let V, W be non empty VectSpStr of 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 *')

now

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 4
.= (a *') * ((f . (v,w)) *') by COMPLFLD:90
.= (a *') * ((f *') . (v,w)) by Def8
.= ((a *') * (f *')) . (v,w) by BILINEAR:def 4 ; :: thesis: verum

end;

hence (a * f) *' = (a *') * (f *') by BINOP_1:2; :: thesis: verum