reserve
  x for object, X for set,
  i, n, m for Nat,
  r, s for Real,
  c, c1, c2, d for Complex,
  f, g for complex-valued Function,
  g1 for n-element complex-valued FinSequence,
  f1 for n-element real-valued FinSequence,
  T for non empty TopSpace,
  p for Element of TOP-REAL n;

theorem
  for f being Function of X,TOP-REAL n holds <->f is Function of X,TOP-REAL n
  proof
    let f be Function of X,TOP-REAL n;
    set g = <->f;
A1: dom g = dom f by VALUED_2:def 33;
A2: dom f = X by FUNCT_2:def 1;
    for x st x in X holds g.x in the carrier of TOP-REAL n
    proof
      let x;
      assume
A3:   x in X;
      then reconsider X as non empty set;
      reconsider x as Element of X by A3;
      reconsider f as Function of X,TOP-REAL n;
A4:   -((f.x) qua real-valued Function) = -f.x;
      g.x = -((f.x) qua real-valued Function) by A1,A2,VALUED_2:def 33;
      hence thesis by A4;
    end;
    hence thesis by A1,A2,FUNCT_2:3;
  end;
