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

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

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

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

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

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

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

now :: thesis: for v being Vector of V

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

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

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

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

thus (f - f) . (v,w) = (f . (v,w)) - (f . (v,w)) by Def7

.= 0. K by RLVECT_1:15

.= (NulForm (V,W)) . (v,w) by FUNCOP_1:70 ; :: thesis: verum

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

thus (f - f) . (v,w) = (f . (v,w)) - (f . (v,w)) by Def7

.= 0. K by RLVECT_1:15

.= (NulForm (V,W)) . (v,w) by FUNCOP_1:70 ; :: thesis: verum