let K be non empty add-associative addLoopStr ; :: thesis: for V, W being non empty ModuleStr over K
for f, g, h being Form of V,W holds (f + g) + h = f + (g + h)
let V, W be non empty ModuleStr over K; :: thesis: for f, g, h being Form of V,W holds (f + g) + h = f + (g + h)
let f, g, h be Form of V,W; :: thesis: (f + g) + h = f + (g + h)

now :: thesis: for v being Vector of V
for w being Vector of W holds ((f + g) + h) . (v,w) = (f + (g + h)) . (v,w)

let v be Vector of V; :: thesis: for w being Vector of W holds ((f + g) + h) . (v,w) = (f + (g + h)) . (v,w)
let w be Vector of W; :: thesis: ((f + g) + h) . (v,w) = (f + (g + h)) . (v,w)
thus ((f + g) + h) . (v,w) = ((f + g) . (v,w)) + (h . (v,w)) by Def2
.= ((f . (v,w)) + (g . (v,w))) + (h . (v,w)) by Def2
.= (f . (v,w)) + ((g . (v,w)) + (h . (v,w))) by RLVECT_1:def 3
.= (f . (v,w)) + ((g + h) . (v,w)) by Def2
.= (f + (g + h)) . (v,w) by Def2 ; :: thesis: verum

end;

hence (f + g) + h = f + (g + h) ; :: thesis: verum