let w be Vector of V; :: according to ZMATRLIN:def 24 :: 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 8 :: thesis: for b₁ being Element of the carrier of INT.Ring holds (FunctionalFAF ((f + g),w)) . (b₁ * v) = b₁ * ((FunctionalFAF ((f + g),w)) . v)
let a be Element of INT.Ring; :: 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 BLTh13
.= ((FunctionalFAF (f,w)) . (a * v)) + ((FunctionalFAF (g,w)) . (a * v)) by HAHNBAN1:def 3
.= (a * ((FunctionalFAF (f,w)) . v)) + ((FunctionalFAF (g,w)) . (a * v)) by HAHNBAN1:def 8
.= (a * ((FunctionalFAF (f,w)) . v)) + (a * ((FunctionalFAF (g,w)) . v)) by HAHNBAN1:def 8
.= a * (((FunctionalFAF (f,w)) . v) + ((FunctionalFAF (g,w)) . v))
.= a * (((FunctionalFAF (f,w)) + (FunctionalFAF (g,w))) . v) by HAHNBAN1:def 3
.= a * ((FunctionalFAF ((f + g),w)) . v) by BLTh13 ; :: thesis: verum