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 Th2:
  for X being non empty TopSpace, M being non empty MetrSpace, f
being Function of X,TopSpaceMetr(M) st f is continuous
 for r being Real,u
being Element of M,P being Subset of TopSpaceMetr(M) st P=Ball(u
  ,r) holds f"P is open
proof
  let X be non empty TopSpace, M be non empty MetrSpace, f be Function of X,
  TopSpaceMetr(M);
  assume
A1: f is continuous;
    let r be Real,u be Element of M,P be Subset of TopSpaceMetr
    (M);
    reconsider P9=P as Subset of TopSpaceMetr(M);
    assume P=Ball(u,r);
    then [#]TopSpaceMetr M <> {} & P9 is open by TOPMETR:14;
    hence thesis by A1,TOPS_2:43;
end;
