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_2:71;

for x being object st x in X holds

(f </> g) . x in the carrier of (TOP-REAL n)

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_2:71;

for x being object st x in X holds

(f </> g) . x in the carrier of (TOP-REAL n)

proof

hence
f </> g is Function of X,(TOP-REAL n)
by A2, FUNCT_2:3; :: thesis: verum
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_2:72;

hence (f </> g) . x in the carrier of (TOP-REAL n) ; :: thesis: verum

end;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_2:72;

hence (f </> g) . x in the carrier of (TOP-REAL n) ; :: thesis: verum