theorem
  for f being Function of TOP-REAL m,TOP-REAL n st f is continuous holds
  f(-) is continuous Function of TOP-REAL m,TOP-REAL n
  proof
    let f be Function of TOP-REAL m,TOP-REAL n;
    assume
A1: f is continuous;
    reconsider g = f(-) as Function of TOP-REAL m,TOP-REAL n by Th34;
    for p being Point of TOP-REAL m, r being positive Real
    ex s being positive Real st g.:Ball(p,s) c= Ball(g.p,r)
    proof
      let p be Point of TOP-REAL m;
      let r be positive Real;
      consider s being positive Real such that
A2:   f.:Ball(-p,s) c= Ball(f.-p,r) by A1,TOPS_4:20;
      take s;
      let y be Element of TOP-REAL n;
      assume y in g.:Ball(p,s);
      then consider x being Element of TOP-REAL m such that
A3:   x in Ball(p,s) and
A4:   g.x = y by FUNCT_2:65;
      dom g = the carrier of TOP-REAL m by FUNCT_2:def 1;
      then
A5:   g.x = f.-x & g.p = f.-p by VALUED_2:def 34;
      -x in Ball(-p,s) by A3,Th23;
      then f.-x in f.:Ball(-p,s) by FUNCT_2:35;
      hence y in Ball(g.p,r) by A2,A4,A5;
    end;
    hence thesis by TOPS_4:20;
  end;
