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 Th25:
  (seq is divergent_to+infty or seq is divergent_to-infty) implies
  abs(seq) is divergent_to+infty
proof
  assume
A1: seq is divergent_to+infty or seq is divergent_to-infty;
  let r;
  now
    per cases by A1;
    suppose
      seq is divergent_to+infty;
      then consider n such that
A2:   for m st n<=m holds |.r.|<seq.m;
      take n;
      let m;
      assume n<=m;
      then r<=|.r.| & |.r.|<seq.m by A2,ABSVALUE:4;
      then
A3:   r<seq.m by XXREAL_0:2;
      seq.m<=|.seq.m.| by ABSVALUE:4;
      then seq.m<=abs(seq).m by SEQ_1:12;
      hence r<abs(seq).m by A3,XXREAL_0:2;
    end;
    suppose
      seq is divergent_to-infty;
      then consider n such that
A4:   for m st n<=m holds seq.m<-r;
      take n;
      let m;
      -|.seq.m.|<=seq.m by ABSVALUE:4;
      then
A5:   -abs(seq).m<=seq.m by SEQ_1:12;
      assume n<=m;
      then seq.m<-r by A4;
      then -abs(seq).m<-r by A5,XXREAL_0:2;
      hence r<abs(seq).m by XREAL_1:24;
    end;
  end;
  hence thesis;
end;
