reserve
  n, m for Nat,
  T for non empty TopSpace,
  M, M1, M2 for non empty MetrSpace;

theorem
  for f being Function of TOP-REAL m, TOP-REAL n holds f is continuous iff
  for p being Point of TOP-REAL m, r being positive Real
  ex s being positive Real st f.:Ball(p,s) c= Ball(f.p,r)
  proof
    let f be Function of TOP-REAL m, TOP-REAL n;
A1: the TopStruct of TOP-REAL m = TopSpaceMetr Euclid m &
    the TopStruct of TOP-REAL n = TopSpaceMetr Euclid n by EUCLID:def 8;
    then reconsider f1 = f as Function of TopSpaceMetr Euclid m,
    TopSpaceMetr Euclid n;
    hereby
      assume
A2:   f is continuous;
      let p be Point of TOP-REAL m;
      let r be positive Real;
      reconsider p1 = p as Point of Euclid m by EUCLID:67;
      reconsider q1 = f.p as Point of Euclid n by EUCLID:67;
      f1 is continuous by A1,A2,YELLOW12:36;
      then consider s being positive Real such that
A3:   f1.:Ball(p1,s) c= Ball(q1,r) by Th17;
      take s;
      Ball(p1,s) = Ball(p,s) & Ball(q1,r) = Ball(f.p,r) by TOPREAL9:13;
      hence f.:Ball(p,s) c= Ball(f.p,r) by A3;
    end;
    assume
A4: for p being Point of TOP-REAL m, r being positive Real
    ex s being positive Real st f.:Ball(p,s) c= Ball(f.p,r);
    for p being Point of Euclid m, q being Point of Euclid n,
    r being positive Real st q = f1.p
    ex s being positive Real st f1.:Ball(p,s) c= Ball(q,r)
    proof
      let p be Point of Euclid m, q be Point of Euclid n,
          r be positive Real such that
A5:   q = f1.p;
      reconsider p1 = p as Point of TOP-REAL m by EUCLID:67;
      consider s being positive Real such that
A6:   f.:Ball(p1,s) c= Ball(f.p1,r) by A4;
      take s;
      Ball(p1,s) = Ball(p,s) & Ball(f.p1,r) = Ball(q,r) by A5,TOPREAL9:13;
      hence thesis by A6;
    end;
    then f1 is continuous by Th17;
    hence thesis by A1,YELLOW12:36;
  end;
