let V, W be non empty ModuleStr over INT.Ring ; :: thesis: for f, g being FrForm of V,W
for v being Vector of V holds FrFunctionalFAF ((f - g),v) = (FrFunctionalFAF (f,v)) - (FrFunctionalFAF (g,v))
let f, g be FrForm of V,W; :: thesis: for v being Vector of V holds FrFunctionalFAF ((f - g),v) = (FrFunctionalFAF (f,v)) - (FrFunctionalFAF (g,v))
let w be Vector of V; :: thesis: FrFunctionalFAF ((f - g),w) = (FrFunctionalFAF (f,w)) - (FrFunctionalFAF (g,w))

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

let v be Vector of W; :: thesis: (FrFunctionalFAF ((f - g),w)) . v = ((FrFunctionalFAF (f,w)) - (FrFunctionalFAF (g,w))) . v
thus (FrFunctionalFAF ((f - g),w)) . v = (f - g) . (w,v) by HTh8
.= (f . (w,v)) - (g . (w,v)) by Def7
.= ((FrFunctionalFAF (f,w)) . v) - (g . (w,v)) by HTh8
.= ((FrFunctionalFAF (f,w)) . v) - ((FrFunctionalFAF (g,w)) . v) by HTh8
.= ((FrFunctionalFAF (f,w)) . v) + ((- (FrFunctionalFAF (g,w))) . v) by HDef4
.= ((FrFunctionalFAF (f,w)) - (FrFunctionalFAF (g,w))) . v by HDef3 ; :: thesis: verum

end;

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