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 v being Vector of V holds FrFunctionalFAF ((a * f),v) = a * (FrFunctionalFAF (f,v))
let f be FrForm of V,W; :: thesis: for a being Element of F_Real
for v being Vector of V holds FrFunctionalFAF ((a * f),v) = a * (FrFunctionalFAF (f,v))
let a be Element of F_Real; :: thesis: for v being Vector of V holds FrFunctionalFAF ((a * f),v) = a * (FrFunctionalFAF (f,v))
let w be Vector of V; :: thesis: FrFunctionalFAF ((a * f),w) = a * (FrFunctionalFAF (f,w))

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

let v be Vector of W; :: thesis: (FrFunctionalFAF ((a * f),w)) . v = (a * (FrFunctionalFAF (f,w))) . v
thus (FrFunctionalFAF ((a * f),w)) . v = (a * f) . (w,v) by HTh8
.= a * (f . (w,v)) by Def3
.= a * ((FrFunctionalFAF (f,w)) . v) by HTh8
.= (a * (FrFunctionalFAF (f,w))) . v by HDef6 ; :: thesis: verum

end;

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