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

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

let v be Vector of V; :: thesis: (FrFunctionalSAF ((f - g),w)) . v = ((FrFunctionalSAF (f,w)) - (FrFunctionalSAF (g,w))) . v
thus (FrFunctionalSAF ((f - g),w)) . v = (f - g) . (v,w) by HTh9
.= (f . (v,w)) - (g . (v,w)) by Def7
.= ((FrFunctionalSAF (f,w)) . v) - (g . (v,w)) by HTh9
.= ((FrFunctionalSAF (f,w)) . v) - ((FrFunctionalSAF (g,w)) . v) by HTh9
.= ((FrFunctionalSAF (f,w)) . v) + ((- (FrFunctionalSAF (g,w))) . v) by HDef4
.= ((FrFunctionalSAF (f,w)) - (FrFunctionalSAF (g,w))) . v by HDef3 ; :: thesis: verum

end;

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