let V, W be non empty ModuleStr over INT.Ring ; :: thesis: for f being FrForm of V,W
for v being Vector of V holds FrFunctionalFAF ((- f),v) = - (FrFunctionalFAF (f,v))
let f be FrForm of V,W; :: thesis: for v being Vector of V holds FrFunctionalFAF ((- f),v) = - (FrFunctionalFAF (f,v))
let w be Vector of V; :: thesis: FrFunctionalFAF ((- f),w) = - (FrFunctionalFAF (f,w))

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

let v be Vector of W; :: thesis: (FrFunctionalFAF ((- f),w)) . v = (- (FrFunctionalFAF (f,w))) . v
thus (FrFunctionalFAF ((- f),w)) . v = (- f) . (w,v) by HTh8
.= - (f . (w,v)) by Def4
.= - ((FrFunctionalFAF (f,w)) . v) by HTh8
.= (- (FrFunctionalFAF (f,w))) . v by HDef4 ; :: thesis: verum

end;

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