reserve n,m,k for Element of NAT;
reserve x, X,X1,Z,Z1 for set;
reserve s,g,r,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
  X c= dom f & f|X is continuous implies (abs f)|X is continuous & (-f)|
  X is continuous
proof
  assume
A1: X c= dom f;
  assume
A2: f|X is continuous;
  thus (abs f)|X is continuous
  proof
    let r;
    assume
A3: r in dom((abs f)|X);
    then r in dom abs f by RELAT_1:57;
    then
A4: r in dom f by VALUED_1:def 11;
    r in X by A3;
    then
A5: r in dom(f|X) by A4,RELAT_1:57;
    then
A6: f|X is_continuous_in r by A2;
    thus (abs(f))|X is_continuous_in r
    proof
      let s1;
      assume that
A7:   rng s1 c= dom ((abs(f))|X) and
A8:   s1 is convergent & lim s1 = r;
      rng s1 c= dom (abs(f)) /\ X by A7,RELAT_1:61;
      then rng s1 c= dom f /\ X by VALUED_1:def 11;
      then
A9:   rng s1 c= dom (f|X) by RELAT_1:61;
      now
        let n;
A10:    s1.n in rng s1 by VALUED_0:28;
        then s1.n in dom (f|X) by A9;
        then s1.n in dom f /\ X by RELAT_1:61;
        then
A11:    s1.n in X by XBOOLE_0:def 4;
        thus (abs((f|X)/*s1)).n = |.((f|X)/*s1).n.| by SEQ_1:12
          .=|.(f|X).(s1.n).| by A9,FUNCT_2:108
          .=|.f.(s1.n).| by A9,A10,FUNCT_1:47
          .=(abs(f)).(s1.n) by VALUED_1:18
          .=((abs(f))|X).(s1.n) by A11,FUNCT_1:49
          .=(((abs(f))|X)/*s1).n by A7,FUNCT_2:108;
      end;
      then
A12:  abs((f|X)/*s1) = ((abs(f))|X)/*s1 by FUNCT_2:63;
A13:  |.(f|X).r.| = |.f.r.| by A5,FUNCT_1:47
        .= (abs(f)).r by VALUED_1:18
        .= ((abs(f))|X).r by A3,FUNCT_1:47;
A14:  (f|X)/*s1 is convergent by A6,A8,A9;
      hence ((abs(f))|X)/*s1 is convergent by A12,SEQ_4:13;
      (f|X).r = lim ((f|X)/*s1) by A6,A8,A9;
      hence thesis by A14,A12,A13,SEQ_4:14;
    end;
  end;
  thus thesis by A1,A2,Th20;
end;
