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
  for N,M being non empty MetrSpace, f being Function of N,M, g being
Function of TopSpaceMetr(N),TopSpaceMetr(M) st f=g & f is uniformly_continuous
  holds g is continuous
proof
  let N,M be non empty MetrSpace, f be Function of N,M, g be Function of
  TopSpaceMetr(N),TopSpaceMetr(M);
  assume that
A1: f=g and
A2: f is uniformly_continuous;
  for r being Real,u being Element of N,u1 being
  Element of M st r>0 & u1=g.u ex s be Real st s>0 & for w being Element
  of N, w1 being Element of M st w1=g.w & dist(u,w)<s holds dist(u1,w1)<r
  proof
    let r be Real, u be Element of N,u1 be Element of M;
    reconsider r9=r as Real;
    assume that
A3: r>0 and
A4: u1=g.u;
    consider s be Real such that
A5: 0<s and
A6: for wu1,wu2 being Element of N st dist(wu1,wu2) < s holds dist((f
    /. wu1),(f/.wu2)) < r by A2,A3;
    for w being Element of N, w1 being Element of M st w1=g.w & dist(u,w)<
    s holds dist(u1,w1)<r
    proof
      let w be Element of N, w1 be Element of M;
      assume that
A7:   w1=g.w and
A8:   dist(u,w)<s;
A9:   u1=f/.u by A1,A4;
      w1=f/.w by A1,A7;
      hence thesis by A6,A8,A9;
    end;
    hence thesis by A5;
  end;
  hence thesis by Th3;
end;
