let w be Vector of W; :: according to ZMODLAT1:def 30 :: thesis: FrFunctionalSAF ((f + g),w) is homogeneous
set Ffg = FrFunctionalSAF ((f + g),w);
set Ff = FrFunctionalSAF (f,w);
set Fg = FrFunctionalSAF (g,w);
let v be Vector of V; :: according to ZMODLAT1:def 22 :: thesis: for r being Scalar of holds (FrFunctionalSAF ((f + g),w)) . (r * v) = r * ((FrFunctionalSAF ((f + g),w)) . v)
let a be Scalar of ; :: thesis: (FrFunctionalSAF ((f + g),w)) . (a * v) = a * ((FrFunctionalSAF ((f + g),w)) . v)
thus (FrFunctionalSAF ((f + g),w)) . (a * v) = ((FrFunctionalSAF (f,w)) + (FrFunctionalSAF (g,w))) . (a * v) by HTh12
.= ((FrFunctionalSAF (f,w)) . (a * v)) + ((FrFunctionalSAF (g,w)) . (a * v)) by HDef3
.= (a * ((FrFunctionalSAF (f,w)) . v)) + ((FrFunctionalSAF (g,w)) . (a * v)) by HDef8
.= (a * ((FrFunctionalSAF (f,w)) . v)) + (a * ((FrFunctionalSAF (g,w)) . v)) by HDef8
.= a * (((FrFunctionalSAF (f,w)) . v) + ((FrFunctionalSAF (g,w)) . v))
.= a * (((FrFunctionalSAF (f,w)) + (FrFunctionalSAF (g,w))) . v) by HDef3
.= a * ((FrFunctionalSAF ((f + g),w)) . v) by HTh12 ; :: thesis: verum