let K be non empty addLoopStr ; :: thesis: for V, W being non empty VectSpStr of 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 VectSpStr of 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

let v be Vector of W; :: thesis: (FunctionalFAF (- f),w) . v = (- (FunctionalFAF f,w)) . v
thus (FunctionalFAF (- f),w) . v = (- f) . w,v by Th9
.= - (f . w,v) by Def5
.= - ((FunctionalFAF f,w) . v) by Th9
.= (- (FunctionalFAF f,w)) . v by HAHNBAN1:def 7 ; :: thesis: verum

end;

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