let K be non empty addLoopStr ; :: thesis: for V, W being non empty ModuleStr over 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 ModuleStr over 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))

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 ModuleStr over 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 :: thesis: for v being Vector of V holds (FunctionalSAF ((- f),w)) . v = (- (FunctionalSAF (f,w))) . v

hence
FunctionalSAF ((- f),w) = - (FunctionalSAF (f,w))
by FUNCT_2:63; :: thesis: verumlet v be Vector of V; :: thesis: (FunctionalSAF ((- f),w)) . v = (- (FunctionalSAF (f,w))) . v

thus (FunctionalSAF ((- f),w)) . v = (- f) . (v,w) by Th9

.= - (f . (v,w)) by Def4

.= - ((FunctionalSAF (f,w)) . v) by Th9

.= (- (FunctionalSAF (f,w))) . v by HAHNBAN1:def 4 ; :: thesis: verum

end;thus (FunctionalSAF ((- f),w)) . v = (- f) . (v,w) by Th9

.= - (f . (v,w)) by Def4

.= - ((FunctionalSAF (f,w)) . v) by Th9

.= (- (FunctionalSAF (f,w))) . v by HAHNBAN1:def 4 ; :: thesis: verum