let w be Vector of W; :: according to BILINEAR:def 14 :: 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 8 :: thesis: for b_{1} being Element of the carrier of K holds (FunctionalSAF ((- f),w)) . (b_{1} * v) = b_{1} * ((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 Th16

.= - ((FunctionalSAF (f,w)) . (a * v)) by HAHNBAN1:def 4

.= - (a * ((FunctionalSAF (f,w)) . v)) by HAHNBAN1:def 8

.= a * (- ((FunctionalSAF (f,w)) . v)) by VECTSP_1:8

.= a * ((- (FunctionalSAF (f,w))) . v) by HAHNBAN1:def 4

.= a * ((FunctionalSAF ((- f),w)) . v) by Th16 ; :: thesis: verum

set Ffg = FunctionalSAF ((- f),w);

set Ff = FunctionalSAF (f,w);

let v be Vector of V; :: according to HAHNBAN1:def 8 :: thesis: for b

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 Th16

.= - ((FunctionalSAF (f,w)) . (a * v)) by HAHNBAN1:def 4

.= - (a * ((FunctionalSAF (f,w)) . v)) by HAHNBAN1:def 8

.= a * (- ((FunctionalSAF (f,w)) . v)) by VECTSP_1:8

.= a * ((- (FunctionalSAF (f,w))) . v) by HAHNBAN1:def 4

.= a * ((FunctionalSAF ((- f),w)) . v) by Th16 ; :: thesis: verum