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

theorem Th6:
  for f being Function of TopSpaceMetr(M1),TopSpaceMetr(M2) holds f is open iff
  for p being Point of M1, q being Point of M2, r being positive Real
  st q = f.p
  ex s being positive Real st Ball(q,s) c= f.:Ball(p,r)
  proof
    let f be Function of TopSpaceMetr(M1),TopSpaceMetr(M2);
    thus f is open implies for p being Point of M1, q being Point of M2,
        r being positive Real st q = f.p
    ex s being positive Real st Ball(q,s) c= f.:Ball(p,r)
    proof
      assume
A1:   f is open;
      let p be Point of M1, q be Point of M2, r be positive Real
      such that
A2:   q = f.p;
      reconsider V = Ball(p,r) as Subset of TopSpaceMetr(M1);
A3:   p in V by GOBOARD6:1;
      V is open by TOPMETR:14;
      hence thesis by A1,A2,A3,Th4;
    end;
    assume
A4: for p being Point of M1, q being Point of M2,
    r being positive Real st q = f.p
    ex s being positive Real st Ball(q,s) c= f.:Ball(p,r);
    for p being Point of TopSpaceMetr(M1),
        V being open Subset of TopSpaceMetr(M1),
        q being Point of M2 st q = f.p & p in V
    ex r being positive Real st Ball(q,r) c= f.:V
    proof
      let p be Point of TopSpaceMetr(M1), V be open Subset of TopSpaceMetr(M1),
      q be Point of M2 such that
A5:   q = f.p;
      reconsider p1 = p as Point of M1;
      assume p in V;
      then consider r being Real such that
A6:   r > 0 and
A7:   Ball(p1,r) c= V by TOPMETR:15;
A8:   f.:Ball(p1,r) c= f.:V by A7,RELAT_1:123;
      ex s being positive Real st Ball(q,s) c= f.:Ball(p1,r)
      by A4,A5,A6;
      hence thesis by A8,XBOOLE_1:1;
    end;
    hence thesis by Th4;
  end;
