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 Th37:
  for f being Function of X,TOP-REAL n holds f[#]r is Function of X,TOP-REAL n
  proof
    let f be Function of X,TOP-REAL n;
    set h = f[#]r;
 dom f = X by FUNCT_2:def 1;
then A1: dom h = X by VALUED_2:def 39;
    for x st x in X holds h.x in the carrier of TOP-REAL n
    proof
      let x;
      assume
A2:   x in X;
      then reconsider X as non empty set;
      reconsider x as Element of X by A2;
      reconsider f as Function of X,TOP-REAL n;
A3:   ((f.x) qua real-valued Function)(#)r = f.x(#)r;
      h.x = ((f.x) qua real-valued Function)(#)r by A1,VALUED_2:def 39;
      hence thesis by A3;
    end;
    hence thesis by A1,FUNCT_2:3;
  end;
