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))

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

let 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;

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