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:24;
hence (FunctionalFAF ((FormFunctional (f,g)),v)) . (r * y) = (f . v) * (g . (r * y)) by HAHNBAN1:def 6
.= (f . v) * ((r *') * (g . y)) by Def1
.= (r *') * ((f . v) * (g . y))
.= (r *') * ((FunctionalFAF ((FormFunctional (f,g)),v)) . y) by A1, HAHNBAN1:def 6 ;
:: thesis: verum