let v be Vector of V; :: according to HERMITAN:def 4 :: thesis: FunctionalFAF (FormFunctional f,g),v is cmplxhomogeneous
set fg = FormFunctional f,g;
set F = FunctionalFAF (FormFunctional f,g),v;
let y be Vector of W; :: according to HERMITAN:def 1 :: thesis: for a being Scalar of holds (FunctionalFAF (FormFunctional f,g),v) . (a * y) = (a *' ) * ((FunctionalFAF (FormFunctional f,g),v) . y)
let r be Scalar of ; :: thesis: (FunctionalFAF (FormFunctional f,g),v) . (r * y) = (r *' ) * ((FunctionalFAF (FormFunctional f,g),v) . y)
A1: FunctionalFAF (FormFunctional f,g),v = (f . v) * g by BILINEAR:25;
hence (FunctionalFAF (FormFunctional f,g),v) . (r * y) = (f . v) * (g . (r * y)) by HAHNBAN1:def 9
.= (f . v) * ((r *' ) * (g . y)) by Def1
.= (r *' ) * ((f . v) * (g . y))
.= (r *' ) * ((FunctionalFAF (FormFunctional f,g),v) . y) by A1, HAHNBAN1:def 9 ;
:: thesis: verum