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

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 Def3
.= ((f . v,w) + (g . v,w)) + (h . v,w) by Def3
.= (f . v,w) + ((g . v,w) + (h . v,w)) by RLVECT_1:def 6
.= (f . v,w) + ((g + h) . v,w) by Def3
.= (f + (g + h)) . v,w by Def3 ; :: thesis: verum

end;

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