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

now

let v be Vector of W; :: thesis: (FunctionalFAF ((f - g),w)) . v = ((FunctionalFAF (f,w)) - (FunctionalFAF (g,w))) . v
thus (FunctionalFAF ((f - g),w)) . v = (f - g) . (w,v) by Th9
.= (f . (w,v)) - (g . (w,v)) by Def8
.= ((FunctionalFAF (f,w)) . v) - (g . (w,v)) by Th9
.= ((FunctionalFAF (f,w)) . v) - ((FunctionalFAF (g,w)) . v) by Th9
.= ((FunctionalFAF (f,w)) . v) + (- ((FunctionalFAF (g,w)) . v)) by RLVECT_1:def 14
.= ((FunctionalFAF (f,w)) . v) + ((- (FunctionalFAF (g,w))) . v) by HAHNBAN1:def 7
.= ((FunctionalFAF (f,w)) - (FunctionalFAF (g,w))) . v by HAHNBAN1:def 6 ; :: thesis: verum

end;

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