let v be Vector of V; :: according to BILINEAR:def 13 :: thesis: FunctionalFAF ((f *'),v) is homogeneous

let w be Vector of W; :: according to HAHNBAN1:def 8 :: thesis: for b_{1} being Element of the carrier of F_Complex holds (FunctionalFAF ((f *'),v)) . (b_{1} * w) = b_{1} * ((FunctionalFAF ((f *'),v)) . w)

let r be Scalar of ; :: thesis: (FunctionalFAF ((f *'),v)) . (r * w) = r * ((FunctionalFAF ((f *'),v)) . w)

set fg = FunctionalFAF ((f *'),v);

thus (FunctionalFAF ((f *'),v)) . (r * w) = (f *') . (v,(r * w)) by BILINEAR:8

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

.= ((r *') * (f . (v,w))) *' by Th27

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

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

.= r * ((FunctionalFAF ((f *'),v)) . w) by BILINEAR:8 ; :: thesis: verum

let w be Vector of W; :: according to HAHNBAN1:def 8 :: thesis: for b

let r be Scalar of ; :: thesis: (FunctionalFAF ((f *'),v)) . (r * w) = r * ((FunctionalFAF ((f *'),v)) . w)

set fg = FunctionalFAF ((f *'),v);

thus (FunctionalFAF ((f *'),v)) . (r * w) = (f *') . (v,(r * w)) by BILINEAR:8

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

.= ((r *') * (f . (v,w))) *' by Th27

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

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

.= r * ((FunctionalFAF ((f *'),v)) . w) by BILINEAR:8 ; :: thesis: verum