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

now

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 Def3
.= (g + f) . v,w by Def3 ; :: thesis: verum

end;

hence f + g = g + f by BINOP_1:2; :: thesis: verum