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 12
.= ((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