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 w being Vector of W holds FunctionalSAF (f - g),w = (FunctionalSAF f,w) - (FunctionalSAF g,w)
let V, W be non empty VectSpStr of 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

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 Th10
.= (f . v,w) - (g . v,w) by Def8
.= ((FunctionalSAF f,w) . v) - (g . v,w) by Th10
.= ((FunctionalSAF f,w) . v) - ((FunctionalSAF g,w) . v) by Th10
.= ((FunctionalSAF f,w) . v) + (- ((FunctionalSAF g,w) . v)) by RLVECT_1:def 14
.= ((FunctionalSAF f,w) . v) + ((- (FunctionalSAF g,w)) . v) by HAHNBAN1:def 7
.= ((FunctionalSAF f,w) - (FunctionalSAF g,w)) . v by HAHNBAN1:def 6 ; :: thesis: verum

end;

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