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

for f being Form of V,W

for v being Vector of V holds FunctionalFAF ((- f),v) = - (FunctionalFAF (f,v))

let V, W be non empty ModuleStr over K; :: thesis: for f being Form of V,W

for v being Vector of V holds FunctionalFAF ((- f),v) = - (FunctionalFAF (f,v))

let f be Form of V,W; :: thesis: for v being Vector of V holds FunctionalFAF ((- f),v) = - (FunctionalFAF (f,v))

let w be Vector of V; :: thesis: FunctionalFAF ((- f),w) = - (FunctionalFAF (f,w))

for f being Form of V,W

for v being Vector of V holds FunctionalFAF ((- f),v) = - (FunctionalFAF (f,v))

let V, W be non empty ModuleStr over K; :: thesis: for f being Form of V,W

for v being Vector of V holds FunctionalFAF ((- f),v) = - (FunctionalFAF (f,v))

let f be Form of V,W; :: thesis: for v being Vector of V holds FunctionalFAF ((- f),v) = - (FunctionalFAF (f,v))

let w be Vector of V; :: thesis: FunctionalFAF ((- f),w) = - (FunctionalFAF (f,w))

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

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

thus (FunctionalFAF ((- f),w)) . v = (- f) . (w,v) by Th8

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

.= - ((FunctionalFAF (f,w)) . v) by Th8

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

end;thus (FunctionalFAF ((- f),w)) . v = (- f) . (w,v) by Th8

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

.= - ((FunctionalFAF (f,w)) . v) by Th8

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