reserve n,m for Element of NAT;
reserve x, X,X1,Z,Z1 for set;
reserve s,g,r,t,p,x0,x1,x2 for Real;
reserve s1,s2,q1 for Real_Sequence;
reserve Y for Subset of REAL;
reserve f,f1,f2 for PartFunc of REAL,REAL;

theorem Th2:
  f|X is uniformly_continuous & X1 c= X implies f|X1 is uniformly_continuous
proof
  assume that
A1: f|X is uniformly_continuous and
A2: X1 c= X;
  now
    let r;
    assume 0<r;
    then consider s such that
A3: 0<s and
A4: for x1,x2 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;
    assume that
A5: x1 in dom(f|X1) and
A6: x2 in dom(f|X1) and
A7: |.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,A6,A7;
  end;
  hence thesis by Th1;
end;
