let V, W be non empty ModuleStr over INT.Ring ; for f being Form of V,W
for w being Vector of W holds
( dom (FunctionalSAF (f,w)) = the carrier of V & rng (FunctionalSAF (f,w)) c= the carrier of INT.Ring & ( for v being Vector of V holds (FunctionalSAF (f,w)) . v = f . (v,w) ) )
let f be Form of V,W; for w being Vector of W holds
( dom (FunctionalSAF (f,w)) = the carrier of V & rng (FunctionalSAF (f,w)) c= the carrier of INT.Ring & ( for v being Vector of V holds (FunctionalSAF (f,w)) . v = f . (v,w) ) )
let w be Vector of W; ( dom (FunctionalSAF (f,w)) = the carrier of V & rng (FunctionalSAF (f,w)) c= the carrier of INT.Ring & ( for v being Vector of V holds (FunctionalSAF (f,w)) . v = f . (v,w) ) )
set F = FunctionalSAF (f,w);
dom f = [: the carrier of V, the carrier of W:]
by FUNCT_2:def 1;
then A1:
ex g being Function st
( (curry' f) . w = g & dom g = the carrier of V & rng g c= rng f & ( for y being object st y in the carrier of V holds
g . y = f . (y,w) ) )
by FUNCT_5:32;
hence
( dom (FunctionalSAF (f,w)) = the carrier of V & rng (FunctionalSAF (f,w)) c= the carrier of INT.Ring )
; for v being Vector of V holds (FunctionalSAF (f,w)) . v = f . (v,w)
let v be Vector of V; (FunctionalSAF (f,w)) . v = f . (v,w)
thus
(FunctionalSAF (f,w)) . v = f . (v,w)
by A1; verum