let K be non empty 1-sorted ; :: thesis: for V, W being non empty ModuleStr over K

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 K & ( for v being Vector of V holds (FunctionalSAF (f,w)) . v = f . (v,w) ) )

let V, W be non empty ModuleStr over K; :: thesis: 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 K & ( for v being Vector of V holds (FunctionalSAF (f,w)) . v = f . (v,w) ) )

let f be Form of V,W; :: thesis: for w being Vector of W holds

( dom (FunctionalSAF (f,w)) = the carrier of V & rng (FunctionalSAF (f,w)) c= the carrier of K & ( for v being Vector of V holds (FunctionalSAF (f,w)) . v = f . (v,w) ) )

let w be Vector of W; :: thesis: ( dom (FunctionalSAF (f,w)) = the carrier of V & rng (FunctionalSAF (f,w)) c= the carrier of K & ( 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 K ) ; :: thesis: for v being Vector of V holds (FunctionalSAF (f,w)) . v = f . (v,w)

let v be Vector of V; :: thesis: (FunctionalSAF (f,w)) . v = f . (v,w)

thus (FunctionalSAF (f,w)) . v = f . (v,w) by A1; :: thesis: verum

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 K & ( for v being Vector of V holds (FunctionalSAF (f,w)) . v = f . (v,w) ) )

let V, W be non empty ModuleStr over K; :: thesis: 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 K & ( for v being Vector of V holds (FunctionalSAF (f,w)) . v = f . (v,w) ) )

let f be Form of V,W; :: thesis: for w being Vector of W holds

( dom (FunctionalSAF (f,w)) = the carrier of V & rng (FunctionalSAF (f,w)) c= the carrier of K & ( for v being Vector of V holds (FunctionalSAF (f,w)) . v = f . (v,w) ) )

let w be Vector of W; :: thesis: ( dom (FunctionalSAF (f,w)) = the carrier of V & rng (FunctionalSAF (f,w)) c= the carrier of K & ( 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 K ) ; :: thesis: for v being Vector of V holds (FunctionalSAF (f,w)) . v = f . (v,w)

let v be Vector of V; :: thesis: (FunctionalSAF (f,w)) . v = f . (v,w)

thus (FunctionalSAF (f,w)) . v = f . (v,w) by A1; :: thesis: verum