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

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

let v be Vector of V; :: thesis: (FrFunctionalSAF ((- f),w)) . v = (- (FrFunctionalSAF (f,w))) . v
thus (FrFunctionalSAF ((- f),w)) . v = (- f) . (v,w) by HTh9
.= - (f . (v,w)) by Def4
.= - ((FrFunctionalSAF (f,w)) . v) by HTh9
.= (- (FrFunctionalSAF (f,w))) . v by HDef4 ; :: thesis: verum

end;

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