r fpf{-}cap(${\it ds}$;IdDeq;$x$;Top))
\\[0ex]\& (\=$\forall$$e$:es{-}E(${\it es}$).\+
\\[0ex](es{-}kind(${\it es}$; $e$) = $k$ $\in$ Knd)
\\[0ex]$\Rightarrow$ (es{-}loc(${\it es}$; $e$) = source($l$) $\in$ Id)
\\[0ex]$\Rightarrow$ (es{-}valtype(${\it es}$; $e$) $\subseteq$r $A$)))
\-\\[0ex]c$\wedge$ \=(alle{-}at(${\it es}$;source($l$);$e$.(es{-}kind(${\it es}$; $e$) = $k$ $\in$ Knd)\+
\\[0ex]$\Rightarrow$ ($\uparrow$isl($f$($e$)))
\\[0ex]$\Rightarrow$ ($\exists$\=${\it e'}$:es{-}E(${\it es}$)\+
\\[0ex](es{-}kind(${\it es}$; ${\it e'}$) = rcv($l$,${\it tg}$) $\in$ Knd \& es{-}sender(${\it es}$; ${\it e'}$) = $e$ $\in$ es{-}E(${\it es}$))))
\-\\[0ex]\& (\=$\forall$${\it e'}$:es{-}E(${\it es}$).\+
\\[0ex](es{-}kind(${\it es}$; ${\it e'}$) = rcv($l$,${\it tg}$) $\in$ Knd)
\\[0ex]$\Rightarrow$ \=((es{-}kind(${\it es}$; es{-}sender(${\it es}$; ${\it e'}$)) = $k$ $\in$ Knd)\+
\\[0ex]c$\wedge$ ($\uparrow$isl($f$(es{-}sender(${\it es}$; ${\it e'}$))))
\\[0ex]c$\wedge$ (es{-}val(${\it es}$; ${\it e'}$) = outl($f$(es{-}sender(${\it es}$; ${\it e'}$))) $\in$ $B$)))
\-\-\\[0ex]\& (\=$\forall$${\it e'}$:es{-}E(${\it es}$).\+
\\[0ex](es{-}kind(${\it es}$; ${\it e'}$) = rcv($l$,${\it tg}$) $\in$ Knd)
\\[0ex]$\Rightarrow$ \=($\forall$${\it e''}$:es{-}E(${\it es}$).\+
\\[0ex](es{-}kind(${\it es}$; ${\it e''}$) = rcv($l$,${\it tg}$) $\in$ Knd)
\\[0ex]$\Rightarrow$ (es{-}sender(${\it es}$; ${\it e''}$) = es{-}sender(${\it es}$; ${\it e'}$) $\in$ es{-}E(${\it es}$))
\\[0ex]$\Rightarrow$ (${\it e''}$ = ${\it e'}$ $\in$ es{-}E(${\it es}$)))))
\-\-\-\-
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