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

theorem Th17:
  for f being Function of TopSpaceMetr(M1),TopSpaceMetr(M2) holds
  f is continuous 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 f.:Ball(p,s) c= Ball(q,r)
  proof
    let f be Function of TopSpaceMetr(M1),TopSpaceMetr(M2);
    hereby
      assume
A1:   f is continuous;
      let p be Point of M1;
      let q be Point of M2;
      let r be positive Real;
      assume that
A2:   q = f.p;
      consider s being Real such that
A3:   s > 0 and
A4:   for w being Element of M1, w1 being Element of M2 st
      w1 = f.w & dist(p,w) < s holds dist(q,w1) < r by A1,A2,UNIFORM1:4;
      reconsider s as positive Real by A3;
      take s;
      thus f.:Ball(p,s) c= Ball(q,r)
      proof
        let y be object;
        assume y in f.:Ball(p,s);
        then consider x being Point of TopSpaceMetr(M1) such that
A5:     x in Ball(p,s) and
A6:     f.x = y by FUNCT_2:65;
        reconsider x1 = x as Point of M1;
        reconsider y1 = y as Point of M2 by A6;
        dist(p,x1) < s by A5,METRIC_1:11;
        then dist(q,y1) < r by A6,A4;
        hence y in Ball(q,r) by METRIC_1:11;
      end;
    end;
    assume
A7: 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 f.:Ball(p,s) c= Ball(q,r);
    for r being Real, u being Element of M1, u1 being Element of M2
    st r > 0 & u1 = f.u
    ex s being Real st s > 0 & for w being Element of M1,
    w1 being Element of M2 st w1 = f.w & dist(u,w)<s holds dist(u1,w1)<r
    proof
      let r be Real, u be Element of M1, u1 be Element of M2;
      assume r > 0 & u1 = f.u;
      then consider s being positive Real such that
A8:   f.:Ball(u,s) c= Ball(u1,r) by A7;
      take s;
      thus s > 0;
      let w be Element of M1, w1 be Element of M2 such that
A9:   w1 = f.w;
      assume dist(u,w) < s;
      then w in Ball(u,s) by METRIC_1:11;
      then f.w in f.:Ball(u,s) by FUNCT_2:35;
      hence thesis by A8,A9,METRIC_1:11;
    end;
    hence thesis by UNIFORM1:3;
  end;
