let w be Vector of W; :: according to BILINEAR:def 15 :: thesis: FunctionalSAF (- f),w is homogeneous
set Ffg = FunctionalSAF (- f),w;
set Ff = FunctionalSAF f,w;
let v be Vector of V; :: according to HAHNBAN1:def 12 :: thesis: for b₁ being Element of the carrier of K holds (FunctionalSAF (- f),w) . (b₁ * v) = b₁ * ((FunctionalSAF (- f),w) . v)
let a be Scalar of ; :: thesis: (FunctionalSAF (- f),w) . (a * v) = a * ((FunctionalSAF (- f),w) . v)
thus (FunctionalSAF (- f),w) . (a * v) = (- (FunctionalSAF f,w)) . (a * v) by Th17
.= - ((FunctionalSAF f,w) . (a * v)) by HAHNBAN1:def 7
.= - (a * ((FunctionalSAF f,w) . v)) by HAHNBAN1:def 12
.= a * (- ((FunctionalSAF f,w) . v)) by VECTSP_1:40
.= a * ((- (FunctionalSAF f,w)) . v) by HAHNBAN1:def 7
.= a * ((FunctionalSAF (- f),w) . v) by Th17 ; :: thesis: verum