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,T holds f is open iff
  for p being Point of TOP-REAL m, r being positive Real
  ex W being open Subset of T st f.p in W & W c= f.:Ball(p,r)
  proof
    let f be Function of TOP-REAL m,T;
A1: the TopStruct of TOP-REAL m = TopSpaceMetr Euclid m by EUCLID:def 8;
    then reconsider f1 = f as Function of TopSpaceMetr Euclid m,T;
A2: the TopStruct of T = the TopStruct of T;
    thus f is open implies
    for p being Point of TOP-REAL m, r being positive Real
    ex W being open Subset of T st f.p in W & W c= f.:Ball(p,r)
    proof
      assume
A3:   f is open;
      let p be Point of TOP-REAL m, r be positive Real;
      reconsider q = p as Point of Euclid m by EUCLID:67;
      f1 is open by A3,A1,A2,Th1;
      then consider W be open Subset of T such that
A4:   f1.p in W & W c= f1.:Ball(q,r) by Th5;
      Ball(p,r) = Ball(q,r) by TOPREAL9:13;
      hence thesis by A4;
    end;
    assume
A5: for p being Point of TOP-REAL m, r being positive Real
    ex W being open Subset of T st f.p in W & W c= f.:Ball(p,r);
    for p being Point of Euclid m, r being positive Real
    ex W being open Subset of T st f1.p in W & W c= f1.:Ball(p,r)
    proof
      let p be Point of Euclid m, r be positive Real;
      reconsider q = p as Point of TOP-REAL m by EUCLID:67;
      consider W being open Subset of T such that
A6:   f.q in W & W c= f.:Ball(q,r) by A5;
      Ball(p,r) = Ball(q,r) by TOPREAL9:13;
      hence thesis by A6;
    end;
    then f1 is open by Th5;
    hence thesis by A1,A2,Th1;
  end;
