reserve r,r1,g for Real,
  n,m,k for Nat,
  seq,seq1, seq2 for Real_Sequence,
  f,f1,f2 for PartFunc of REAL,REAL,
  x for set;
reserve r,r1,r2,g,g1,g2 for Real;

theorem
  f is convergent_in+infty implies abs(f) is convergent_in+infty &
  lim_in+infty abs(f)=|.lim_in+infty f.|
proof
  assume
A1: f is convergent_in+infty;
A2: now
    let seq;
    assume that
A3: seq is divergent_to+infty and
A4: rng seq c=dom abs(f);
A5: rng seq c=dom f by A4,VALUED_1:def 11;
    then
A6: lim(f/*seq)=lim_in+infty f by A1,A3,Def12;
A7: f/*seq is convergent by A1,A3,A5;
    then abs(f/*seq) is convergent;
    hence (abs f)/*seq is convergent by A5,RFUNCT_2:10;
    thus lim((abs f)/*seq)=lim abs(f/*seq) by A5,RFUNCT_2:10
      .=|.lim_in+infty f.| by A7,A6,SEQ_4:14;
  end;
  now
    let r;
    consider g such that
A8: r<g & g in dom f by A1;
    take g;
    thus r<g & g in dom abs(f) by A8,VALUED_1:def 11;
  end;
  hence abs(f) is convergent_in+infty by A2;
  hence thesis by A2,Def12;
end;
