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