let V, W be non empty ModuleStr over INT.Ring ; :: thesis: for f being FrForm of V,W
for w being Vector of W holds
( dom (FrFunctionalSAF (f,w)) = the carrier of V & rng (FrFunctionalSAF (f,w)) c= the carrier of F_Real & ( for v being Vector of V holds (FrFunctionalSAF (f,w)) . v = f . (v,w) ) )
let f be FrForm of V,W; :: thesis: for w being Vector of W holds
( dom (FrFunctionalSAF (f,w)) = the carrier of V & rng (FrFunctionalSAF (f,w)) c= the carrier of F_Real & ( for v being Vector of V holds (FrFunctionalSAF (f,w)) . v = f . (v,w) ) )
let w be Vector of W; :: thesis: ( dom (FrFunctionalSAF (f,w)) = the carrier of V & rng (FrFunctionalSAF (f,w)) c= the carrier of F_Real & ( for v being Vector of V holds (FrFunctionalSAF (f,w)) . v = f . (v,w) ) )
set F = FrFunctionalSAF (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 (FrFunctionalSAF (f,w)) = the carrier of V & rng (FrFunctionalSAF (f,w)) c= the carrier of F_Real ) ; :: thesis: for v being Vector of V holds (FrFunctionalSAF (f,w)) . v = f . (v,w)
let v be Vector of V; :: thesis: (FrFunctionalSAF (f,w)) . v = f . (v,w)
thus (FrFunctionalSAF (f,w)) . v = f . (v,w) by A1; :: thesis: verum