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

theorem
  for f being Function of T,TOP-REAL m holds f is open iff
  for p being Point of T, V being open Subset of T st p in V
  ex r being positive Real st Ball(f.p,r) c= f.:V
  proof
    let f be Function of T,TOP-REAL m;
A1: the TopStruct of TOP-REAL m = TopSpaceMetr Euclid m by EUCLID:def 8;
    then reconsider f1 = f as Function of T,TopSpaceMetr Euclid m;
A2: the TopStruct of T = the TopStruct of T;
    thus f is open implies
    for p being Point of T, V being open Subset of T st p in V
    ex r being positive Real st Ball(f.p,r) c= f.:V
    proof
      assume
A3:   f is open;
      let p be Point of T, V be open Subset of T;
      assume
A4:   p in V;
      reconsider fp = f.p as Point of Euclid m by EUCLID:67;
      f1 is open by A3,A1,A2,Th1;
      then consider r being positive Real such that
A5:   Ball(fp,r) c= f1.:V by A4,Th4;
      Ball(f.p,r) = Ball(fp,r) by TOPREAL9:13;
      hence thesis by A5;
    end;
    assume
A6: for p being Point of T, V being open Subset of T st p in V
    ex r being positive Real st Ball(f.p,r) c= f.:V;
    for p being Point of T, V being open Subset of T,
    q being Point of Euclid m st q = f1.p & p in V
    ex r being positive Real st Ball(q,r) c= f1.:V
    proof
      let p be Point of T, V be open Subset of T,
          q be Point of Euclid m such that
A7:   q = f1.p;
      assume p in V;
      then consider r being positive Real such that
A8:   Ball(f.p,r) c= f.:V by A6;
      Ball(f.p,r) = Ball(q,r) by A7,TOPREAL9:13;
      hence thesis by A8;
    end;
    then f1 is open by Th4;
    hence thesis by A1,A2,Th1;
  end;
