let w be Vector of V; :: according to BILINEAR:def 11 :: thesis: FunctionalFAF ((a * f),w) is additive
set Ffg = FunctionalFAF ((a * f),w);
set Ff = FunctionalFAF (f,w);
let v, y be Vector of W; :: according to VECTSP_1:def 19 :: thesis: (FunctionalFAF ((a * f),w)) . (v + y) = ((FunctionalFAF ((a * f),w)) . v) + ((FunctionalFAF ((a * f),w)) . y)
A1: FunctionalFAF (f,w) is additive by Def11;
thus (FunctionalFAF ((a * f),w)) . (v + y) = (a * (FunctionalFAF (f,w))) . (v + y) by Th15
.= a * ((FunctionalFAF (f,w)) . (v + y)) by HAHNBAN1:def 6
.= a * (((FunctionalFAF (f,w)) . v) + ((FunctionalFAF (f,w)) . y)) by A1
.= (a * ((FunctionalFAF (f,w)) . v)) + (a * ((FunctionalFAF (f,w)) . y)) by VECTSP_1:def 2
.= ((a * (FunctionalFAF (f,w))) . v) + (a * ((FunctionalFAF (f,w)) . y)) by HAHNBAN1:def 6
.= ((a * (FunctionalFAF (f,w))) . v) + ((a * (FunctionalFAF (f,w))) . y) by HAHNBAN1:def 6
.= ((FunctionalFAF ((a * f),w)) . v) + ((a * (FunctionalFAF (f,w))) . y) by Th15
.= ((FunctionalFAF ((a * f),w)) . v) + ((FunctionalFAF ((a * f),w)) . y) by Th15 ; :: thesis: verum