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

set fg = FormFunctional (f,g);

set F = FunctionalFAF ((FormFunctional (f,g)),v);

let y be Vector of W; :: according to HAHNBAN1:def 8 :: thesis: for b_{1} being Element of the carrier of K holds (FunctionalFAF ((FormFunctional (f,g)),v)) . (b_{1} * y) = b_{1} * ((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 Th24;

hence (FunctionalFAF ((FormFunctional (f,g)),v)) . (r * y) = (f . v) * (g . (r * y)) by HAHNBAN1:def 6

.= (f . v) * (r * (g . y)) by HAHNBAN1:def 8

.= r * ((f . v) * (g . y)) by GROUP_1:def 3

.= r * ((FunctionalFAF ((FormFunctional (f,g)),v)) . y) by A1, HAHNBAN1:def 6 ;

:: thesis: verum

set fg = FormFunctional (f,g);

set F = FunctionalFAF ((FormFunctional (f,g)),v);

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

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 Th24;

hence (FunctionalFAF ((FormFunctional (f,g)),v)) . (r * y) = (f . v) * (g . (r * y)) by HAHNBAN1:def 6

.= (f . v) * (r * (g . y)) by HAHNBAN1:def 8

.= r * ((f . v) * (g . y)) by GROUP_1:def 3

.= r * ((FunctionalFAF ((FormFunctional (f,g)),v)) . y) by A1, HAHNBAN1:def 6 ;

:: thesis: verum