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
  f|X is uniformly_continuous implies (abs f)|X is uniformly_continuous
proof
  assume
A1: f|X is uniformly_continuous;
  now
    let r;
    assume 0<r;
    then consider s such that
A2: 0<s and
A3: 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 A2;
    let x1,x2;
    assume that
A4: x1 in dom((abs f)|X) and
A5: x2 in dom((abs f)|X) and
A6: |.x1-x2.|<s;
    x2 in dom(abs f) by A5,RELAT_1:57;
    then
A7: x2 in dom f by VALUED_1:def 11;
    x2 in X by A5,RELAT_1:57;
    then
A8: x2 in dom(f|X) by A7,RELAT_1:57;
    |.(abs(f)).x1-(abs(f)).x2.| = |.|.f.x1.|-(abs(f)).x2.| by VALUED_1:18
      .= |.|.f.x1.|-|.f.x2.|.| by VALUED_1:18;
    then
A9: |.(abs(f)).x1-(abs(f)).x2.| <= |.f.x1-f.x2.| by COMPLEX1:64;
    x1 in dom(abs f) by A4,RELAT_1:57;
    then
A10: x1 in dom f by VALUED_1:def 11;
    x1 in X by A4,RELAT_1:57;
    then x1 in dom(f|X) by A10,RELAT_1:57;
    then |.f.x1-f.x2.|<r by A3,A6,A8;
    hence |.(abs f).x1 - (abs f).x2.| < r by A9,XXREAL_0:2;
  end;
  hence thesis by Th1;
end;
