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 open iff
  for p being Point of TOP-REAL m, r being positive Real
  ex s being positive Real st Ball(f.p,s) c= f.:Ball(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;
    thus f is open implies
    for p being Point of TOP-REAL m, r being positive Real
    ex s being positive Real st Ball(f.p,s) c= f.:Ball(p,r)
    proof
      assume
A2:   f is open;
      let p be Point of TOP-REAL m, 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 open by A1,A2,Th1;
      then consider s being positive Real such that
A3:   Ball(q1,s) c= f1.:Ball(p1,r) by Th6;
      Ball(p1,r) = Ball(p,r) & Ball(q1,s) = Ball(f.p,s) by TOPREAL9:13;
      hence thesis by A3;
    end;
    assume
A4: for p being Point of TOP-REAL m, r being positive Real
    ex s being positive Real st Ball(f.p,s) c= f.:Ball(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 Ball(q,s) c= f1.:Ball(p,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;
      reconsider q1 = q as Point of TOP-REAL n by EUCLID:67;
      consider s being positive Real such that
A6:   Ball(q1,s) c= f.:Ball(p1,r) by A4,A5;
      Ball(p1,r) = Ball(p,r) & Ball(q1,s) = Ball(q,s) by TOPREAL9:13;
      hence thesis by A6;
    end;
    then f1 is open by Th6;
    hence thesis by A1,Th1;
  end;
