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))

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

hence
FunctionalSAF ((f + g),w) = (FunctionalSAF (f,w)) + (FunctionalSAF (g,w))
by FUNCT_2:63; :: thesis: verumlet 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 Def2

.= ((FunctionalSAF (f,w)) . v) + (g . (v,w)) by Th9

.= ((FunctionalSAF (f,w)) . v) + ((FunctionalSAF (g,w)) . v) by Th9

.= ((FunctionalSAF (f,w)) + (FunctionalSAF (g,w))) . v by HAHNBAN1:def 3 ; :: thesis: verum

end;thus (FunctionalSAF ((f + g),w)) . v = (f + g) . (v,w) by Th9

.= (f . (v,w)) + (g . (v,w)) by Def2

.= ((FunctionalSAF (f,w)) . v) + (g . (v,w)) by Th9

.= ((FunctionalSAF (f,w)) . v) + ((FunctionalSAF (g,w)) . v) by Th9

.= ((FunctionalSAF (f,w)) + (FunctionalSAF (g,w))) . v by HAHNBAN1:def 3 ; :: thesis: verum