reserve x,y for set;
reserve s,s1,s2,s4,r,r1,r2 for Real;
reserve n,m,i,j for Element of NAT;

theorem Th4:
  for N, M being non empty MetrSpace, f being Function of
  TopSpaceMetr(N),TopSpaceMetr(M) st f is continuous
 for r being Real,u
being Element of N,u1 being Element of M st r>0 & u1=f.u ex s st
  s>0 & for w being Element of N, w1 being Element of M st w1=f.w & dist(u,w)<s
  holds dist(u1,w1)<r
proof
  let N, M be non empty MetrSpace, f be Function of TopSpaceMetr(N),
  TopSpaceMetr(M);
  assume
A1: f is continuous;
  let r be Real,u be Element of N,u1 be Element of M;
  reconsider P=Ball(u1,r) as Subset of TopSpaceMetr(M) by TOPMETR:12;
A2: the carrier of N=the carrier of TopSpaceMetr(N) & dom f=the carrier of
  TopSpaceMetr(N) by FUNCT_2:def 1,TOPMETR:12;
  assume r>0 & u1=f.u;
  then f.u in P by GOBOARD6:1;
  then
A3: u in f"P by A2,FUNCT_1:def 7;
  f"P is open by A1,Th2;
  then consider s1 be Real such that
A4: s1>0 and
A5: Ball(u,s1) c= f"P by A3,TOPMETR:15;
  reconsider s1 as Real;
  for w being Element of N, w1 being Element of M st w1=f.w & dist(u,w)<s1
  holds dist(u1,w1)<r
  proof
    let w be Element of N, w1 be Element of M;
    assume that
A6: w1=f.w and
A7: dist(u,w)<s1;
    w in Ball(u,s1) by A7,METRIC_1:11;
    then f.w in P by A5,FUNCT_1:def 7;
    hence thesis by A6,METRIC_1:11;
  end;
  hence thesis by A4;
end;
