let w be Vector of V; :: according to ZMODLAT1:def 27 :: thesis: FrFunctionalFAF ((f + g),w) is additive
set Ffg = FrFunctionalFAF ((f + g),w);
set Ff = FrFunctionalFAF (f,w);
set Fg = FrFunctionalFAF (g,w);
let v, y be Vector of W; :: according to VECTSP_1:def 19 :: thesis: (FrFunctionalFAF ((f + g),w)) . (v + y) = ((FrFunctionalFAF ((f + g),w)) . v) + ((FrFunctionalFAF ((f + g),w)) . y)
A1: FrFunctionalFAF (f,w) is additive ;
A2: FrFunctionalFAF (g,w) is additive ;
thus (FrFunctionalFAF ((f + g),w)) . (v + y) = ((FrFunctionalFAF (f,w)) + (FrFunctionalFAF (g,w))) . (v + y) by HTh13
.= ((FrFunctionalFAF (f,w)) . (v + y)) + ((FrFunctionalFAF (g,w)) . (v + y)) by HDef3
.= (((FrFunctionalFAF (f,w)) . v) + ((FrFunctionalFAF (f,w)) . y)) + ((FrFunctionalFAF (g,w)) . (v + y)) by A1
.= (((FrFunctionalFAF (f,w)) . v) + ((FrFunctionalFAF (f,w)) . y)) + (((FrFunctionalFAF (g,w)) . v) + ((FrFunctionalFAF (g,w)) . y)) by A2
.= ((((FrFunctionalFAF (f,w)) . v) + ((FrFunctionalFAF (g,w)) . v)) + ((FrFunctionalFAF (f,w)) . y)) + ((FrFunctionalFAF (g,w)) . y)
.= ((((FrFunctionalFAF (f,w)) + (FrFunctionalFAF (g,w))) . v) + ((FrFunctionalFAF (f,w)) . y)) + ((FrFunctionalFAF (g,w)) . y) by HDef3
.= (((FrFunctionalFAF (f,w)) + (FrFunctionalFAF (g,w))) . v) + (((FrFunctionalFAF (f,w)) . y) + ((FrFunctionalFAF (g,w)) . y))
.= (((FrFunctionalFAF (f,w)) + (FrFunctionalFAF (g,w))) . v) + (((FrFunctionalFAF (f,w)) + (FrFunctionalFAF (g,w))) . y) by HDef3
.= ((FrFunctionalFAF ((f + g),w)) . v) + (((FrFunctionalFAF (f,w)) + (FrFunctionalFAF (g,w))) . y) by HTh13
.= ((FrFunctionalFAF ((f + g),w)) . v) + ((FrFunctionalFAF ((f + g),w)) . y) by HTh13 ; :: thesis: verum