let X be set ; :: thesis: for n being Nat

for f being Function of X,(TOP-REAL n) holds <-> f is Function of X,(TOP-REAL n)

let n be Nat; :: thesis: for f being Function of X,(TOP-REAL n) holds <-> f is Function of X,(TOP-REAL n)

let f be Function of X,(TOP-REAL n); :: thesis: <-> f is Function of X,(TOP-REAL n)

set g = <-> f;

A1: dom (<-> f) = dom f by VALUED_2:def 33;

A2: dom f = X by FUNCT_2:def 1;

for x being object st x in X holds

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

for f being Function of X,(TOP-REAL n) holds <-> f is Function of X,(TOP-REAL n)

let n be Nat; :: thesis: for f being Function of X,(TOP-REAL n) holds <-> f is Function of X,(TOP-REAL n)

let f be Function of X,(TOP-REAL n); :: thesis: <-> f is Function of X,(TOP-REAL n)

set g = <-> f;

A1: dom (<-> f) = dom f by VALUED_2:def 33;

A2: dom f = X by FUNCT_2:def 1;

for x being object st x in X holds

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

proof

hence
<-> f is Function of X,(TOP-REAL n)
by A1, A2, FUNCT_2:3; :: thesis: verum
let x be object ; :: thesis: ( x in X implies (<-> f) . x in the carrier of (TOP-REAL n) )

assume A3: x in X ; :: thesis: (<-> f) . 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) = - (f . x) ;

(<-> f) . x = - (f . x) by A1, A2, VALUED_2:def 33;

hence (<-> f) . x in the carrier of (TOP-REAL n) by A4; :: thesis: verum

end;assume A3: x in X ; :: thesis: (<-> f) . 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) = - (f . x) ;

(<-> f) . x = - (f . x) by A1, A2, VALUED_2:def 33;

hence (<-> f) . x in the carrier of (TOP-REAL n) by A4; :: thesis: verum