let V, W be non empty ModuleStr over INT.Ring ; :: thesis: for f being FrForm of V,W
for a being Element of F_Real
for w being Vector of W holds FrFunctionalSAF ((a * f),w) = a * (FrFunctionalSAF (f,w))
let f be FrForm of V,W; :: thesis: for a being Element of F_Real
for w being Vector of W holds FrFunctionalSAF ((a * f),w) = a * (FrFunctionalSAF (f,w))
let a be Element of F_Real; :: thesis: for w being Vector of W holds FrFunctionalSAF ((a * f),w) = a * (FrFunctionalSAF (f,w))
let w be Vector of W; :: thesis: FrFunctionalSAF ((a * f),w) = a * (FrFunctionalSAF (f,w))

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

let v be Vector of V; :: thesis: (FrFunctionalSAF ((a * f),w)) . v = (a * (FrFunctionalSAF (f,w))) . v
thus (FrFunctionalSAF ((a * f),w)) . v = (a * f) . (v,w) by HTh9
.= a * (f . (v,w)) by Def3
.= a * ((FrFunctionalSAF (f,w)) . v) by HTh9
.= (a * (FrFunctionalSAF (f,w))) . v by HDef6 ; :: thesis: verum

end;

hence FrFunctionalSAF ((a * f),w) = a * (FrFunctionalSAF (f,w)) by FUNCT_2:63; :: thesis: verum