let X be set ; :: thesis: for n being Nat
for f being Function of X,(TOP-REAL n)
for g being Function of X,R^1 holds f <+> g is Function of X,(TOP-REAL n)
let n be Nat; :: thesis: for f being Function of X,(TOP-REAL n)
for g being Function of X,R^1 holds f <+> g is Function of X,(TOP-REAL n)
let f be Function of X,(TOP-REAL n); :: thesis: for g being Function of X,R^1 holds f <+> g is Function of X,(TOP-REAL n)
let g be Function of X,R^1; :: thesis: f <+> g is Function of X,(TOP-REAL n)
set h = f <+> g;
A1: dom f = X by FUNCT_2:def 1;
dom g = X by FUNCT_2:def 1;
then A2: dom (f <+> g) = X by A1, VALUED_2:def 41;
for x being object st x in X holds
(f <+> g) . x in the carrier of (TOP-REAL n)

proof

let x be object ; :: thesis: ( x in X implies (f <+> g) . x in the carrier of (TOP-REAL n) )
assume A3: x in X ; :: thesis: (f <+> g) . x in the carrier of (TOP-REAL n)
then reconsider X = X as non empty set ;
reconsider x = x as Element of X by A3;
reconsider f = f as Function of X,(TOP-REAL n) ;
(f <+> g) . x = (f . x) + (g . x) by A2, VALUED_2:def 41;
hence (f <+> g) . x in the carrier of (TOP-REAL n) ; :: thesis: verum

end;

hence f <+> g is Function of X,(TOP-REAL n) by A2, FUNCT_2:3; :: thesis: verum