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 v being Vector of V holds

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

let V, W be non empty ModuleStr over K; :: thesis: for f being Form of V,W

for v being Vector of V holds

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

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

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

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

set F = FunctionalFAF (f,v);

dom f = [: the carrier of V, the carrier of W:] by FUNCT_2:def 1;

then A1: ex g being Function st

( (curry f) . v = g & dom g = the carrier of W & rng g c= rng f & ( for y being object st y in the carrier of W holds

g . y = f . (v,y) ) ) by FUNCT_5:29;

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

let y be Vector of W; :: thesis: (FunctionalFAF (f,v)) . y = f . (v,y)

thus (FunctionalFAF (f,v)) . y = f . (v,y) by A1; :: thesis: verum

for f being Form of V,W

for v being Vector of V holds

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

let V, W be non empty ModuleStr over K; :: thesis: for f being Form of V,W

for v being Vector of V holds

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

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

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

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

set F = FunctionalFAF (f,v);

dom f = [: the carrier of V, the carrier of W:] by FUNCT_2:def 1;

then A1: ex g being Function st

( (curry f) . v = g & dom g = the carrier of W & rng g c= rng f & ( for y being object st y in the carrier of W holds

g . y = f . (v,y) ) ) by FUNCT_5:29;

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

let y be Vector of W; :: thesis: (FunctionalFAF (f,v)) . y = f . (v,y)

thus (FunctionalFAF (f,v)) . y = f . (v,y) by A1; :: thesis: verum