\begin{tabbing}
$\forall$$A$:Type, $B$:($A$$\rightarrow$Type), ${\it eq}$:EqDecider($A$), $f$,$g$:fpf($A$; $a$.$B$($a$)), $x$:$A$.
\\[0ex]($\uparrow$fpf{-}dom(${\it eq}$; $x$; fpf{-}join(${\it eq}$; $f$; $g$)))
\\[0ex]$\Rightarrow$ (\=fpf{-}ap(fpf{-}join(${\it eq}$; $f$; $g$); ${\it eq}$; $x$)\+
\\[0ex]=
\\[0ex]if fpf{-}dom(${\it eq}$; $x$; $f$) then fpf{-}ap($f$; ${\it eq}$; $x$) else fpf{-}ap($g$; ${\it eq}$; $x$) fi 
\\[0ex]$\in$ $B$($x$))
\-
\end{tabbing}