let K be non empty 1-sorted ; :: thesis: for V, W being non empty VectSpStr of 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 VectSpStr of 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:] & rng f c= the carrier of K ) by FUNCT_2:def 1;
then consider g being Function such that
A1: ( (curry f) . v = g & dom g = the carrier of W & rng g c= rng f ) and
A2: for y being set st y in the carrier of W holds
g . y = f . v,y by FUNCT_5:36;
thus ( dom (FunctionalFAF f,v) = the carrier of W & rng (FunctionalFAF f,v) c= the carrier of K ) by A1; :: 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, A2; :: thesis: verum