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