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)

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 BLDef2
.= (g + f) . (v,w) by BLDef2 ; :: thesis: verum

end;

hence f + g = g + f ; :: thesis: verum