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

theorem Th4:
  for f being Function of T,TopSpaceMetr(M) holds f is open iff
  for p being Point of T, V being open Subset of T,
      q being Point of M st q = f.p & p in V
  ex r being positive Real st Ball(q,r) c= f.:V
  proof
    let f be Function of T,TopSpaceMetr(M);
    thus f is open implies for p being Point of T, V being open Subset of T,
        q being Point of M st q = f.p & p in V
    ex r being positive Real st Ball(q,r) c= f.:V
    proof
      assume
A1:   f is open;
      let p be Point of T, V be open Subset of T,
          q be Point of M such that
A2:   q = f.p;
      assume p in V;
      then consider W being open Subset of TopSpaceMetr(M) such that
A3:   f.p in W and
A4:   W c= f.:V by A1,Th3;
      ex r being Real st r > 0 & Ball(q,r) c= W by A2,A3,TOPMETR:15;
      hence thesis by A4,XBOOLE_1:1;
    end;
    assume
A5: for p being Point of T, V being open Subset of T,
        q being Point of M st q = f.p & p in V
    ex r being positive Real st Ball(q,r) c= f.:V;
    for p being Point of T, V being open Subset of T st p in V
    ex W being open Subset of TopSpaceMetr(M) st f.p in W & W c= f.:V
    proof
      let p be Point of T;
      let V be open Subset of T;
      reconsider q = f.p as Point of M;
      assume p in V;
      then consider r being positive Real such that
A6:   Ball(q,r) c= f.:V by A5;
      reconsider W = Ball(q,r) as open Subset of TopSpaceMetr(M) by TOPMETR:14;
      take W;
      thus f.p in W by GOBOARD6:1;
      thus W c= f.:V by A6;
    end;
    hence thesis by Th3;
  end;
