let V, W be non empty ModuleStr over F_Complex ; :: thesis: for f being Form of V,W

for a being Element of F_Complex holds (a * f) *' = (a *') * (f *')

let f be Form of V,W; :: thesis: for a being Element of F_Complex holds (a * f) *' = (a *') * (f *')

let a be Element of F_Complex; :: thesis: (a * f) *' = (a *') * (f *')

for a being Element of F_Complex holds (a * f) *' = (a *') * (f *')

let f be Form of V,W; :: thesis: for a being Element of F_Complex holds (a * f) *' = (a *') * (f *')

let a be Element of F_Complex; :: thesis: (a * f) *' = (a *') * (f *')

now :: thesis: for v being Vector of V

for w being Vector of W holds ((a * f) *') . (v,w) = ((a *') * (f *')) . (v,w)

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

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

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

thus ((a * f) *') . (v,w) = ((a * f) . (v,w)) *' by Def8

.= (a * (f . (v,w))) *' by BILINEAR:def 3

.= (a *') * ((f . (v,w)) *') by COMPLFLD:54

.= (a *') * ((f *') . (v,w)) by Def8

.= ((a *') * (f *')) . (v,w) by BILINEAR:def 3 ; :: thesis: verum

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

thus ((a * f) *') . (v,w) = ((a * f) . (v,w)) *' by Def8

.= (a * (f . (v,w))) *' by BILINEAR:def 3

.= (a *') * ((f . (v,w)) *') by COMPLFLD:54

.= (a *') * ((f *') . (v,w)) by Def8

.= ((a *') * (f *')) . (v,w) by BILINEAR:def 3 ; :: thesis: verum