let v be Vector of V; :: according to BILINEAR:def 12 :: thesis: FunctionalFAF (FormFunctional f,g),v is additive
set fg = FormFunctional f,g;
set F = FunctionalFAF (FormFunctional f,g),v;
let y, y9 be Vector of W; :: according to GRCAT_1:def 13 :: thesis: (FunctionalFAF (FormFunctional f,g),v) . (y + y9) = ((FunctionalFAF (FormFunctional f,g),v) . y) + ((FunctionalFAF (FormFunctional f,g),v) . y9)
A1: FunctionalFAF (FormFunctional f,g),v = (f . v) * g by Th25;
hence (FunctionalFAF (FormFunctional f,g),v) . (y + y9) = (f . v) * (g . (y + y9)) by HAHNBAN1:def 9
.= (f . v) * ((g . y) + (g . y9)) by GRCAT_1:def 13
.= ((f . v) * (g . y)) + ((f . v) * (g . y9)) by VECTSP_1:def 11
.= ((f . v) * (g . y)) + ((FunctionalFAF (FormFunctional f,g),v) . y9) by A1, HAHNBAN1:def 9
.= ((FunctionalFAF (FormFunctional f,g),v) . y) + ((FunctionalFAF (FormFunctional f,g),v) . y9) by A1, HAHNBAN1:def 9 ;
:: thesis: verum