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

theorem
  for g being Function of I[01],TOP-REAL n, f being Function of
  Closed-Interval-MSpace(0,1),Euclid n st g is continuous & f=g holds f is
  uniformly_continuous
proof
  let g be Function of I[01],TOP-REAL n, f being Function of
  Closed-Interval-MSpace(0,1),Euclid n such that
A1: g is continuous and
A2: f=g;
A3: the TopStruct of TOP-REAL n = TopSpaceMetr Euclid n by EUCLID:def 8;
  then reconsider h=g as Function of I[01], TopSpaceMetr Euclid n;
  h is continuous by A1,A3,PRE_TOPC:33;
  hence f is uniformly_continuous by A2,Lm1,Th7,TOPMETR:20;
end;
