let f, g be Form of V,W; :: thesis: f + g = g + f

now :: thesis: for v being Vector of V

for w being Vector of W holds (f + g) . (v,w) = (g + f) . (v,w)

hence
f + g = g + f
; :: thesis: verumfor w being Vector of W holds (f + g) . (v,w) = (g + f) . (v,w)

let v be Vector of V; :: thesis: for w being Vector of W holds (f + g) . (v,w) = (g + f) . (v,w)

let w be Vector of W; :: thesis: (f + g) . (v,w) = (g + f) . (v,w)

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

.= (g + f) . (v,w) by Def2 ; :: thesis: verum

end;let w be Vector of W; :: thesis: (f + g) . (v,w) = (g + f) . (v,w)

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

.= (g + f) . (v,w) by Def2 ; :: thesis: verum