let K be non empty left-distributive doubleLoopStr ; :: thesis: for V, W being non empty ModuleStr over K

for a, b being Element of K

for f being Form of V,W holds (a + b) * f = (a * f) + (b * f)

let V, W be non empty ModuleStr over K; :: thesis: for a, b being Element of K

for f being Form of V,W holds (a + b) * f = (a * f) + (b * f)

let r, s be Element of K; :: thesis: for f being Form of V,W holds (r + s) * f = (r * f) + (s * f)

let f be Form of V,W; :: thesis: (r + s) * f = (r * f) + (s * f)

for a, b being Element of K

for f being Form of V,W holds (a + b) * f = (a * f) + (b * f)

let V, W be non empty ModuleStr over K; :: thesis: for a, b being Element of K

for f being Form of V,W holds (a + b) * f = (a * f) + (b * f)

let r, s be Element of K; :: thesis: for f being Form of V,W holds (r + s) * f = (r * f) + (s * f)

let f be Form of V,W; :: thesis: (r + s) * f = (r * f) + (s * f)

now :: thesis: for v being Vector of V

for w being Vector of W holds ((r + s) * f) . (v,w) = ((r * f) + (s * f)) . (v,w)

hence
(r + s) * f = (r * f) + (s * f)
; :: thesis: verumfor w being Vector of W holds ((r + s) * f) . (v,w) = ((r * f) + (s * f)) . (v,w)

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

let w be Vector of W; :: thesis: ((r + s) * f) . (v,w) = ((r * f) + (s * f)) . (v,w)

thus ((r + s) * f) . (v,w) = (r + s) * (f . (v,w)) by Def3

.= (r * (f . (v,w))) + (s * (f . (v,w))) by VECTSP_1:def 3

.= ((r * f) . (v,w)) + (s * (f . (v,w))) by Def3

.= ((r * f) . (v,w)) + ((s * f) . (v,w)) by Def3

.= ((r * f) + (s * f)) . (v,w) by Def2 ; :: thesis: verum

end;let w be Vector of W; :: thesis: ((r + s) * f) . (v,w) = ((r * f) + (s * f)) . (v,w)

thus ((r + s) * f) . (v,w) = (r + s) * (f . (v,w)) by Def3

.= (r * (f . (v,w))) + (s * (f . (v,w))) by VECTSP_1:def 3

.= ((r * f) . (v,w)) + (s * (f . (v,w))) by Def3

.= ((r * f) . (v,w)) + ((s * f) . (v,w)) by Def3

.= ((r * f) + (s * f)) . (v,w) by Def2 ; :: thesis: verum