let w be Vector of W; :: according to ZMODLAT1:def 28 :: thesis: FrFunctionalSAF ((a * f),w) is additive
set Ffg = FrFunctionalSAF ((a * f),w);
set Ff = FrFunctionalSAF (f,w);
let v, y be Vector of V; :: according to VECTSP_1:def 19 :: thesis: (FrFunctionalSAF ((a * f),w)) . (v + y) = ((FrFunctionalSAF ((a * f),w)) . v) + ((FrFunctionalSAF ((a * f),w)) . y)
A1: FrFunctionalSAF (f,w) is additive ;
thus (FrFunctionalSAF ((a * f),w)) . (v + y) = (a * (FrFunctionalSAF (f,w))) . (v + y) by HTh14
.= a * ((FrFunctionalSAF (f,w)) . (v + y)) by HDef6
.= a * (((FrFunctionalSAF (f,w)) . v) + ((FrFunctionalSAF (f,w)) . y)) by A1
.= (a * ((FrFunctionalSAF (f,w)) . v)) + (a * ((FrFunctionalSAF (f,w)) . y))
.= ((a * (FrFunctionalSAF (f,w))) . v) + (a * ((FrFunctionalSAF (f,w)) . y)) by HDef6
.= ((a * (FrFunctionalSAF (f,w))) . v) + ((a * (FrFunctionalSAF (f,w))) . y) by HDef6
.= ((FrFunctionalSAF ((a * f),w)) . v) + ((a * (FrFunctionalSAF (f,w))) . y) by HTh14
.= ((FrFunctionalSAF ((a * f),w)) . v) + ((FrFunctionalSAF ((a * f),w)) . y) by HTh14 ; :: thesis: verum