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 r being non negative Real
  for f being continuous Function of Tcircle(0.TOP-REAL(m+1),r),TOP-REAL m
  holds f(-) is continuous Function of Tcircle(0.TOP-REAL(m+1),r),TOP-REAL m
  proof
    let r be non negative Real;
    set T = Tcircle(0.TOP-REAL(m+1),r);
    let f be continuous Function of T,TOP-REAL m;
    reconsider g = f(-) as Function of T,TOP-REAL m by Th61;
    for p being Point of T, r being positive Real
    ex W being open Subset of T st p in W & g.:W c= Ball(g.p,r)
    proof
      let p be Point of T;
      let r be positive Real;
      reconsider q = -p as Point of T by Th60;
      consider W being open Subset of T such that
A1:   q in W and
A2:   f.:W c= Ball(f.q,r) by TOPS_4:18;
      reconsider W1 = (-)W as open Subset of T;
      take W1;
      -q in W1 by A1,Def3;
      hence p in W1;
      let y be Element of TOP-REAL m;
      assume y in g.:W1;
      then consider x being Element of T such that
A3:   x in W1 and
A4:   g.x = y by FUNCT_2:65;
      dom g = the carrier of T by FUNCT_2:def 1;
      then
A5:   g.x = f.-x & g.p = f.-p by VALUED_2:def 34;
      -x in (-)W1 by A3,Def3;
      then f.-x in f.:W by FUNCT_2:35;
      hence y in Ball(g.p,r) by A2,A4,A5;
    end;
    hence thesis by TOPS_4:18;
  end;
