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

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

let v be Vector of V; :: thesis: (FunctionalSAF ((f - g),w)) . v = ((FunctionalSAF (f,w)) - (FunctionalSAF (g,w))) . v
thus (FunctionalSAF ((f - g),w)) . v = (f - g) . (v,w) by Th9
.= (f . (v,w)) - (g . (v,w)) by Def7
.= ((FunctionalSAF (f,w)) . v) - (g . (v,w)) by Th9
.= ((FunctionalSAF (f,w)) . v) - ((FunctionalSAF (g,w)) . v) by Th9
.= ((FunctionalSAF (f,w)) . v) + (- ((FunctionalSAF (g,w)) . v)) by RLVECT_1:def 11
.= ((FunctionalSAF (f,w)) . v) + ((- (FunctionalSAF (g,w))) . v) by HAHNBAN1:def 4
.= ((FunctionalSAF (f,w)) - (FunctionalSAF (g,w))) . v by HAHNBAN1:def 3 ; :: thesis: verum

end;

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