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 Def3
.= ((FunctionalSAF f,w) . v) + (g . v,w) by Th10
.= ((FunctionalSAF f,w) . v) + ((FunctionalSAF g,w) . v) by Th10
.= ((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