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
  x0 in dom f & f is_continuous_in x0 implies abs(f) is_continuous_in x0
  & -f is_continuous_in x0
proof
  assume
A1: x0 in dom f;
  assume
A2: f is_continuous_in x0;
  now
    let s1;
    assume that
A3: rng s1 c= dom(abs(f)) and
A4: s1 is convergent & lim s1=x0;
A5: rng s1 c= dom f by A3,VALUED_1:def 11;
    then
A6: f.x0 = lim (f/*s1) by A2,A4;
A7: f/*s1 is convergent by A2,A4,A5;
    then abs(f/*s1) is convergent by SEQ_4:13;
    hence (abs(f))/*s1 is convergent by A5,RFUNCT_2:10;
    thus (abs(f)).x0 = |.f.x0.| by VALUED_1:18
      .= lim (abs(f/*s1)) by A7,A6,SEQ_4:14
      .= lim ((abs(f))/*s1) by A5,RFUNCT_2:10;
  end;
  hence abs(f) is_continuous_in x0;
  thus thesis by A1,A2,Th8;
end;
