let K be non empty right_zeroed addLoopStr ; :: thesis: for V, W being non empty ModuleStr over K

for f being Form of V,W holds f + (NulForm (V,W)) = f

let V, W be non empty ModuleStr over K; :: thesis: for f being Form of V,W holds f + (NulForm (V,W)) = f

let f be Form of V,W; :: thesis: f + (NulForm (V,W)) = f

set g = NulForm (V,W);

for f being Form of V,W holds f + (NulForm (V,W)) = f

let V, W be non empty ModuleStr over K; :: thesis: for f being Form of V,W holds f + (NulForm (V,W)) = f

let f be Form of V,W; :: thesis: f + (NulForm (V,W)) = f

set g = NulForm (V,W);

now :: thesis: for v being Vector of V

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

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

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

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

thus (f + (NulForm (V,W))) . (v,w) = (f . (v,w)) + ((NulForm (V,W)) . (v,w)) by Def2

.= (f . (v,w)) + (0. K) by FUNCOP_1:70

.= f . (v,w) by RLVECT_1:def 4 ; :: thesis: verum

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

thus (f + (NulForm (V,W))) . (v,w) = (f . (v,w)) + ((NulForm (V,W)) . (v,w)) by Def2

.= (f . (v,w)) + (0. K) by FUNCOP_1:70

.= f . (v,w) by RLVECT_1:def 4 ; :: thesis: verum