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)
& (for r ex g st g<r & g in dom f & f.g<>0) implies f^ is convergent_in-infty &
  lim_in-infty(f^)=0
proof
  assume that
A1: f is divergent_in-infty_to+infty or f is divergent_in-infty_to-infty and
A2: for r ex g st g<r & g in dom f & f.g<>0;
A3: dom(f^)=dom f\f"{0} by RFUNCT_1:def 2;
A4: now
    let r;
    consider g such that
A5: g<r & g in dom f and
A6: f.g<>0 by A2;
    take g;
    not f.g in {0} by A6,TARSKI:def 1;
    then not g in f"{0} by FUNCT_1:def 7;
    hence g<r & g in dom(f^) by A3,A5,XBOOLE_0:def 5;
  end;
  now
    per cases by A1;
    suppose
A7:   f is divergent_in-infty_to+infty;
A8:   now
        let seq such that
A9:     seq is divergent_to-infty and
A10:    rng seq c=dom(f^);
        dom(f^) c=dom f by A3,XBOOLE_1:36;
        then rng seq c=dom f by A10;
        then f/*seq is divergent_to+infty by A7,A9;
        then (f/*seq)" is convergent & lim((f/*seq)")=0 by Th34;
        hence (f^)/*seq is convergent & lim((f^)/*seq)=0 by A10,RFUNCT_2:12;
      end;
      hence f^ is convergent_in-infty by A4;
      hence thesis by A8,Def13;
    end;
    suppose
A11:  f is divergent_in-infty_to-infty;
A12:  now
        let seq such that
A13:    seq is divergent_to-infty and
A14:    rng seq c=dom(f^);
        dom(f^) c=dom f by A3,XBOOLE_1:36;
        then rng seq c=dom f by A14;
        then f/*seq is divergent_to-infty by A11,A13;
        then (f/*seq)" is convergent & lim((f/*seq)")=0 by Th34;
        hence (f^)/*seq is convergent & lim((f^)/*seq)=0 by A14,RFUNCT_2:12;
      end;
      hence f^ is convergent_in-infty by A4;
      hence thesis by A12,Def13;
    end;
  end;
  hence thesis;
end;
