reserve s1,s2,q1 for Real_Sequence;

theorem
  for X,X1 be set,
      Y be RealNormSpace,
      f be PartFunc of REAL,the carrier of Y
  holds
  f|X is uniformly_continuous & X1 c= X implies f|X1 is uniformly_continuous
proof
  let X,X1 be set,
      Y be RealNormSpace,
      f be PartFunc of REAL,the carrier of Y;
  assume that
A1: f|X is uniformly_continuous and
A2: X1 c= X;
  now
    let r be Real;
    assume 0<r;
    then consider s be Real such that
A3: 0<s and
A4: for x1,x2 be Real st x1 in dom(f|X) & x2 in dom(f|X) & |. x1-x2 .|<s holds
    ||. f/.x1-f/.x2 .||<r by A1,Th1;
    take s;
    thus 0<s by A3;
    let x1,x2 be Real;
    assume that
A5: x1 in dom(f|X1) & x2 in dom(f|X1) & |. x1-x2 .|<s;
    f|X1 c= f|X by A2,RELAT_1:75;
    then dom(f|X1) c= dom(f|X) by RELAT_1:11;
    hence ||. f/.x1-f/.x2 .||<r by A4,A5;
  end;
  hence thesis by Th1;
end;
