let w be Vector of W; :: according to ZMATRLIN:def 23 :: 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 VECTSP_1:def 19 :: thesis: (FunctionalSAF ((a * f),w)) . (v + y) = ((FunctionalSAF ((a * f),w)) . v) + ((FunctionalSAF ((a * f),w)) . y)
A1: FunctionalSAF (f,w) is additive ;
thus (FunctionalSAF ((a * f),w)) . (v + y) = (a * (FunctionalSAF (f,w))) . (v + y) by BLTh14
.= a * ((FunctionalSAF (f,w)) . (v + y)) by HAHNBAN1:def 6
.= a * (((FunctionalSAF (f,w)) . v) + ((FunctionalSAF (f,w)) . y)) by A1
.= (a * ((FunctionalSAF (f,w)) . v)) + (a * ((FunctionalSAF (f,w)) . y))
.= ((a * (FunctionalSAF (f,w))) . v) + (a * ((FunctionalSAF (f,w)) . y)) by HAHNBAN1:def 6
.= ((a * (FunctionalSAF (f,w))) . v) + ((a * (FunctionalSAF (f,w))) . y) by HAHNBAN1:def 6
.= ((FunctionalSAF ((a * f),w)) . v) + ((a * (FunctionalSAF (f,w))) . y) by BLTh14
.= ((FunctionalSAF ((a * f),w)) . v) + ((FunctionalSAF ((a * f),w)) . y) by BLTh14 ; :: thesis: verum