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 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 VectSpStr of 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:] & rng f c= the carrier of K ) by FUNCT_2:def 1;
then consider g being Function such that
A1: ( (curry' f) . w = g & dom g = the carrier of V & rng g c= rng f ) and
A2: for y being set st y in the carrier of V holds
g . y = f . y,w by FUNCT_5:39;
thus ( dom (FunctionalSAF f,w) = the carrier of V & rng (FunctionalSAF f,w) c= the carrier of K ) by A1; :: 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, A2; :: thesis: verum