let K be non empty associative 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 * (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 * (b * f)

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

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

for a, b being Element of K

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

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

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

now :: thesis: for v being Vector of V

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

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

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

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

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

.= r * (s * (f . (v,w))) by GROUP_1:def 3

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

.= (r * (s * f)) . (v,w) by Def3 ; :: thesis: verum

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

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

.= r * (s * (f . (v,w))) by GROUP_1:def 3

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

.= (r * (s * f)) . (v,w) by Def3 ; :: thesis: verum