let X be set ; :: thesis: for n being Nat
for r being real number
for f being Function of X,(TOP-REAL n) holds f [#] r is Function of X,(TOP-REAL n)
let n be Nat; :: thesis: for r being real number
for f being Function of X,(TOP-REAL n) holds f [#] r is Function of X,(TOP-REAL n)
let r be real number ; :: thesis: for f being Function of X,(TOP-REAL n) holds f [#] r is Function of X,(TOP-REAL n)
let f be Function of X,(TOP-REAL n); :: thesis: f [#] r is Function of X,(TOP-REAL n)
set h = f [#] r;
dom f = X by FUNCT_2:def 1;
then A2: dom (f [#] r) = X by VALUED_2:def 39;
for x being set st x in X holds
(f [#] r) . x in the carrier of (TOP-REAL n)

proof

let x be set ; :: thesis: ( x in X implies (f [#] r) . x in the carrier of (TOP-REAL n) )
assume A3: x in X ; :: thesis: (f [#] r) . 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) ;
A4: (f . x) (#) r = (f . x) (#) r ;
(f [#] r) . x = (f . x) (#) r by A2, VALUED_2:def 39;
hence (f [#] r) . x in the carrier of (TOP-REAL n) by A4; :: thesis: verum

end;

hence f [#] r is Function of X,(TOP-REAL n) by A2, FUNCT_2:5; :: thesis: verum