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 P being Subset of TOP-REAL n, Q being non empty Subset of Euclid n
  , g being Function of I[01],(TOP-REAL n)|P, f being Function of
Closed-Interval-MSpace(0,1),((Euclid n)|Q) st P=Q & g is continuous & f=g holds
  f is uniformly_continuous
proof
  let P be Subset of TOP-REAL n, Q be non empty Subset of Euclid n, g be
Function of I[01],(TOP-REAL n)|P, f be Function of Closed-Interval-MSpace(0,1),
  ((Euclid n)|Q);
  assume that
A1: P=Q and
A2: g is continuous & f=g;
  (TOP-REAL n)|P = TopSpaceMetr((Euclid n)|Q) by A1,EUCLID:63;
  hence thesis by A2,Lm1,Th7,TOPMETR:20;
end;
