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 Th9:
  X c= dom f & f|X is uniformly_continuous implies f|X is continuous
proof
  assume
A1: X c= dom f;
  assume
A2: f|X is uniformly_continuous;
  now
    let x0,r be Real;
    assume that
A3: x0 in X and
A4: 0<r;
A5: x0 in dom(f|X) by A1,A3,RELAT_1:62;
    consider s such that
A6: 0<s and
A7: for x1,x2 st x1 in dom(f|X) & x2 in dom(f|X) & |.x1-x2.| < s
    holds |.f.x1-f.x2.| < r by A2,A4,Th1;
    reconsider s as Real;
    take s;
    thus 0<s by A6;
    let x1 be Real;
    assume that
A8: x1 in X and
A9: |.x1-x0.| < s;
    x1 in dom(f|X) by A1,A8,RELAT_1:62;
    hence |.f.x1 - f.x0.| < r by A7,A9,A5;
  end;
  hence thesis by A1,FCONT_1:14;
end;
