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 divergent_in-infty_to+infty or f is divergent_in-infty_to-infty)
  implies abs(f) is divergent_in-infty_to+infty
proof
  assume
A1: f is divergent_in-infty_to+infty or f is divergent_in-infty_to-infty;
  now
    per cases by A1;
    suppose
A2:   f is divergent_in-infty_to+infty;
A3:   now
        let seq;
        assume that
A4:     seq is divergent_to-infty and
A5:     rng seq c=dom abs(f);
A6:     rng seq c=dom f by A5,VALUED_1:def 11;
        then f/*seq is divergent_to+infty by A2,A4;
        then abs(f/*seq) is divergent_to+infty by Th25;
        hence (abs f)/*seq is divergent_to+infty by A6,RFUNCT_2:10;
      end;
      now
        let r;
        consider g such that
A7:     g<r & g in dom f by A2;
        take g;
        thus g<r & g in dom abs(f) by A7,VALUED_1:def 11;
      end;
      hence thesis by A3;
    end;
    suppose
A8:   f is divergent_in-infty_to-infty;
A9:   now
        let seq;
        assume that
A10:    seq is divergent_to-infty and
A11:    rng seq c=dom abs(f);
A12:    rng seq c=dom f by A11,VALUED_1:def 11;
        then f/*seq is divergent_to-infty by A8,A10;
        then abs(f/*seq) is divergent_to+infty by Th25;
        hence (abs f)/*seq is divergent_to+infty by A12,RFUNCT_2:10;
      end;
      now
        let r;
        consider g such that
A13:    g<r & g in dom f by A8;
        take g;
        thus g<r & g in dom abs(f) by A13,VALUED_1:def 11;
      end;
      hence thesis by A9;
    end;
  end;
  hence thesis;
end;
