let w be Vector of W; :: according to BILINEAR:def 13 :: thesis: FunctionalSAF ((a * f),w) is additive
set Ffg = FunctionalSAF ((a * f),w);
set Ff = FunctionalSAF (f,w);
let v, y be Vector of V; :: according to GRCAT_1:def 13 :: thesis: (FunctionalSAF ((a * f),w)) . (v + y) = ((FunctionalSAF ((a * f),w)) . v) + ((FunctionalSAF ((a * f),w)) . y)
A1: FunctionalSAF (f,w) is additive by Def13;
thus (FunctionalSAF ((a * f),w)) . (v + y) = (a * (FunctionalSAF (f,w))) . (v + y) by Th15
.= a * ((FunctionalSAF (f,w)) . (v + y)) by HAHNBAN1:def 9
.= a * (((FunctionalSAF (f,w)) . v) + ((FunctionalSAF (f,w)) . y)) by A1, GRCAT_1:def 13
.= (a * ((FunctionalSAF (f,w)) . v)) + (a * ((FunctionalSAF (f,w)) . y)) by VECTSP_1:def 11
.= ((a * (FunctionalSAF (f,w))) . v) + (a * ((FunctionalSAF (f,w)) . y)) by HAHNBAN1:def 9
.= ((a * (FunctionalSAF (f,w))) . v) + ((a * (FunctionalSAF (f,w))) . y) by HAHNBAN1:def 9
.= ((FunctionalSAF ((a * f),w)) . v) + ((a * (FunctionalSAF (f,w))) . y) by Th15
.= ((FunctionalSAF ((a * f),w)) . v) + ((FunctionalSAF ((a * f),w)) . y) by Th15 ; :: thesis: verum