let w be Vector of W; :: according to BILINEAR:def 12 :: thesis: FunctionalSAF ((- f),w) is additive
set Ffg = FunctionalSAF ((- f),w);
set Ff = FunctionalSAF (f,w);
let v, y be Vector of V; :: according to VECTSP_1:def 19 :: thesis: (FunctionalSAF ((- f),w)) . (v + y) = ((FunctionalSAF ((- f),w)) . v) + ((FunctionalSAF ((- f),w)) . y)
A1: FunctionalSAF (f,w) is additive by Def12;
thus (FunctionalSAF ((- f),w)) . (v + y) = (- (FunctionalSAF (f,w))) . (v + y) by Th16
.= - ((FunctionalSAF (f,w)) . (v + y)) by HAHNBAN1:def 4
.= - (((FunctionalSAF (f,w)) . v) + ((FunctionalSAF (f,w)) . y)) by A1
.= (- ((FunctionalSAF (f,w)) . v)) - ((FunctionalSAF (f,w)) . y) by RLVECT_1:30
.= ((- (FunctionalSAF (f,w))) . v) - ((FunctionalSAF (f,w)) . y) by HAHNBAN1:def 4
.= ((- (FunctionalSAF (f,w))) . v) + (- ((FunctionalSAF (f,w)) . y)) by RLVECT_1:def 11
.= ((- (FunctionalSAF (f,w))) . v) + ((- (FunctionalSAF (f,w))) . y) by HAHNBAN1:def 4
.= ((FunctionalSAF ((- f),w)) . v) + ((- (FunctionalSAF (f,w))) . y) by Th16
.= ((FunctionalSAF ((- f),w)) . v) + ((FunctionalSAF ((- f),w)) . y) by Th16 ; :: thesis: verum