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 Th61:
  for r being non negative Real
  for f being Function of Tcircle(0.TOP-REAL(n+1),r),TOP-REAL n holds
  f(-) is Function of Tcircle(0.TOP-REAL(n+1),r),TOP-REAL n
  proof
    let r be non negative Real;
    set X = the carrier of Tcircle(0.TOP-REAL(n+1),r);
    let f be Function of X,TOP-REAL n;
    set g = f(-);
A1: dom g = dom f by VALUED_2:def 34;
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 x in X;
      then reconsider x as Element of X;
      reconsider y = -x as Element of X by Th60;
      reconsider f as Function of X,TOP-REAL n;
      g.x = f.y by A1,A2,VALUED_2:def 34;
      hence thesis;
    end;
    hence thesis by A1,A2,FUNCT_2:3;
  end;
