let K be non empty doubleLoopStr ; :: thesis: for V, W being non empty ModuleStr over K

for f being Form of V,W

for a being Element of K

for w being Vector of W holds FunctionalSAF ((a * f),w) = a * (FunctionalSAF (f,w))

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

for a being Element of K

for w being Vector of W holds FunctionalSAF ((a * f),w) = a * (FunctionalSAF (f,w))

let f be Form of V,W; :: thesis: for a being Element of K

for w being Vector of W holds FunctionalSAF ((a * f),w) = a * (FunctionalSAF (f,w))

let a be Element of K; :: thesis: for w being Vector of W holds FunctionalSAF ((a * f),w) = a * (FunctionalSAF (f,w))

let w be Vector of W; :: thesis: FunctionalSAF ((a * f),w) = a * (FunctionalSAF (f,w))

for f being Form of V,W

for a being Element of K

for w being Vector of W holds FunctionalSAF ((a * f),w) = a * (FunctionalSAF (f,w))

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

for a being Element of K

for w being Vector of W holds FunctionalSAF ((a * f),w) = a * (FunctionalSAF (f,w))

let f be Form of V,W; :: thesis: for a being Element of K

for w being Vector of W holds FunctionalSAF ((a * f),w) = a * (FunctionalSAF (f,w))

let a be Element of K; :: thesis: for w being Vector of W holds FunctionalSAF ((a * f),w) = a * (FunctionalSAF (f,w))

let w be Vector of W; :: thesis: FunctionalSAF ((a * f),w) = a * (FunctionalSAF (f,w))

now :: thesis: for v being Vector of V holds (FunctionalSAF ((a * f),w)) . v = (a * (FunctionalSAF (f,w))) . v

hence
FunctionalSAF ((a * f),w) = a * (FunctionalSAF (f,w))
by FUNCT_2:63; :: thesis: verumlet v be Vector of V; :: thesis: (FunctionalSAF ((a * f),w)) . v = (a * (FunctionalSAF (f,w))) . v

thus (FunctionalSAF ((a * f),w)) . v = (a * f) . (v,w) by Th9

.= a * (f . (v,w)) by Def3

.= a * ((FunctionalSAF (f,w)) . v) by Th9

.= (a * (FunctionalSAF (f,w))) . v by HAHNBAN1:def 6 ; :: thesis: verum

end;thus (FunctionalSAF ((a * f),w)) . v = (a * f) . (v,w) by Th9

.= a * (f . (v,w)) by Def3

.= a * ((FunctionalSAF (f,w)) . v) by Th9

.= (a * (FunctionalSAF (f,w))) . v by HAHNBAN1:def 6 ; :: thesis: verum