let K be non empty addLoopStr ; :: thesis: for V, W being non empty VectSpStr of K
for f being Form of V,W
for w being Vector of W holds FunctionalSAF (- f),w = - (FunctionalSAF f,w)
let V, W be non empty VectSpStr of K; :: thesis: for f being Form of V,W
for w being Vector of W holds FunctionalSAF (- f),w = - (FunctionalSAF f,w)
let f be Form of V,W; :: thesis: for w being Vector of W holds FunctionalSAF (- f),w = - (FunctionalSAF f,w)
let w be Vector of W; :: thesis: FunctionalSAF (- f),w = - (FunctionalSAF f,w)

now

let v be Vector of V; :: thesis: (FunctionalSAF (- f),w) . v = (- (FunctionalSAF f,w)) . v
thus (FunctionalSAF (- f),w) . v = (- f) . v,w by Th10
.= - (f . v,w) by Def5
.= - ((FunctionalSAF f,w) . v) by Th10
.= (- (FunctionalSAF f,w)) . v by HAHNBAN1:def 7 ; :: thesis: verum

end;

hence FunctionalSAF (- f),w = - (FunctionalSAF f,w) by FUNCT_2:113; :: thesis: verum