let w be Vector of V; :: according to ZMODLAT1:def 29 :: thesis: FrFunctionalFAF ((a * f),w) is homogeneous
set Ffg = FrFunctionalFAF ((a * f),w);
set Ff = FrFunctionalFAF (f,w);
let v be Vector of W; :: according to ZMODLAT1:def 22 :: thesis: for r being Scalar of holds (FrFunctionalFAF ((a * f),w)) . (r * v) = r * ((FrFunctionalFAF ((a * f),w)) . v)
let b be Scalar of ; :: thesis: (FrFunctionalFAF ((a * f),w)) . (b * v) = b * ((FrFunctionalFAF ((a * f),w)) . v)
thus (FrFunctionalFAF ((a * f),w)) . (b * v) = (a * (FrFunctionalFAF (f,w))) . (b * v) by HTh15
.= a * ((FrFunctionalFAF (f,w)) . (b * v)) by HDef6
.= a * (b * ((FrFunctionalFAF (f,w)) . v)) by HDef8
.= b * (a * ((FrFunctionalFAF (f,w)) . v))
.= b * ((a * (FrFunctionalFAF (f,w))) . v) by HDef6
.= b * ((FrFunctionalFAF ((a * f),w)) . v) by HTh15 ; :: thesis: verum