let w be Vector of V; :: according to BILINEAR:def 14 :: thesis: FunctionalFAF (f + g),w is homogeneous
set Ffg = FunctionalFAF (f + g),w;
set Ff = FunctionalFAF f,w;
set Fg = FunctionalFAF g,w;
let v be Vector of W; :: according to HAHNBAN1:def 12 :: thesis: for b₁ being Element of the carrier of K holds (FunctionalFAF (f + g),w) . (b₁ * v) = b₁ * ((FunctionalFAF (f + g),w) . v)
let a be Scalar of ; :: thesis: (FunctionalFAF (f + g),w) . (a * v) = a * ((FunctionalFAF (f + g),w) . v)
thus (FunctionalFAF (f + g),w) . (a * v) = ((FunctionalFAF f,w) + (FunctionalFAF g,w)) . (a * v) by Th14
.= ((FunctionalFAF f,w) . (a * v)) + ((FunctionalFAF g,w) . (a * v)) by HAHNBAN1:def 6
.= (a * ((FunctionalFAF f,w) . v)) + ((FunctionalFAF g,w) . (a * v)) by HAHNBAN1:def 12
.= (a * ((FunctionalFAF f,w) . v)) + (a * ((FunctionalFAF g,w) . v)) by HAHNBAN1:def 12
.= a * (((FunctionalFAF f,w) . v) + ((FunctionalFAF g,w) . v)) by VECTSP_1:def 11
.= a * (((FunctionalFAF f,w) + (FunctionalFAF g,w)) . v) by HAHNBAN1:def 6
.= a * ((FunctionalFAF (f + g),w) . v) by Th14 ; :: thesis: verum